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%\vskip .5cm
\Centerline{\Large{\bf{ABSTRACTS / R\'ESUM\'ES DE
CONF\'ERENCES}}}
\smallskip
\noindent
Here are the abstracts, as of November 6, 1998. After November 6,
updates to the abstracts will
appear in the printed copy of the meeting programme only.
Participants are asked to consult the meeting programme available
during on-site registration.
\noindent
Voici les r\'esum\'es de conf\'erences,
en date du 6 novembre 1998. Apr\`es cette date,
tout changement \`a cette liste ne sera apport\'e qu'\`a la copie imprim\'e du
programme de la r\'eunion. Les participants sont encourag\'es \'a
consulter le programme, disponible \'a la table d'inscription.
\session{\bf{COXETER-JAMES LECTURE / CONF\'ERENCE COXETER-JAMES}}
\abs{Henri Darmon}
{McGill University, Montreal, Quebec~~H3A 2K6, Canada}
{Recent progress in the theory of elliptic curves}
After Wiles' proof of the Shimura-Taniyama conjecture, the Birch and
Swinnerton-Dyer conjecture has become the outstanding open problem in
the arithmetic theory of elliptic curves. In the late 80's, the work
of
Kolyvagin led to an almost complete proof of the Birch and
Swinnerton-Dyer conjecture for elliptic curves of (analytic) rank at
most one. The higher rank case remains shrouded in mystery. Recently
Bertolini and I have been able to prove part of a p-adic variant of
the
Birch and Swinnerton Dyer conjecture which applies to curves of higher
rank. The proof combines ideas that arose in the work of Wiles and
earlier work of Ribet on Fermat's Last Theorem with the methods of
Thaine and Kolyvagin.
%\vskip .4cm
\session{\bf{DOCTORAL PRIZE LECTURE / CONF\'ERENCE DU PRIX DOCTORAL}}
\abs{Yuri Berest}
{University of California, Berkeley, California~~94720,
USA}
{Lacunae for hyperbolic differential operators with
variable coefficients}
Lacuna of a linear hyperbolic differential operator is a domain inside
its propagation cone where a proper fundamental solution vanishes
identically. Huygens' principle for the classical wave equation is
the
simplest important example of such a phenomenon. The study of lacunas
for hyperbolic equations of arbitrary order was initiated by
I.~G.~Petrovsky (1945). Extending and clarifying his results,
M.~Atiyah, R.~Bott and L.~G\aa rding (1970--73) created a profound and
complete theory for hyperbolic operators with constant coefficients.
In contrast, much less is known about lacunas for operators with
variable coefficients. The purpose of the present talk is to report
on
new developments in this direction. We start with a review of
classical
results on lacunas (related mostly to the famous Hadamard's conjecture
for second order operators). We present new examples and give a
solution of (a restricted version of) Hadamard's problem for wave-type
operators on Minkowski spaces. In the second part of the talk we
explain how these results can be extended to the case of higher order
hyperbolic operators. Our goal is to give a generalization of the
Petrovsky-Atiyah-Bott-G\aa rding theory to certain classes of partial
differential operators with singular coefficients. The final part of
our talk will be devoted to the discussion of various connections with
algebraic geometry (structure of rings of differential operators on
singular varieties), representation theory (finite reflection groups)
and integrable systems.
%\vskip .4cm
\session{\bf{PUBLIC LECTURE / CONF\'ERENCE PUBLIQUE}}
\abs{H. S. M. Coxeter}
{Department of Mathematics, University of Toronto, Toronto,
Ontario~~M5S 3G3, Canada}
{The Descartes circle theorem and Fibonacci numbers}
The {\it numerical distance\/} between two circles in the Euclidean
plane is defined to be the number
\[
\frac{c^2 - a^2 - b^2}{2ab},
\]
where $a$ and $b$ are their radii while $c$ is the ordinary distance
between their centres. An infinite sequence of circles is defined to
be {\it loxodromic\/} if every four consecutive members are mutually
tangent. $D_n$ denotes the numerical distance between the $m$th and
$(m+n)$-th circles (the same for all $m$). Obviously $D_{-n} = D_n$.
Since the numerical distance is $-1$ when $a=b$ and $c=0$ so that the
two circles coincide, $D_0 = -1$. Since it is $1$ when $a+b = c$ so
that the circles are externally tangent, $D_1 = D_2 = D_3 = 1$. Any
number of further values of $D_n$ can be determined successively by
the
recurrence equation
\[
D_m + D_{m+4} = 2(D_{m+1} + D_{m+2} + D_{m+3}) .
\]
There is also an explicit formula
\[
D_n = \sum^{[n/2]}_{\nu=0} {n \choose 2 \nu} f_{n- \nu - 2},
\]
in terms of binomial coefficients and Fibonacci numbers.
%\vskip .4cm
\session{\bf{PLENARY SPEAKERS / CONF\'ERENCIERS PRINCIPAUX}}
\abs{Mikl\'os Cs\"org\H{o}}
{Carleton University, Ottawa, Ontario~~K1S 5B6, Canada}
{Random walking around financial mathematics}
The 1997 Nobel Prize in Economic Sciences was awarded to Robert
C.~Merton and Myron S.~Scholes who, in collaboration with the late
Fischer Black, developed a pioneering formula for the valuation of
stock options. In 1973, Black and Scholes published what has come to
be known as the Black-Scholes formula. In order to derive and
properly
appreciate this formula, taking a historical route, we will first
review some of the fundamental notions of Brownian motion-Wiener
process, as well as some elements of It\^o calculus. Consequently, we
will summarize the derivation of the Black-Scholes formula for the
European and American options. Though the solution is in terms of
geometric Brownian motion, the latter will be highlighted also in
terms
of geometric fractional Brownian motion. Time permitting, we will
also
describe some long time path properties of various geometric processes
of the {\it call on average\/} (Asian) option via those of the {\it
call on maximum\/} option. The latter will be based on ongoing work
with Endre Cs\'aki, Ant\'onia F\"oldes and P\'al R\'ev\'esz.
\abs{Zoltan F\"uredi}
{University of Illinois at Urbana, Urbana,
Illinois~~61801-2975, USA}
{Lotto, footballpool and other covering radius problems}
The aim of this talk is to review connections between Tur\'an's
(hyper)graph problem and other parts of Combinatorics, like Steiner
systems, packings and coverings, constant weight codes, Kneser graphs.
The code $C$ is called a {\it covering code\/} of $X$ with radius $r$
if every element of $X$ is within Hamming distance $r$ from at least
one codeword from $C$. Given $X$ we are interested in a minimum
sized
$C$. Continuing a work of Hanani, Ornstein and S\'os, and Brouwer we
determine the Lottery number, $L(n,k,p,2)$, the minimum number of
$k$-subsets of an $n$-set such that all the ${n \choose p}$ $p$-sets
are intersected by one of them in at least $2$ elements, for all $n>
n_0(k,p)$. Answering a question of H\"am\"al\"ainen, Honkala, Litsyn
and \"Ostergard we show further connections between Tur\'an theorem
and constant weight covering codes.
\abs{William M. Goldman}
{University of Maryland, College Park, Maryland~~20742, USA}
{Topology and dynamics of moduli spaces of geometric
structures
on surfaces}
The space of representations of the fundamental group of a surface
into
a Lie group is a natural object which exhibits rich geometric
structure
and rich symmetry. In particular these moduli spaces are interesting
examples of symplectic geometry. The ergodicity of the modular group
acting on moduli space (for certain compact $G$) will be discussed, as
the relationship to uniformization of geometric structures. When $G$
is noncompact, we conjecture that there are components where the
action
is ergodic as well as components where the action is properly
discontinuous (arising from uniformizations). Uniformizations by
convex domains in the real projective plane, as well as complex
hyperbolic Kleinian groups will also be discussed. Using singular
hyperbolic structures, one can interpret arbitrary surface group
homomorphisms in terms of uniformizations, which provides a geometric
interpretation to the dynamical questions about moduli spaces.
\abs{Don O'Shea}
{Mt. Holyoke College, South Hadley, Massachusetts~~01075, USA}
{The glory and perversity of us}
Mathematics has an image problem. I contend that much of that problem
stems from myths that we have internalized about the discipline, the
profession, and ourselves. These myths go largely unexamined and
unchallenged, and often work great harm.
In particular, I talk about the notion that mathematics is inherently
hierarchical, that there is a right way of teaching a topic, that
mathematics is a young man's game or a sacred calling. I analyse the
rather limited ways in which we talk about what we do and what we
teach
and ought to teach, and I study how these constrict our curricular
offerings. I give examples of what seems to work and what does not,
of
when we excel and when we shoot ourselves in the foot.
%\vskip 1.0cm
%\hrule
%\vskip .3cm
\session{
{\bf Algebraic Geometry / G\'eom\'etrie alg\'ebrique}}
\org{P.~Milman, Organizer}
%\hrule
%\vskip .4cm
\abs{Fedor Bogomolov}
{Courant Institute, New~York University, New~York, USA}
{On the fundamental groups and universal coverings of
complex projective surfaces and symplectic fourfolds}
In the talk I will consider the problem of holomorphic convexity of
the
universal coverings of complex projective surfaces and the problem of
representation of a finitely presented group as a fundamental group of
complex projective surface or symplectic Lefschetz pencil.
\abs{Andrew Hwang}
{University of Toronto}
{Construction of Circle-Invariant Kahler Metrics}
In the total space of an Hermitian disk bundle, it is possible to
construct a Kahler metric from a single function of one variable
by using an ansatz of Calabi. A change of coordinates closely related
to a Legendre transform converts the scalar curvature operator for
such a metric into a very simple linear second-order operator; it is
thereby possible to construct metrics of specified scalar curvature.
I will describe these methods and their limitations. (Mostly joint
work with Michael A. Singer, University of Edinburgh.)
\abs{Lisa Jeffrey}
{Department of Mathematics, University of Toronto, Toronto,
Ontario~~M5S 3G3, Canada}
{The Verlinde formula for moduli spaces of parabolic bundles}
The moduli space $M(n,d)$ is an algebraic variety parametrizing those
representations of the fundamental group of a punctured Riemann surface
into the Lie group $\SU(n)$ for which a loop around the boundary is
sent to a particular $n$-th root of unity multiplied by the identity
matrix. If $n$ and $d$ are coprime it is in fact a Kaehler manifold.
One may relax the constraint and study moduli spaces $M(a)$
parametrizing those representations for which the loop around the
boundary is sent to an element conjugate to $a$, if $a$ is some element
in $\SU(n)$, and these are also Kaehler manifolds for a suitable class
of $a$.
The Verlinde formula calculates the dimension of the space of
holomorphic sections of certain line bundles over the spaces $M(n,d)$
and $M(a)$: these dimensions are in a sense the dimensions of the
quantizations of these spaces. We recall how a new proof of the
Verlinde formula for $M(n,d)$, given in joint work with F.~Kirwan (Ann.
Math., 1998) may be obtained, and show how to modify this proof to
obtain a proof of the variant of the Verlinde formula which applies to
$M(a)$.
\abs{Askold Khovansky}
{Department of Mathematics, University of Toronto, Toronto,
Ontario~~M5S 3G3, Canada}
{Algebraic geometry and geometry of convex polyhedra}
Newton polyhedra provide a connection between Algebraic Geometry and
Geometry of Convex Polyhedra. This connection is useful in both
directions. On the one hand it gives visible and understandable
answers on the numerous questions which came from Algebraic Geometry.
On the other hand it suggests algebraic intuition in the Geometry of
Convex polyhedra. The talk will contain a review of the subject
including some new results.
\abs{Don O'Shea}
{Mt. Holyoke College, South Hadley, Massachusetts~~01075, USA}
{Limits of tangent spaces to real surfaces}
There is a lovely theory, due to Henry, Le, Teissier and others,
describing the space of limits of tangent spaces at singular points of
complex surfaces. This space encodes much delicate geometric
information about the surface in a neighbourhood of the singular
point. Such a theory would be highly desirable over the reals, but
the
complex results do not specialize in a straightforward manner. In the
talk, we describe the complex theory briefly, and discuss some recent
work which has cleared up a number of open questions in the real case.
In particular, let $X,0$ be a real surface in $\Bbb{R}^n,0$. We
investigate the tangent semicone $C^+$ to the surface (by which we
mean
the set of all vectors which can be obtained as a limit of a sequence
$t_ix_i$ with $t_i > 0$ and $x_i \in X$ where the $x_i$ tend to $0$)
and the Nash space $K$ of the surface (the set of all planes which can
be obtained as a limit of tangent planes to $X$ at smooth points of
$x$
tending to $0$).
We prove a structure theorem for $K$ analogous to, but different in
some interesting respects, than that over the complexes established by
L\^e. We show that there is a sharp (and, to us, unexpected)
dichotomy
between exceptional rays in $C^+$ which are tangent to the singular
locus of $X$ and those that aren't. In the latter case, we determine
precisely when a ray lying in the singular part of $C^+$ must be
exceptional, and show that the set of elements in $K$ containing the
exceptional ray cannot contain discrete elements---in fact, we can
give
a lower bound on the size of this set. This suggests a possible
algorithm for determining when and where $C^+$ (and the geometric
tangent cone $C = C^+ \cup -C^+$) fails to be algebraic and, more
speculatively, an algorithm for computing $C^+$.
\abs{Evgenii Shustin}
{Tel-Aviv University, Tel-Aviv, Israel}
{Asymptotically proper bounds in the geometry of equisingular
families of curves}
Given a smooth complex algebraic hypersurface and an ample linear
system on it, we consider the family of irreducible curves in this
linear system having isolated singular points of given topological or
analytic types, and we ask if this family is nonempty, smooth,
irreducible and has expected dimension. We present sufficient
conditions for the above ``good'' properties of equisingular families,
expressed as upper bounds to the sums of certain invariants of
singularities, and we show that these bounds are asymptotically proper,
{\it i.e.} if the sum of the same singularity invariants exceeds the
upper bound multiplied by an absolute constant, then the equisingular
family is empty, or not smooth, or reducible, or has a nonexpected
dimension.
Our approach consists in reducing the problem to $h^1$-vanishing for
the ideal sheaves of zero-dimensional schemes associated to
singularities, and in applying various $h^1$-vanishing criteria which
come from Riemann-Roch, Kodaira theorem, Bogomolov's theory of unstable
rank two vector bundles on surfaces, Castelnuovo function theory.
(Joint work with G.-M.~Greuel and C.~Lossen.)
\abs{Yosef Yomdin}
{Weizmann Institute of Science, Rehovot, Israel}
{A tractable problem on the way (hopefully) to counting
limit cycles of polynomial vector fields}
We show that a certain infinitesimal version of the problem of bounding
the number of limit cycles of a polynomial vector field on the plane
leads to an interesting question, which can be formulated very simply:
Let $P(x)$ and $q(x)$ be two polynomials. Consider the function
$H(x,y)=\int_0^x q(t)/\bigl(1-yP(t)\bigr)$.
1.~~Find all $x$ for which $H(x,y)$ is identically zero in $y$.
2. For any fixed x estimate the number of (real) zeroes of
$H(x,y)$ in $y$.
These questions turn out to be closely related to several fields in
Analysis and Algebra, in particular to the Generalized moments problem,
to the algebra of polynomials under composition and to $D$-modules. We
plan to present some of these relations, an answer to the questions 1
and 2 and to discuss some additional results and problems in the above
direction.
%\vskip 1.0cm
%\hrule
%\vskip .3cm
\session{
{\bf Discrete Geometry / G\'eom\'etrie discr\`ete}}
\org{R.~Connelly, R.~Erdahl, M.~Senechal and W.~Whiteley, Organizers}
%\hrule
%\vskip .4cm
\abs{Victor Alexandrov}
{Sobolev Institute of Mathematics, Novosibirsk,
630090~~Russia}
{Sufficient conditions for the extendability of an $N$-th
order flex of polyhedra}
A very long-standing (and still open) problem is to prove that a
smooth
compact surface in Euclidean $3$-space is rigid. Classical attempts
to
solve this problem were based on the notion of an infinitesimal flex.
Since a counterexample was constructed by R.~Connelly in the class of
polyhedral surfaces, particular attention was given to studying
infinitesimal flexes of polyhedra.
We give a new approach to describing a high order flex of polyhedra.
The main results provide some sufficient conditions under which an
infinitesimal flex of a polyhedron can be extended to an authentic
flex. We discuss also results of computer experiments demonstrating
that one set of our sufficient conditions is fulfilled for the Bricard
octahedra.
More precisely, let $Q$ be a closed polyhedron with $v$ vertexes, $e$
edges and triangular faces. A special bilinear map $B\colon\bbd R^{3v}
\times \bbd R^{3v} \to \bbd R^{e}$ is constructed in such a way that
the vector
$$
X_0+tX_1+\cdots +t^kX_k \quad (X_j\in\Bbb R^{3v}, j=0,1,\dots
,k)\leqno(1)
$$
is the $k$-th order infinitesimal flex of the polyhedron $X_0=Q$ if
and
only if the equation
$$
\sum_{j=0}^{n} B(X_j,X_{n-j})=0
$$
holds for each $1\leq n\leq k$. Put $A_0X=B(X, X_0)+ B(X_0 ,X)$.
Suppose an infinitesimal flex $X_0+tX_1+\cdots +t^nX_n$ of the
polyhedron $Q$ is given. To extend this flex into a higher order one,
it is necessary to solve the linear equation
$$
A_0X_{n+1} =\sum_{j=1}^{n} B(X_j,X_{n+1-j}).
$$
The following sufficient condition for algorithmic verification of the
flexibility of polyhedra is proven:
\lproclaim{Theorem}{
Let a polyhedron $Q=X_0$ has an infinitesimal flex $X_0+tX_1+\cdots
+t^nX_n$ and let $L_n\subset\bbd R^{3v}$ be the linear span of the
vectors $X_0, X_1, \dots , X_n$. Suppose the equation $A_0X=-B(X_i,
X_j)-B(X_j,X_i)$ has a solution $X\in L_n$ for all $1\leq i\leq n$,
$1\leq j\leq n$. Then $Q$ is flexible.}
Computer experiments show that the conditions of the Theorem are
fulfilled with $n=5$ for Bricard's octahedron of the second type.
\abs{Valery Alexeev}
{Department of Mathematics, University of Georgia, Athens,
Georgia~~30605 USA}
{Families of algebraic varieties associated with cell
decompositions}
Fix a finite set $S$ of integral points in a lattice $L$ and let $Q$ be
their convex hull. To any cell decomposition of $Q$ with vertices in
$S$ we associate: 1)~~a family of projective algebraic varieties, and
2)~~a family of pairs of projective algebraic varieties together with
Cartier divisors.
Similarly, we construct families of the above two types for any
periodic cell decomposition with integral vertices.
The parameter spaces of families of the second type for various cell
decompositions fit together into a moduli space, a proper algebraic
variety itself. One application of this construction is a canonical
geometric compactification of the moduli of principally polarized
abelian varieties.
\abs{P. Atela}
{Department of Mathematics, Smith College, North Hampton,
Massachusetts~~01063-0001, USA}
{Periodicity in geometric dynamical models in Phyllotaxis}
We will discuss stable periodic configurations that arise
in the analysis of a dynamical geometric model of phyllotaxis.
\abs{Lynn M. Batten}
{Department of Mathematics, University of Manitoba, Winnipeg,
Manitoba~~R3T 2N2, Canada}
{Blocking sets and security}
There is much literature on blocking sets in projective planes. The
abundance of blocking sets of various types in this setting makes them
amenable to use in security systems. We explain how this can be done,
and describe the reliability issues involved.
\abs{Margaret M. Bayer}
{University of Kansas, Lawrence, Kansas~~66045-2142, USA}
{Eulerian partially ordered sets}
The face lattices of convex polytopes belong to the class of Eulerian
partially ordered sets. In every interval of these ranked posets, the
number of elements of even rank equals the number of elements of odd
rank. The flag vector of a ranked poset gives the numbers of chains
for the various rank sets. This talk discusses the closed convex cone
of flag vectors of Eulerian posets. (The linear span is determined by
the generalized Dehn-Sommerville equations.) The approach is based on
work of Billera and Hetyei on flag vectors of ranked posets and uses
``half-Eulerian'' partially ordered sets.
\abs{Andr\'as Bezdek}
{The Mathematical Institute of the Hungarian Academy of
Science, Budapest, Hungary, and Department of Mathematics, Auburn
University, Auburn, Alabama~~36849-5310, USA}
{A Sylwester type theorem on circles}
In spite of the specific title, generalizations of two combinatorial
geometric problems will be disscussed. The classical result of
Sylvester shows that $n$ points in the plane, not all on a common
line,
there exists an ordinary line (a line through exactly two points). The
similar question about unit circles will be disscussed. A problem of
V.~Boltyanski (proved by A.~V.~Bogomolnaya, A.~V.~Nazarov and
S.~Rukshin) claims that if $n$ points in a convex $n$-sided polygon
are
given, then using the sides and the given points $n$ side-point pairs
can be formed such that the triangles determined by the pairs cover
the
convex polygon. We strengthen this result by proving that the
assumption according to which the given points are inside of the
polygon is redundant. We also give a bound for the number of disjoint
such triangles.
\abs{Karoly Bezdek}
{Department of Geometry, Eotvos University, H-1088~~Budapest,
Hungary and Department of Mathematics, Cornell University Ithaca,
New~York~~14853-7901, USA}
{On a stronger form of Rogers' lemma and the minimum surface
area of Voronoi cells in unit ball packings}
Rogers' lemma is an essential tool for estimating the density of unit
ball packings in Euclidean space. Also, it motivates several other
techniques of the classical theory of packing. In this talk we prove a
strengthening of Rogers' lemma and apply it to estimate the minimum
surface area of Voronoi cells in unit ball packings. We prove a lower
bound for the surface area of Voronoi cells of unit ball packings in
$d$-dimensional Euclidean space. This bound is sharp for $d=2$ and
implies Rogers' upper bound for the density of unit ball packings in
$d$-space for all $d>1$. Finally, we strengthen these results for
$d=3$.
\abs{T. Bisztriczky}
{Department of Mathematics, University of
Calgary, Calgary, Alberta~~T2N 2N4, Canada}
{A Signature Theorem for uniform oriented matroids}
Let $M$ be an uniform oriented matroid over a set $E$ of $n$ elements
$e(1), e(2),\dots,e(n)$ endowed with the linear order
$e(1) 2$ are identified.
Besides hyper-simplices and hyper-octahedra, they are exactly those
with bipartite skeleton: hyper-cubes, cubic lattices and $8$, $2$,
$1$
tilings of hyperbolic $3$-, $4$-, $5$-space (only two, $435$ and
$4335$, of those $11$ are compact).
abs{Robert Erdahl}
{Queen's University, Kingston, Ontario~~K7L 3N6, Canada}
{Voronoi's hypothesis on perfect domains}
It was implicit in Voronoi's famous last two geometrical memoirs that
he felt the partition of the cone of metrical forms into lattice type
domains is a refinement of the partition into perfect domains. This
property of perfect domains is now referred to as Voronoi's
Hypothesis.
The first of these memoirs was on perfect forms and their associated
domains, and in the second he introduced his famous theory of lattice
types; so this hypothesis links these last two memoirs. Voronoi's
hypothesis has played an important role in the programs to classify
higher dimensional lattices that followed. More precisely, this
hypothesis played a decisive role in the fundamental work on
classification of Delaunay, and more recently of Ryshkov and
Baranovski. Ryshkov and Baranovsky were able to show that Voronoi's
hypothesis holds up to dimension $5$. I will show that this
convenient
relationship does not hold in general by demonstrating that the
$L$-type partition in $6$ dimensions is not a refinement of the
perfect
partition. This work is joint with Konstantin Rybnikov.
\abs{Ferenc Fodor}
{Department of Mathematics, Auburn University, Auburn,
Alabama~~36849-5310, USA}
{Large polygons in convex sets and polygons with large
perimeter}
In this talk we investigate the problem that how large convex $n$-gons
can be inscribed in a convex disc ${\cal K}$ in its Minkowski metric.
We show that in any convex disc we can always inscribe a triangle of
minimal side length $2/3$, and in a centrally symmetric disc a
triangle
with sides not shorter than $0.773\dots\,$. We also give a short
construction for an inscribed triangle whose sides are at least $3/4$
in the Minkowski metric induced by ${\cal K}$.
We also consider the question of finding the maximum perimeter $p_{n}$
of a convex $n$-gon of unit diameter. We prove that the diameter graph
of such an extremal polygon must be connected. Therefore if $n=2^{k}$
no regular polygon is extremal. We shall also present applications of
these facts in connection with finite circle packing problems and
diameter minimalization of finite point sets with mutual distances at
least one.
Some of the above results are joint works with A.~Bezdek and
I.~Talata.
\abs{Deborah S. Franzblau}
{Department of Mathematics, CUNY/College of Staten Island,
Staten Island, New~York~~10314, USA}
{Generic rigidity of molecular graphs}
A hard open problem is to find a purely combinatorial algorithm to
determine whether a graph, representing a structure made of rigid bars
and flexible joints, is generically rigid in $3$ dimensions. A {\it
molecular graph\/} or {\it square\/} of a graph, in which an edge is
added between each pair of neigboring edges, models an atom-bond
network with constraints on bond angles as well as bond lengths, which
is a special case important in materials science, especially for the
study of glasses. In this talk I will discuss the computation of
generic degrees of freedom, the standard measure of rigidity, of
molecular graphs.
\abs{Christopher Gold}
{Chair in Geomatics, Laval University, Quebec City,
Quebec~~G1K 7P4, Canada}
{Voronoi methods in geomatics---the importance of the
spatial model}
Geomatics covers surveying, remote sensing, geographic information
systems ($\GIS$) and other topics concerning the capture and
management
of spatial information. Thus the restricted (discrete) model of space
used in the computer is of central importance. While known for many
years, the simple Voronoi diagram is only now being recognized as a
central organizing concept for many types of data. This has been
helped
by the discipline of Computational Geometry, and is now being
recognized within Geomatics. This talk will focus on applications
based on simple concepts, and show how the Voronoi model is ideally
suited to both algorithm development and geographic data input,
management and analysis.
The plenary address will survey some of the classical examples of
these
moduli spaces and their structures. The special session talk will
describe some of the more technical details of the proof.
\abs{Chaim Goodman-Strauss}
{Department of Mathematics, University of Arkansas,
Fayetteville,
Arkansas~~72701, USA}
{Addressing and substitution tilings}
We will discuss recent work concerning addressing in substitution
tilings. Such addresses allow control over a variety of structures in
such tilings, including adjancies, the combinatorics and the geometry
itself of the tiling. In essence, there is a correspondence between
the combinatorical structure of a substitution species and a
automatically defined quotient on a regular language viewed as a
topological Cantor set.
\abs{Timothy F. Havel}
{Biological Chemistry and Molecular Pharmacology, Harvard
Medical
School, Boston, Massachusetts~~02115, USA}
{The role of tensegrity in distance geometry}
Distance geometry is an invariant-theoretic approach to Euclidean (and
classical non-Euclidean) geometry, which was developed in large part
by
A.~Cayley, K.~Menger, I.~J.~Schoenberg, J.~J.~Seidel, and
L.~M.~Blumenthal. This approach has proved useful in mathematical
studies of the kinematics of mechanisms, and in numerical studies of
the conformations of molecules. Tensegrity frameworks are an
architectural novelty introduced by K.~Snelson and popularized by
B.~Fuller, whose constructions and mathematical properties have been
extensively studied by R.~Connelly and colleagues. A key result has
been an algebraic criterion which can identify tensegrity frameworks
that are globally rigid, meaning that all realizations of the
framework
are congruent. This talk will describe the insights that Connelly's
results have brought to the applications of distance geometry to
molecular conformation, and conversely, some new insights which the
perspective of distance geometry brings to the theory of tensegrity
frameworks.
\abs{Donald Jacobs}
{Department of Physics and Astronomy, Michigan State
University,
East Lansing, Michigan 48824-1116, USA}
{Graph rigidity: applications to material science and
proteins}
The mechanical stability of network glasses and proteins can be
effectively studied by modeling the microstructure as a generic
bar-joint framework. Graph rigidity for bar-joint frameworks in two
dimensions is completely characterized by Laman's theorem [1].
Recently, an efficient algorithm has been constructed to study generic
rigidity percolation in two dimensions [2]. The desire to construct a
similar algorithm in three dimensions is impeded by the lack of an
analogous theorem. It has been proposed [3] that an all subgraph
constraint counting characterization of generic rigidity is recovered
in three-dimensional bar-joint networks having no implied hinge
joints.
Based on this proposition, an efficient combinatorial algorithm has
been constructed for bond-bending networks, which have no
implied-hinge
joints. Complete agreement is found with exact calculations involving
diagonalization of dynamical matrices, for systems up to $1000$
degrees
of freedom.
{\centering
{\bf References}
\par}
{\bf 1.}~~G.~Laman, J.~Engrg. Math. {\bf 4}(1970), 331.
{\bf 2.}~~D.J. Jacobs and B. Hendrickson, J.~Comput. Phys.
{\bf 137}(1997), 346.
{\bf 3.}~~D.J. Jacobs, J.~Phys. A: Math. Gen. {\bf 31}(1998), 6653.
\abs{G\'abor Kert\'esz}
{Department of Geometry, E\"otv\"os Lor\'and University,
H-1088~~Budapest, Hungary}
{The Dido problem on planes of constant curvature}
What is the tract of land of maximum area that can be fenced by a
given
collection of fence segments? The segments should be the sides of a
chord
polygon. The proof is simple if the segments can not cross. However
the proof of the general case was unknown for decades. Fifteen years
ago I found a proof. Almost the same considerations show that this so
called Dido type theorem for segments holds even in the hyperbolic
plane.
\abs{W\l odzimierz Kuperburg}
{Department of Mathematics, Auburn University, Auburn,
Alabama~~36849-5310, USA}
{Covering the cube with equal balls}
Define $c^d(k)$, the {\it covering radius}, as the minimum radius of
$k$ congruent balls that can cover the unit $d$-dimensional cube.
(The
packing radius is defined similarly.) In general, the problem of
determining the values of $c^d(k)$ as well as the corresponding
packing
problem seem to be extremely difficult even in dimension $2$. On the
packing problem, several results have been obtained in dimension $2$
by
various authors, and a few results in dimension $3$, mainly by
J.~Schaer. We discuss the covering problem and we determine $c^d(2)$,
$c^3(3)$, $c^3(4)$, $c^3(8)$, $c^4(4)$, and $c^4(16)$ along with the
optimal configurations of balls that produce them. Also, we state
conjectures on the remaining values of $c^3(k)$ and their ball
configurations for $k\le 12$.
\abs{Barry Monson}
{Department of Mathematics and Statistics, University of
New Brunswick, Fredericton, New~Brunswick~~E3B 5A3, Canada}
{Realizations of regular abstract polytopes}
A {\it regular abstract polytope\/} ${\cal P}$ is a poset having the
`essential' structural features of the face lattice of a regular
convex
polytope, including transitivity of $\Aut ({\cal P})$ on flags. Other
examples include regular tessellations and honeycombs, star-polyhedra
and many less familiar beasts. Since ${\cal P}$ need not actually be
a lattice or have a particularly nice geometric realization, it is
interesting to describe all {\it realizations\/} for ${\cal P}$. The
basic theory of realizations was developed by McMullen (1989), who
used
geometric methods to describe a crucial family of real representations
for $\Aut ({\cal P})$.
After an explicit, but brief, description of these objects, I will
discuss work with Asia Ivi\'{c} Weiss, in which we construct families
of regular polytopes parametrized by certain rings of algebraic
integers. Typically, our understanding of the realization cones for
these polytopes is very limited.
\abs{Konstantin Rybnikov}
{Department of Mathematics and Statistics, Queen's University,
Kingston, Ontario~~K7L 3N6, Canada}
{On traces of $d$-stresses in the skeletons of lower
dimensions of homology manifolds}
We show how $d$-stresses on piecewise-linear realizations of
$d$-manifolds in ${\bbd R}^d$ induce $k$-stresses, $0 \le k \le d-1$,
and discuss possible connections between the constructed polynomial
mappings and $g$-vectors of manifolds. This is a joint work with
R.~Erdahl and S.~Ryshkov.
\abs{Idzhad Kh. Sabitov}
{Faculty of Mechanics and Mathematics, Moscow State
University,
119899~~Moscow, Russia}
{Solution of polyhedra}
By analogy with the chapter of elementary mathematics named ``the
solution of triangles'' we can propose an idea for finding all metric
characteristics of a polyhedron based on the knowledge of its metric
and combinatorial structure. The essential moment of the proposed
approach is the generalized Heron's formule for volumes of isometric
polyhedra established in [1]. Namely we affirm that the volumes of all
isometric polyhedra with a fixed combinatorial simplicial structure
$K$
and given lengths $(l)$ of its edges are roots of a polynomial
equation
$$
Q(V)=V^{2N}+a_1(l)V^{2N-2}+\cdots +a_N(l)=0,
$$
whose coefficients are, in one's turn, polynomials in edges' lengths
with some numerical coefficients depending only on the combinatorial
structure $K$. For the construction of a such equation one can
indicate some corresponding algorithm. It turns out moreover that for
the polyhedra in general position we can find only a finite number of
values of its diagonals so we have a finite algorithm for the
construction of isometric polyhedra. Some simple cases of this
algorihm
are realized on the computer.
{\centering
{\bf References}
\par}
{\bf 1.}~~ I.~Kh.~Sabitov, Fund. i Prikl. Mat. {\bf 2}(1996),
1235--1246.
\abs{Peter Schmitt}
{Institut f\"ur Mathematik, Universit\"at Wien, A-1090~~Wien,
Austria}
{The versatility of (small) sets of prototiles}
A set of prototiles (which are usually assumed to be closed
topological
disks or balls) may or may not admit a tiling of the plane (or of
$n$-space). The set of all (distinct) tilings admitted represents
the
{\it versatility\/} of a set of prototiles. It may be large or small,
and the tilings may have quite different properties. The talk
provides
an survey of the versatility of small sets, with emphasis on
periodicity properties, and describes some general constructions.
\abs{Istvan Talata}
{Department of Mathematics, Auburn University, Auburn,
Alabama~~36849-5310, USA}
{On translative coverings of a convex body with
its homothetic copies of given total volume}
Let $K$ be a $d$-dimensional convex body. Denote by $h(K)$ the
minimum
number of smaller homothetic copies of $K$ which are needed to cover
$K$. Furthermore, denote by $hv(K)$ the smallest real number with the
property that every sequence of positive homothetic copies of $K$ with
total volume at least $hv(K) \vol(K)$ permints a translative covering
of $K$. It is clear that $h(K)\leq hv(K)+1$.
It is proved by Rogers [1957] that for any $d$-dimensional convex body
$K$ there exists a covering of $R^d$ with translates of $K$ with
density at most $\delta_d=d\log d+d\log \log d+5d$. As already Rogers
observed in 1967, this result implies that $h(K)\leq 2^d\delta_d$ for
centrally symmetric convex bodies. Similarly, $h(K)\leq 4^d\delta_d$
was proved for arbitrary convex bodies.
In this talk we show that the method used by Rogers can be extended
for
homothetic copies of $K$ with different coefficients. This way we can
improve on the upper bound $hv(K)\leq (d+1)^d-1$ of Januszewski [1998]
proving $hv(K)\leq 2^{d+o(d)}$ for centrally symmetric convex bodies,
and $hv(K)\leq 4^{d+o(d)}$ for arbitrary $d$-dimensional convex
bodies.
\abs{Anke Walz}
{Cornell University, Ithaca, New~York~~14853, USA}
{The Bellows conjecture in dimension four}
I will talk about the following generalized version of the Bellows
Conjecture: {\it ``If $S_t$ is a flex of an orientable singular cycle
in Euclidean $4$-space, then the volume of $S_t$ is constant.''} A
{\it flex\/} is a continuous motion of a configuration of vertices
that
fixes the lengths of the edges of all triangles of the cycle.
I.~Sabitov first proved this result for dimension $d=3$ by using
resultants to show that the volume of a polyhedron in 3-space is (up
to
a constant factor) integral over the squares of the edgelengths of the
polyhedron. R.~Connelly and A.~Walz modified his proof, looking at
places instead of resultants. I will show that this approach can be
generalized to dimension $4$. This provides a proof of the Bellows
Conjecture in dimension $4$.
\abs{Asia Weiss}
{York University, Toronto, Ontario~~M3J 1P3, Canada}
{To be announced}
\abs{Walter Whiteley}
{Department of Mathematics and Statistics, York University,
Toronto, Ontario~~M3J 1P3, Canada}
{Constraining a spherical polyhedron with dihedral angles}
The theorems of Cauchy, Dehn and Alexandrov describe which distance
constraints on edges and face diagonals of spherical polyhedra are
independent and which are sufficient for (local) rigidity of convex
polyhedra. We consider alternate constraints: the face-vertex
incidences and the dihedral angles along edges. Specific results
include:
(a)~~the incidences alone are independent, and we then have at most
$\min(|E|-1, 2|F|-3)$ additional independent dihedral angles;
(b)~~dependencies of dihedral angles (or self-stresses of the
corresponding spherical framework on the Gaussian sphere) are
isomorphic to parallel drawings of the polyhedron with a fixed point
(isogonal maps preserving all angles);
(c)~~there are examples of additional dependencies which result in
(infinitesimal) deformations of polyhedron preserving all incidences
and dihedral angles---but not all face angles.
We {\it conjecture\/} that, for almost all proper realizations of any
combinatorial type of spherical polyhedron with all dihedral angles
fixed, there are only isogonal deformations.
%\vskip 1.0cm
%\hrule
%\vskip .3cm
\session{
{\bf Extremal Combinatorics / Combinatoire extr\'emale}}
\org{D.~deCaen, Organizer}
%\hrule
%\vskip .4cm
\abs{Richard Anstee$^\ast$, Ron Ferguson and Attila Sali}
{Department of Mathematics, University of British Columbia,
Vancouver, British Columbia~~V6T 1Z2, Canada; Department of
Mathematics, University of British Columbia, Vancouver, British
Columbia~~V6T 1Z2, Canada and Mathematics Institute of Hungarian
Academy of Sciences, Budapest, Hungary}
{Small forbidden configurations}
An extremal set problem stated in matrix terms considers a $k\times l$
$(0,1)$-matrix $F$ and an $m\times n$ $(0,1)$-matrix $A$ which has no
repeated columns. We assume $A$ has no submatrix which is a row and
column permutation of $F$ and then seek bounds on $n$ in terms of
$F,m$. We wish to report progress in obtaining a number of new exact
bounds for various $2\times l$ $F$. The arguments often use graph
theory in interesting ways. Tur\'an's $m^2/4$ bound arises with a
different twist. It seems possible to get exact bounds for all
$2\times l$ $F$. Progress for $3\times l$ $F$ has also been obtained
but the triple systems are harder to analyze.
\abs{Aart Blokhuis, Ralph Faudree$^\ast$, Andr\'as Gy\'arf\'as and
Mikl\'os Ruszink\'o}
{Department of Mathematics and Computer Science, Technical
University of Eindhoven; Department of Mathematical Sciences,
University of Memphis, Memphis, Tennessee~~38152, USA; Computer and
Automation Research Institute, Hungarian Academy of Sciences,
Budapest, Hungary and Computer and Automation Research Institute,
Hungarian Academy of Sciences, Budapest, Hungary}
{Anti-Ramsey colorings in several rounds}
For positive integers $k\le n$ and $t$ let $\chi^t(k,n)$ denote the
minimum number of colors such that at least in one of the $t$
colorings
of edges of $K_n$ all $k\choose 2$ edges of every $K_k\subseteq K_n$
get different colors. Generalizing a result of K\"orner and Simonyi,
it
is shown in this paper that $\chi^t(3,n)=\Theta (n^{1/t})$. Also
two-round colorings in cases $k>3$ are investigated. Tight bounds for
$\chi ^2(k,n)$ for all values of $k$ except for $k=5$ are obtained.
Conversely, let $t(k,n)$ denote the minimum number of colorings such
that---having the same $k\choose 2$ colors in each coloring---at least
in one of the total $t$ colorings of $K_n$ all $k\choose 2$ edges of
every $K_k\subseteq K_n$ get different colors. It is also shown, that
for $k=n/2$ $t(k,n)$ is exponentially large. Several related
questions are investigated.
\abs{Jason Brown}
{Department of Mathematics and Statistics, Dalhousie
University,
Halifax, Nova Scotia~~B3H 3J5, Canada}
{The inducibility of complete bipartite graphs}
Given a fixed graph $G$, it is a natural extremal problem to ask:
among all graphs of order $n$, which have the maximum number of
induced
subgraphs isomorphic to $G$? Furthermore, what is this maximum
number? How can it be estimated? We examine here these problems for
complete bipartite graphs. Surprisingly, the extremal graphs are not
always complete bipartite graphs with equal (or nearly equal) parts.
Some extensions to the multipartite case will also be noted.
\abs{David Fisher}
{Department of Mathematics, University of Colorado at Denver,
Denver, Colorado~~80217-3364, USA}
{The minimum number of triangles in a graph}
{\it Given $0\le e\le{n\choose2}$, what is the minimum number of
triangles in a graph with $n$ vertices and $e$ edges?} Let $t_{n,e}$
be
the answer. For example, $G$ has $7$ vertices, $13$ edges, and $3$
triangles.
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\put(-3,0){\makebox(2,5)[r]{$G^c=$}}
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\par}
Since any graph with $7$ vertices and $13$ edges has at least $3$
triangles, we have $t_{7,13}=3$.
Trivially, $t_{n,e}=0$ if $e\le{n^2\over4}$. McKay (1963) proved
$t_{n,e}\ge{(4e-n^2)e\over3n}$. Bollob\'as (1978) improved this by
showing $t_{n,e}$ is at least the piecewise linear function with nodes
$\Bigl({k\choose2}({n\over k})^2, {k\choose3}({n\over k})^3\Bigr)$ for
$k=1,2,\dots\,$. Fisher (1989) found a better bound when
${n^2\over4}\le e\le{n^2\over3}$ by proving $t_{n,e}
\ge{9en-2n^3-2(n^2-3e)^{3/2}\over27}$. Nikiforov and Khadzhiivanov
(1981) and Lov\'asz and Simonovits (1983) independently found
$t_{n,e}=
(e- \lfloor{n^2\over4}\rfloor) \lfloor{n\over2}\rfloor$ for
$\lfloor{n^2\over4}\rfloor\le e<
\lfloor{n^2\over4}\rfloor+\lfloor{n\over2}\rfloor$.
I have a conjecture: {\it If $G$ has $n$ vertices, $e\ge{n^2\over4}$
edges, and $t_{n,e}$ triangles, then $G^c$ is not connected}.
Barnhart
and I verified this when $n\le14$. If true for all $n$, one can
recursively find $t_{n,e}$. This would give analytic proofs of all
results above plus a new bound similar to Fisher's when
$e>{n^2\over3}$.
\abs{Jerrold Griggs}
{Department of Mathematics, University of South Carolina,
Columbia, South Carolina~~29208, USA}
{Extremal graphs with bounded densities of small subgraphs}
Let $\Ex(n,k,\mu)$ denote the maximum number of edges of an $n$-vertex
graph in which every subgraph of $k$ vertices has at most $\mu$ edges.
Here we summarize some known results for the problem of determining
$\Ex(n,k,\mu)$, present simple new proofs, and provide new estimates
and extremal graphs. One of our main aims is to show how the
classical
Tur\'an theory can be applied to such problems. The case
$\mu={k\choose
2}-1$ is the famous result of Tur\'an. This is joint work with
Mikl\'os Simonovits and George Rubin Thomas.
\abs{Penny Haxell}
{University of Waterloo, Waterloo, Ontario~~N2L 3G1, Canada}
{Integer and fractional packings in dense graphs}
Let $H_0$ be any fixed graph. For a graph $G$ we define $\nu_{H_0}(G)$
to be the maximum size of a set of pairwise edge-disjoint copies of
$H_0$ in $G$. We say a function $\psi$ from the set of copies of $H_0$
in $G$ to $[0,1]$ is a {\it fractional ${H_0}$-packing\/} of $G$ if
$\sum_{H\ni e}\psi(H)\leq 1$ for every edge $e$ of $G$. Then
$\nu_{H_0}^\ast(G)$ is defined to be the maximum value of
$\sum_{H\in{G\choose{H_0}}}\psi(H)$ over all fractional
${H_0}$-packings $\psi$ of $G$. We show that
$\nu_{H_0}^\ast(G)-\nu_{H_0}(G)=o(|V(G)|^2)$ for all graphs $G$.
(Joint work with V.~R\"odl.)
\abs{Felix Lazebnik}
{Department of Mathematical Sciences, University of
Delaware, Newark, Delaware~~19716, USA}
{On a class of algebraically defined graphs}
Let $F^n$ denote be the $n$-dimensional vector space over a field
$F$. For $n\ge 2$ and each $i= 1,2,\ldots, n-1$, let $f_i\colon
F^{2i}\to F$ be a function of $2i$ variables. We consider a
bipartite
graph whose vertex partitions $P$ and $L$ are copies of $F^n$ with $p
=
(p_1,p_2,\ldots, p_n)\in P$ and $l = (l_1,l_2,\ldots, l_n)\in L$ being
joined by an edge if and only if the following $n-1$ equalities are
satisfied:
$$
\displaylines{
l_2 + p_2 = f_1(p_1,l_1)\cr
l_3 + p_3 = f_2(p_1,l_1, p_2,l_2)\cr
\vdots\cr
l_n + p_n = f_{n-1}(p_1,l_1, p_2,l_2, \ldots, p_{n-1},l_{n-1})\cr
}
$$
For particular fields $F$ and particular functions $f_i$'s, the
families of graphs defined this way (or slightly modified) possesses
many remarkable properties. They are concerned with forbidden cycles,
girth, graph homomorphism, eigenvalues, and edge-decompositions of
complete graphs and complete bipartite graphs. In this talk we survey
some known and some new results on such graphs based on the work of
A.~J.~Woldar and the speaker.
\abs{Laszlo Sz\'ekely}
{University of South Carolina, Columbia, South
Carolina~~29208-0001, USA}
{Some new Erd\H os-Ko-Rado type problems}
I will survey some recent variants of the classic Erd\H os-Ko-Rado
theorem: intersecting chains of sets, intersecting chains of
subspaces, intersecting chains in posets, higher-order Erd\H
os-Ko-Rado
theorem, Erd\H os-Ko-Rado theorem for arithmetic progressions.
\abs{Bing Zhou}
{Trent University, Peterborough, Ontario, Canada}
{Extremal graphs related to star chromatic number and
fractional
chromatic number}
We present results related to two generalizations of the chromatic
number of a graph: the star chromatic number $\chi^{\ast}$ and
fractional chromatic number $\chi_{f}$.
In general we have the inequality $2\leq\chi_{f}(G)\leq\chi^{\ast}
(G)\leq \chi(G)$ for any graph $G$. We will study the graphs that
have
$\chi^{\ast }(G)=\chi(G)$ in relation with other properties of the
graphs such as girth and planarity. Similar problems will be
considered for the fractional chromatic number as well.
%\vskip 1.0cm
%\hrule
%\vskip .3cm
\session{
{\bf Number Theory / Th\'eorie des nombres}}
\org{R.~Murty and N.~Yui, Organizers}
%\hrule
%\vskip .4cm
\abs{A. Akbary}
{Department of Mathematics and Statistics, Concordia
University,
Montreal, Quebec~~H3G 1M8, Canada}
{On the distribution of the values of symmetric square
$L$-functions in the half plane $\rmRe(s)>\frac{3}{2}$}
Let $L_{{\sym}^2(f)}(s)$ be the symmetric square $L$-function
associated to a newform of weight $2$ and level $N$. We will derive an
asymptotic formula for the average values of $L_{{\sym}^2(f)}(s)$ at a
point $s_0$ in the half plane $\rmRe(s)> \frac{7}{4}$. Assuming the
Riemann hypothesis for the Riemann zeta function, we are able to
extend
our result to the half plane $\rmRe(s)>\frac{3}{2}$.
\abs{Henri Darmon}
{McGill University, Montreal, Quebec~~H3A 2K6, Canada}
{Modularity of hypergeometric abelian varieties}
I will explain how Wiles result can be applied to prove the modularity
of a large class of abelian varieties with real multiplication
occuring
as specialisations of ``rigid families''.
\abs{Chantal David}
{Department of Mathematics and Statistics, Concordia
University,
Montreal, Quebec~~H3G 1M8, Canada}
{To be announced}
\abs{Jacek W. Fabrykowski}
{Department of Mathematics, University of Manitoba, Winnipeg,
Manitoba~~R3T 2N2, Canada}
{To be announced}
\abs{G. Frei}
{Department of Mathematics and Statistics, University of
Laval,
Montreal, Quebec~~G1K 7P4, Canada}
{On the discovery of the reciprocity law by Artin}
\abs{Eyal Goren}
{Department of Mathematics, CICMA Concordia and McGill,
McGill University, Montreal, Quebec~~H3A 2K6, Canada}
{Stratifications of moduli spaces and modular forms}
We present recent results (some in collaboration with E.~Bachmat and
with F.~Oort) on stratification of Hilbert and Drinfel'd modular
varieties in finite characteristic, which is related to modular forms
on these varieties. Special values of $L$-functions play an
interesting
role.
\abs{C. Greither}
{D\'epartement de math\'ematiques et statistique, Universit\'e
de Laval, Montr\'eal, Qu\'ebec~~G1K 7P4, Canada}
{On Brumer's conjecture}
Stickelberger showed around 1890 that a certain ideal $I$ in the
integral group ring ${\bf Z}G$ annihilates the class group of an
abelian extension $L$ with Galois group $G$ over the field of rational
numbers. The ideal $I$ is constructed algebraically, but it has an
interpretation by means of special values of $L$-functions as well.
Stickelberger's proof is constructive in the sense that the required
generators of principal ideals are written down as explicitly as one
could ask for.
If now $L$ is a complex abelian extension of a totally real field $K$,
one can construct a Brumer-Stickelberger ideal $I$ by reversing
history, that is, by starting from special L-values. The Brumer
conjecture then predicts annihilation of the class group $\Cl(K)$ by
$I$ (with some precautions); it is not proved yet in general. The
talk
will explain how to prove this conjecture for some classes of
extensions $L/K$. The essential ingredients are the theory of Fitting
ideals, Iwasawa theory, and the clever method of Wiles to bypass
trivial zeros. Nobody knows how to get the required generators
explicitly; the proof is highly nonconstructive.
At the end of the talk, some numerical and heuristical remarks
concerning the ${\bf Z}G$-structure of class groups will be
presented,
in the spirit of the Cohen-Lenstra heuristics.
\abs{James Huard}
{Department of Mathematics, Canisius College, Buffalo,
New~York~~14208, USA}
{An arithmetic reciprocity relation of Liouville type and
applications}
Let $n$ be a positive integer and let $f\colon {\bf Z}\rightarrow {\bf
C}$ be an even function. Liouville gave the identity
\begin{align*}
\sum_{aA + bB = n} \bigl( f(a-b) -f (a+b)\bigr)
&\null= f(0) \bigl(\sigma(n) - \tau (n)\bigr) + \sum_{d|n} f(d) + 2n
\sum_{d|n} {f(d)\over d}\\
& \qquad\null - \sum_{d|n} df(d) - 2\sum_{d|n} \sum_{1\leq v \leq d}
f(v),
\end{align*}
where the sum on the left hand side is taken over all positive
integers $a$, $A$, $b$, $B$ satisfying $aA + bB =n$. Some
generalizations of this identity will be presented and applications
will be made to the determination of such sums as
\[
\sum^{n-1}_{m=1} \sigma_3 (m)\sigma_3 (n-m)
\]
and to the representation of $n$ by certain positive quaternary
quadratic forms. The proofs of these results require elementary
methods. This is a joint work with Z.~Ou, B.~K.~Spearman, and
K.~S.~Williams.
\def\Q{{\bf Q}}
\abs{Hershey Kisilevsky}
{Concordia University, Montreal, Quebec~~H3G 1M8, Canada}
{Rank of $E(K)$ for cyclic cubic extensions $K/\Q$}
For an elliptic curve, $E$, defined over the rational field, we consider
the rank of $E(K)$ for cyclic cubic extensions $K/\Q$.
\abs{Manfred Kolster}
{Department of Mathematics, McMaster University,
Hamilton, Ontario~~L8S 4K1, Canada}
{Higher relative class number formulas}
Let $F/F^+$ be a $\CM$ extension of number fields, and let $\chi$
denote the non-trivial character of the Galois group of $F/F^+$. The
classical {\it relative class number formula\/} gives the following
relation between the value of the $L$-function of $\chi$ at $0$ and
the
relative class number $h^-$:
$$
L_{F^+}(\chi,0) = \frac{2^{r_1}}{Q} \cdot \frac{h^-}{w(F)},
$$
where $r_1 = [F^+:\bbd Q]$, $Q$ is the so-called $Q$-index, and $w(F)$
is the number of roots of unity in $F$. The purpose of the talk is to
describe analogs of this formula relating the values $L_{F^+}(\chi,
1-n)$ for $n \geq 3$ an odd integer to the orders of relative motivic
cohomology groups and $K$-groups.
\abs{Arne Ledet}
{Department of Mathematics, Queen's University, Kingston,
Ontario~~K7L 3N6, Canada}
{Some small $2$-groups as Galois groups}
In 1936, Witt gave a criterion for embedding a $V_4$-extension into a
$Q_8$-extension in terms of an equivalence of quadratic forms. Using
this result, it is possible to give similar criteria for solving other
embedding problems, starting with embedding a $D_4$-extension into a
$QD_8$-extension, where $D_4$ and $QD_8$ are the dihedral and
quasi-dihedral groups of order $8$ and $16$, resp. This again permits
a
complete description of extensions with certain $2$-groups of
order~$16$ as Galois groups.
\abs{C. Levesque}
{D\'epartement de math\'ematiques et statistique, Universit\'e
de Laval, Montr\'eal, Qu\'ebec~~G1K 7P4, Canada}
{Explicit solutions of a family of Thue diophantine equations}
Looking for units of certain number fields built from modular
coverings
$X_0(m)$, H.~Darmon obtained the family of polynomials
$$
F_a(X,Y)= X^5+2X^4Y+(a+3)X^3Y^2+(2a+3)X^2Y^3+(a+1)XY^4-Y^5.
$$
For $a$ sufficiently large, O. Kihel exhibited a fundamental system of
units of the field ${\bf Q}(\omega)$ where $\omega$ is a root of
$F_a(X,1)=0$. In this lecture we will show that for $a$ sufficiently
large, the family of Thue diophantine equations of degree $5$ given by
$$
F_a(X,Y)=\pm1
$$
has only the six trivial solutions $(x,y)=(0,1),(0,-1), (1,0), (-1,0),
(1,-1), (-1,1)$. The associated Siegel equation leads firstly to a
linear form in four logarithms, but the main ingredient and the
feature
of the proof is to write it as a linear form in two logarithms.
\abs{Kumar Murty}
{Mathematics Department, University of Toronto, Toronto,
Ontario~~M5S 3G3, Canada}
{Zeros of Dedekind zeta functions in towers of fields}
We shall discuss some recent results about the non-existence of
zeros of Dedekind zeta functions near $s=1$ as we vary over a
tower of fields.
\abs{W. Georg Nowak}
{Universit\"at f\"ur Bodenkultur, A-1180~~Vienna, Austria}
{Large convex domains sometimes contain more lattice points
than we would expect}
Let as usual $r(n)$ denote the number of ways to write $n\in{\bf N}$
as
a sum of two squares. Then the quantity
$$
P(t) = \sum_{0\le n\le t} r(n) - \pi t
$$
(the ``lattice rest'' of an origin-centered circle of radius
$\sqrt{t}$) is
well-known to satisfy
$$
\int_0^X \bigl(P(t)\bigr)^2 \,dt \sim C X^{3/2} \leqno (1)
$$
(Cram\'er, Landau),
$$
\liminf_{t\to\infty} {P(t) \over (t\log t)^{1/4}} < 0 \leqno (2)
$$
(Hardy), and
$$
\limsup_{t\to\infty} {P(t) \over t^{1/4}\exp\bigl(c(\log\log
t)^{1/4}(\log\log t)^{-3/4}\bigr)} > 0 \leqno (3)
$$
(Corr\'adi \& K\'atai). Compared to (2), (3) is not only weaker but
also incapable of generalisation to more general domains, its proof
being based on rather special ``arithmetic'' arguments.
The present talk addresses the corresponding problem for cubes: Let
$$
r_3(n) = \#\{(u,v)\in{\bf Z}^2: |u|^3 + |v|^3 = n \} ,
$$
and denote by $P_3(t)$ the error term in the asymptotic formula for
$\sum_{0\le n\le t^{3/2}}r_3(n)$. Combining classic analytic number
theory with some profound algebra and a very recent deep result of
Heath-Brown [1], the authors [2] where able to show that
$$
\limsup_{t\to\infty} {P_3(t) \over (t \log\log t)^{1/4}} > 0.
$$
This is based on the fact that the set
$$
\{(m^{3/2}+n^{3/2})^{2/3} : (m,n)\in{\bf N}^2,\gcd (m,n) = 1\}
$$
contains a rather ``large'' set of numbers which are linearly
independent over the rationals.
The corresponding general problem for $r_k(n)$, $k>3$, remains open at
the present state-of-art.
{\centering
{\bf References}
\par}
{\bf 1.}~~D.~R.~Heath-Brown, {\it The density of rational points on
cubic
surfaces}. Acta Arith. {\bf 79}(1997), 17--30.
{\bf 2.}~~M.~K\"uhleitner, W.~G.~Nowak, J.~Schoi\ss engeier and
T.~Wooley,
{\it On sums of two cubes: An $\Omega_+$-estimate for the error term}.
Acta
Arith. {\bf 85}(1998), 179--195.
\abs{Yannis Petridis}
{McGill University, Montreal, Quebec~~H3A 2K6, Canada}
{Zeros of the Riemann zeta function and central values of
$L$-series of holomorphic cusp forms}
We examine the stability of the zeros of the Riemann zeta function,
which are twice the scattering poles of $\SL(2, Z)$, in relation to
the
central value of the $L$-series of holomorphic cusp forms of weight
$2$
for the congruence subgroups $\Gamma_0(q)$, $q$ prime. We work with
perturbations in characters varieties of $\Gamma_0(q)$ and study the
effects on the spectral and scattering theory of the Laplace operator.
\abs{Gael R\'emond}
{University of Ottawa, Ottawa, Ontario~~K1N 6N5, Canada}
{Theta heights and Kodaira construction}
We prove that in Kodaira construction a height on the base curve is
described by the theta height of the jacobians of the fibres. We make
use of theta heights as introduced by Bost and David who prove a
comparison with Faltings' height. Therefore our result shows that a
suitable effective version of Shafarevich conjecture (a bound for
Faltings' height) implies an effective Mordell.
\abs{D. Roy}
{University of Ottawa, Ottawa, Ontario~~K1N 6N5, Canada}
{Criteria of algebraic independence and approximation
by hypersurfaces}
Given a point $\theta$ in $\bbd C^m$, a fundamental problem is how
close one can approximate $\theta$ by a point of an algebraic variety
of dimension $d$, defined over $\bbd Q$, with degree $\le D$ and
logarithmic height $\le T$. The problem has a different flavor
whether, for a fixed $d$, one wants an estimate valid for a pair
$(D,T)$ or for infinitely many pairs $(D_n,T_n)$ chosen from a given
non-decreasing sequence of positive integers $(D_n)_{n\ge 1}$, and a
given non-decreasing unbounded sequence of positive real numbers
$(T_n)_{n\ge 1}$ with $T_n\ge D_n$ for each $n\ge 1$. In a joint work
with Michel Laurent, we analyze the second type of problem when
$d=m-1$. We show that, for infinitely many indices $n$, there exists
a
nonzero polynomial $P\in \bbd Z[X_1,\dots,X_m]$ of degree $\le D_n$
whose coefficients have absolute value $\le \exp(T_n)$, such that $P$
admits at least one zero $\alpha$ in $\bbd C^m$ with
$$
\|\theta-\alpha\| \le \exp\bigl(-c(m) D_{n-1}^m T_{n-1}\bigr)
$$
for some positive constant $c(m)$ which depends only on $m$.
This follows from a new criteria of algebraic independence with
multiplicities. The object of the talk is to explain the link
between the two results.
\abs{Gary Walsh}
{Department of Mathematics, University of Ottawa, Ottawa,
Ontario~~K1N 6N5, Canada}
{Old and new results on quartic diophantine equations}
During his illustrious career, Wilhelm Ljunggren proved many results
on
the solvability of Diophantine equations of the form $aX^4-bY^2=c$.
In
this talk we present some of Ljunggren's work, some problems remaining
from his work, and recent progress on these outstanding problems. This
is joint work with M.~A.~Bennett.
\abs{Hugh Williams}
{Department of Mathematics, University of Manitoba, Winnipeg,
Manitoba~~R3T 2N2, Canada}
{Computer verification of the Ankeny-Artin-Chowla conjecture
for all $p< 5.10^{10}$}
Let $p$ be a prime congruent to $1$ modulo $4$ and let $t$, $u$ be
rational integers such that $(t+\surd pu)/2$ is the fundamental unit
of the real quadratic field ${\bf Q} (\surd p)$. The
Ankeny-Artin-Chowla conjecture ($\AAC$ conjecture) asserts that $p$
will not divide $u$. This is equivalent to the assertion that $p$
will
not divide $B_{(p-1)/2}$, where $B_n$ denotes the $n$-th Bernoulli
number. Although first published in 1952, this conjecture still
remains unproved today. Indeed, it appears to be most difficult to
prove. Even testing the conjecture can be quite challenging because
of
the size of the numbers $t$, $u$; for example, when $p = 40094470441$,
then both $t$ and $u$ exceed $10^{330000}$. In 1988 the $\AAC$
conjecture was verified by computer for all $p<10^9$. In this paper
we
describe a new technique for testing the $\AAC$ conjecture and we
provide some results of a computer run of the method for all primes
$p$
up to $5.10^{10}$.
This is joint work with Alf van~der~Poorten and Herman te~Riele.
\abs{Kenneth Williams}
{School of Mathematics and Statistics, Carleton University,
Ottawa, Ontario~~K1S 5B6, Canada}
{Values of the Dedekind eta function at quadratic
irrationalities}
The Dedekind eta function $\eta(z)$ is defined by
$$
\eta (z) = e^{\pi i z/12} \prod^\infty_{m=1} (1-e^{2\pi imz})
$$
for $im (z) >0$. Let $d$ be the discriminant of an imaginary
quadratic
field. The value of $\bigl| \eta \bigl((b +\sqrt{d}/2a\bigr)\bigr|$
is
determined for integers $a$, $b$, $c$ satisfying
$$
b^2 -4ac = d, \quad a >0, \ (a,b,c) =1.
$$
This is joint work with A.~J. van~der~Poorten.
%\vskip 1.0cm
%\hrule
%\vskip .3cm
\session{
{\bf Operator Algebras / Alg\`ebres d'op\'erateurs}}
\org{J.~Mingo, Organizer}
%\hrule
%\vskip .4cm
\abs{Berndt Brenken}
{Department of Mathematics and Statistics, University of
Calgary,
Calgary, Alberta~~T2N 1N4, Canada}
{Endomorphisms of finite direct sums of $I_{\infty}$ factors}
The well known correspondence between $*$-representations of Cuntz
algebras and unital $*$-endomorphisms of ${\cal B}({\cal H})$ is
extended to a correspondence between $*$-representations of
Cuntz-Krieger algebras and unital $*$-endomorphisms of finite direct
sums of $I_{\infty}$ factors.
\abs{Kenneth R. Davidson}
{Pure Mathematics Department, University of Waterloo,
Waterloo, Ontario~~N2L 3G1, Canada}
{Principal bimodules of nest algebras}
We classify the $\WOT$-closed bimodules over a pair of nest algebras
which are singly generated as algebraic and as norm-closed bimodules.
The obstructions relate to the finite rank atoms. In particular, if
both nest algebras have infinite multiplicity, then every
$\WOT$-closed
bimodule is (algebraically) principal. Another important special case
is the ideal of strictly upper triangular operators, which is always
principal; and the generator is a sum of commutators. In general,
every countably generated $\WOT$-closed bimodule is singly generated,
and we obtain explicit bounds on the number and norms of the terms in
a
factorization through the generator.
\abs{George Elliott}
{Fields Institute, Toronto, Ontario, Canada}
{An abstract Brown-Douglas-Fillmore absorption theorem, II}
The criterion of Elliott and Kucerovsky (see Part I) for an extension
of a stable, nuclear, separable $C^\ast$-algebra by an arbitrary
separable
$C^\ast$-algebra to be absorbing---namely, that it be what is called
``purely large''---can be verified directly in all known cases.
This is joint work with Dan Kucerovsky.
\abs{Thierry Giordano}
{University of Ottawa, Ottawa, Ontario~~K1N 6N5, Canada}
{Ergodic theory and dimension $G$-spaces}
Let $G$ be a discrete group. The real group algebra $A = \Bbb R G$ has
a natural order structure given by the positive cone $A_+ = \{\sum r_g
g ; r_g = 0$ a.e. and $r_g \ge 0\}$ and is endowed with an
order-preserving action of $G$ (by right multiplication). If $n \in
\Bbb N$, then $A^n$ is a partially ordered vector space with the direct
sum ordering and a $G$-space with the above $G$-action.
\smallskip{\bf Definition.\ } A $G$-dimension space $H$ is a partially ordered
vector space with an action of $G$ (as a group of order automorphisms)
that can be obtained as a direct limit
$$
H = \lim\limits_{\longrightarrow} A^{n(i)} \buildrel \phi_i\over
\longrightarrow A^{n(i+1)} , \leqno(1)
$$
where $\phi_i$ is a positive $A$-module map (for the natural ordering
on $A$).
Corresponding to the inductive limit in (1) is a matrix-valued random
walk on $G$. The harmonic functions associated to this random walk are
in a natural bijection with the states on $H$. A state $\gamma$
corresponding to a bounded harmonic function is called bounded. It
induces a pseudo-norm on $H$ and allows us to associate to $H$ the real
$L^1$-space $L^1(X)$, the ``completion'' of $H$ (as defined by Goodearl
and Handelman). If for all $g \in G$, $g\gamma \le N(g)\gamma$ ($N(g)$
depending on $g$), then $G$ acts on $L^1(X)$.
D.E. Handelman and I have defined the notion of ergodicity and
different generalizations of approximate transitivity for the action of
$G$ on $(H,\gamma)$ which extends to $L^1(X)$ and its dual.
\abs{Dan Kucervosky}
{Fields Institute, Toronto, Ontario, Canada}
{An abstract Brown-Douglas-Fillmore absorption theorem, I}
A common generalization is given of all previous absorption theorems
for $C^\ast$-algebra extensions. We consider extensions of a stable,
nuclear, separable $C^\ast$-algebra by an arbitrary separable
$C^\ast$-algebra, and in this setting, we give a simple intrinsic
criterion for an extension to be absorbing. The criterion involves
only the $C^\ast$-algebra of the extension, together with the
canonical
closed two-sided ideal. This is joint work with George Elliott.
\abs{Michael Lamoureux}
{Department of Mathematics and Statistics, University of
Calgary,
Calgary, Alberta~~T2N 1N4, Canada}
{Crossed product algebra constructions}
By a number of examples, we demonstrate the relationships between
$C^\ast$-crossed product algebras that arise from group actions,
semigroup actions, and partial actions, and the corresponding
nonselfadjoint crossed products. The connection between the
corresponding ideal structures will be discussed.
\abs{Alexandru Nica}
{Department of Pure Mathematics, University of Waterloo,
Waterloo, Ontario~~N2L 3G1, Canada}
{Some minimization problems for the free analogue of the
Fisher information}
We consider the free non-commutative analogue $\Phi^{\ast}$,
introduced
by D.~Voiculescu, of the concept of Fisher information for random
variables. We determine the minimal possible value of $\Phi^{\ast}
(a,a^{\ast})$, if $a$ is a non-commutative random variable subject to
the constraint that the distribution of $a^{\ast}a$ is prescribed.
More
generally, we obtain the minimal possible value of $\Phi^{\ast} (\{
a_{ij},a_{ij}^{\ast}\}_{i,j})$, if $\{a_{ij}\}_{1 \leq i,j \leq d}$ is
a family of non-commutative random variables such that the
distribution
of $A^{\ast}A$ is prescribed, where $A$ is the matrix
$(a_{ij})_{i,j=1}^{d}$. The $d \times d$-generalization is obtained
from the case $d=1$ via a result of independent interest, concerning
the minimal value of $\Phi^{\ast} (\{a_{ij},a_{ij}^{\ast}\}_{i,j})$
when the matrix $A = (a_{ij})_{i,j=1}^{d}$ and its adjoint have a
given
joint distribution. (A version of this result describes the minimal
value of $\Phi^{\ast} (\{ b_{ij} \}_{ij})$ when the matrix $B =
(b_{ij})_{i,j=1}^{d}$ is selfadjoint and has a given distribution.)
We then show how the minimization results obtained for $\Phi^{\ast}$
lead to maximization results concerning the free entropy
$\chi^{\ast}$,
also defined by Voiculescu.
This is joint work with Dimitri Shlyakhtenko and Roland Speicher.
\abs{John Phillips}
{Department of Mathematics and Statistics, University of
Victoria, Victoria, British Columbia~~V8W 3P4, Canada}
{Spectral flow and index in bounded and unbounded
$\theta$-summable Fredholm modules: integral formulas}
In joint work with Alan Carey, we study $\theta$-summable Fredholm
modules $(H,D_0)$ for Banach ${\ast}$-algebras, $A$, and integral
formulas for the pairing of $(H,D_0)$ with $K_1(A)$. In particular, if
$(H, D_0)$ is $\theta$-summable (in Connes' original sense that
$\Tr(e^{-t D_0^2})<\infty$ for all $t>0$) then we prove that if $u$ is
a unitary in $A^1$ with $[u,D_0]$ bounded and $P=\chi_{[0,\infty
)}(D_0)$, then
$$
\rmindex (PuP)={1\over {\sqrt{\pi}}} \int_0^1\Tr \Bigl({d\over {
dt}} (D_t) e^{-D_t^2}\Bigr)\,dt
$$
where $D_t=D_0+t(uD_0u^{\ast}-D_0)$ is the straight line path from
$D_0$ to $uD_0u^{\ast}$. This is the pairing of $[u]$ in $K_1(A)$ with
$[D_0]$ in $K^1(A)$ and can also be interpreted as the spectral flow
of
the path $\{D_t\}$. A proof of this formula was first outlined by
Ezra
Getzler.
Our proof is quite different from the one indicated by Getzler. Our
method is able to handle Connes' new notion of $\theta$-summability
(which we dub $\weak$-$\theta$-summability) {\it i.e.}, $\Tr
(e^{-tD_0^2})<\infty$ for {\it some\/} $t>0$. We can also handle the
various bounded notions of $\theta$-summability (indeed, our method is
based on this). Finally our method simultaneously deals with the
$\hbox{II}_{\infty}$-version of all these results.
Roughly speaking, our proofs involve three steps. First, given $(H
,D_0$) we make a careful analytic study of the map $D\mapsto
F_D:=D(1+D^2 )^{-{1\over 2}}$ defined for $D$ in $D_0+{\cal
B}(H)_{\sa}$. The pair $(H,F_{D_0})$ is a pre-Fredholm module for $A$
which is $\theta$-summable (in a restricted sense) and the $F_D$ vary
in a certain affine space of bounded self-adjoint operators,
$F_{D_0}+{\cal L}_{\sa}$. We show that this map
$$
D\mapsto F_D\colon D_0+{\cal B}(H)_{\sa}\rightarrow
F_{D_0}+{\cal L}_{\sa}
$$
is suitably smooth. Second, we study the various notions of bounded
$\theta$-summability that arise in this context. In particular, one of
our results shows that if $(H,F_0)$ is a pre-Fredholm module which is
$\theta$-summable for $A$ in the strongest sense, then for $F$ in
$F_0+{\cal L}_{\sa}$ and $X$ in ${\cal L}_{\sa}$ (the tangent space to
$F_0+{\cal L}_{\sa}$ at $F$), $\Tr (X e^{-|1-F^2|^{-1}})$ is a closed
(and exact) 1-form on the manifold, $F_0+{\cal L}_{\sa}$. The idea to
use these differential-geometric notions in this context goes back to
I.M. Singer and was also used by Getzler in his proof. The point here
is that now the integral
$$
\int_0^1\Tr \Bigl({d\over {dt}}(F_t)e^{-|1-F_t^2|^{-1}}\Bigr)\,dt
$$
is seen to be independent of path in the space, $F_0+{\cal L}_{\sa}$.
Third, we can now reduce to the case of genuine (bounded,
$\theta$-summable) Fredholm modules $(H,F_0$) ({\it i.e.}, $F_0^2=1$),
and paths $F_t=F_0+t(uF_0u^{\ast}-F_0)$. In this setting, $P=2F_0-1$
and
the formula
$$
\Index (PuP)={1\over C} \int_0^1\Tr \Bigl({d\over {
dt}}(F_t)e^{-|1-F_t^2|^{-1}}\Bigr)\,dt
$$
is still {\it not\/} a triviality.
As mentioned above, our results are proved in such a way that the
type $\hbox{II}_{\infty}$ case is included at all stages.
\abs{Jack Spielberg}
{Department of Mathematics, Arizona State University, Tempe,
Arizona~~85287-1804, USA}
{A new look at $C^\ast$-algebras of infinite graphs}
I will describe a new approach to the generalization of Cuntz-Krieger
algebras from finite to infinite matrices. I will compare this with
other methods.
\abs{Sam Walters}
{Department of Mathematics and Computer Science, University
of Northern British Columbia, Prince George, British
Columbia~~V2K 4A2, Canada}
{$K$-theory of non commutative spheres arising from the
Fourier
automorphism}
It is shown that for a dense $G_\delta$ set of the real number
$\theta$
(containing the rationals) there is an isomorphism
$K_0(A_\theta\rtimes_\sigma\Bbb Z_4)\cong\Bbb Z^9$, where $A_\theta$
is
the rotation $C^\ast$-algebra generated by unitaries $U,V$ satisfying
$VU=e^{2\pi i\theta}UV$ and $\sigma$ is the Fourier automorphism given
by
$\sigma(U)=V$, $\sigma(V)=U^{-1}$. More precisely, a basis consisting
of
nine canonical modules is explicitly given. It is also shown that for
a
dense $G_\delta$ one has $K_1(A_\theta\rtimes_\sigma\Bbb Z_4)=0$.
%\vskip 1.0cm
%\hrule
%\vskip .3cm
\session{
{\bf Probability Theory / Th\'eorie des probabilit\'es}}
\org{M.~Cs\"org\H o, Organizer}
%\hrule
%\vskip .4cm
\abs{Siva Athreya}
{The Fields Institute, 222 College Street, Fields Institute,
Toronto,
Ontario~~M5T 3J1 Canada}
{Existence of Positive Solutions Satisfying the Boundary
Harnack Principle
for a Semi-linear Dirichlet Problem}
Boundary Harnack principle is a key tool in obtaining many results in
classical potential theory. Suppose $D$ is a smooth domain and $u$ and
$v$ are two positive harmonic functions on $D$ that vanish on a subset
$A$ of $\partial D$. The boundary Harnack principle says that $u$ and
$v$ tend to zero at the same rate. Over the past three decades, there
has been a lot of research on extending the principle to very general
domains.
Another natural question is, does the boundary Harnack principle hold
for solutions of elliptic partial differential equations other than
$\triangle u = 0$ ? We shall investigate the above question in our
talk. This was part of my Ph.D. thesis work with Professor K.~Burdzy,
at the University of Washington.
\abs{Claude Belisle}
{D\'epartement de math\'ematiques et statistique, Universit\'e
de Laval, Montr\'eal, Qu\'ebec~~G1K 7P4, Canada}
{The Hit-and-run sampler}
The {\it hit-and-run\/} sampler is a Markov Chain Monte Carlo method
for simulating probability measures.
Let $\pi$ be an absolutely continuous probability measure on ${\bbd
R}^d$. Let $\nu$ be a full dimensional probability measure on the
surface $S$ of the $d$-dimensional unit ball centered at the origin.
Given a {\it current\/} point $X_n \in {\bbd R}^d$, the hit-and-run
sampler chooses a next point $X_{n+1}$ according to the
conditionalization of $\pi$ on the line through $X_n$ and $X_n +
\Theta_{n+1}$. The {\it directions\/} $\Theta_1, \Theta_2,
\Theta_3,\dots$ are independent and identically distributed on $S$,
with distribution $\nu$. Under an appropriate irreducibility
condition, the Markov chain $(X_n; n \ge 0)$ converges in total
variation towards the {\it target\/} distribution $\pi$. In this talk,
I will discuss the convergence properties of this Markov chain.
Related Markov Chain Monte Carlo methods, including the Gibbs sampler,
will also be discussed.
\abs{David Brillinger}
{Statistics Department, University of California, Berkeley,
California~~94720-3860, USA}
{Some aspects of the motion of particles described by
stochastic
differential equations}
Consider a particle moving on the surface of the unit sphere in $R^3$
and heading towards a specific destination with a constant average
speed, but subject to random deviations. The motion is modelled as a
diffusion with drift restricted to the surface of the sphere.
Expressions are set down for various characteristics of the process
including expected travel time to a cap, the limiting distribution,
the
likelihood ratio and some estimates for parameters appearing in the
model.
The cases of motion near the surface of the sphere and where
explanatory variables are present are also considered.
\abs{Murray D. Burke}
{Department of Mathematics and Statistics, University of
Calgary,
Calgary, Alberta~~T2N 1N4, Canada}
{Model checking and estimation: A large sample approach}
A brief survey of some models used in statistical analysis is given.
Estimation and tests of fit considerations often lead to
distribution-free or asymptotically distribution-free statistics.
However, in many types of situations ({\it e.g.} multivariate,
semiparametric models), the resulting processes depend on the
underlying distribution function or family of distribution functions.
Weighted bootstrap methods are considered as an omnibus way of
approximating the limiting distribution of test statistics. Since
percentage points of the resulting statistics do not need to be
calculated explicitly, one can be flexible in the choice of statistic
and achieve greater power against particular alternatives.
\abs{Colleen D. Cutler}
{Department of Statistics and Actuarial Science, University
of Waterloo, Waterloo, Ontario~~N2L 3G1, Canada}
{Scaling structures, chaos, and determinism in time series}
The influence of ideas from chaos theory in numerous fields of science
has led to the development of new interest in and new methods of
searching for underlying structure and deterministic chaos in time
series. Many of these methods have centred around ideas of {\it
scaling\/} (various concepts of fractal dimension, such as correlation
dimension, information dimension, and pointwise dimension) and notions
of {\it embedding\/} (reconstruction of hidden state space from an
observed time series). In this talk we will review these ideas and
methods and relate them to notions of chaos, predictability, and
determinism in time series.
\abs{Andr\'e Dabrowski}
{Department of Mathematics and Statistics, University of
Ottawa,
Ottawa, Ontario~~K1N 6N5, Canada}
{A unified approach to fast teller queues and $\ATM$}
In the design of modern $\ATM$ switches, it is necessary to use
simulation to estimate the probability that a queue within the switch
exceeds a given large value. Since these are extremely small
probabilities, importance sampling methods must be used. Here we
obtain a change of measure for a broad class of models with direct
applicability to $\ATM$ switches.
We consider a model with several independent (not necessarily
identical) sources of cells where each source is modeled by a Markov
renewal point process with batch arrivals. We do not assume that
batch
sizes are independent of the state of the Markov process. These
arrivals join a queue served by multiple independent servers, each
with
service times also modeled as a Markov renewal process. The queue is
viewed as the additive component of a Markov additive chain
constrained
so that the additive component remains nonnegative. We apply the
theory in McDonald (1996) to obtain the asymptotics of the tail of the
distribution of the queue size in steady state and that of the mean
time between large deviations of the queue size.
Joint work with B.~Beck (Universit\'e Catholique de Louvain) and
D.~McDonald (Ottawa).
\abs{Eric Derbez}
{The Fields Institute}
{Generating functions and Integrated SuperBrownian Excursion
($\ISE$)}
Using the lace expansion (first developed by Brydges and Spencer) as
implemented by Hara and Slade, Derbez and Slade have recently proved
that lattice trees (studied in statistical mechanics) converge in an
appropriate scaling limit to a variant of super Brownian motion known
as Integrated SuperBrownian Excursion ($\ISE$). The key to linking
these
two seemingly disconnected models is a rather simple looking
generating
function.
\abs{Shui Feng}
{McMaster University and Fields Institute}
{The behaviour of some degenerate diffusions near boundary
under large deviations}
In this talk we will provide answers to the following question: What
are the possible paths of some degenerate diffusions under large
deviations?
\abs{Ren\'e Ferland}
{Universit\'e du Qu\'ebec \`a Montr\'eal, Montr\'eal,
Qu\'ebec~~H3C 3P8, Canada}
{Propagation of chaos: from Physics to Finance}
The chaos propagation property was first proved by Mark Kac for a
simple model which bear his name since then. We present chaos
propagation results (and fluctuations) for Boltzmann-like equations
(scalar and mollified) and for scalar conservation laws. We also
discuss possible applications of propagation of chaos in mathematical
finance.
\abs{A. F\"oldes}
{The College of Staten Island, CUNY, New~York, New~York, USA}
{About the local time of random walk and Brownian motion}
The local time of the random walk and the local time of the Brownian
motion (just like the processes themselves) share many properties.
Strong approximation results will be discussed for the partial sum
process of i.i.d. sequence of vectors having dependent components,
where the components of the approximating process are independent.
Applications of these results for additive functionals of random walk
in one and two dimensions and for Brownian motion will also be given.
This is a joint work with E.~Cs\'aki.
\abs{Genevi\'eve Gauthier}
{\'Ecole des Hautes \'Etudes Commerciales, Service de
l'enseignement des methodes quantitatives de gestion, 3000, chemin de
la Cote-Sainte-Catherine, Montr\'eal, Quebec~~H3T 2A7, Canada}
{Asymptotic distribution of the $\EMS$ option}
Monte Carlo simulation is a commonly used valuation tool in finance.
The method is useful for computing prices of derivative securities
when
an analytical solution does not exist. Recently, a new simulation
technique, known as empirical martingale simulation ($\EMS$), has been
proposed by Duan and Simonato (1998) as a way of improving simulation
accuracy. $\EMS$ has one drawback, however. Because of the
dependency
among sample paths created by the $\EMS$ adjustment, the standard
error
of the price estimate cannot be obtained by simply using one
simulation
sample and its asymptotic distribution is unknown. In this paper, we
develop a scheme that is capable of overcoming this deficiency. The
$\EMS$ price estimator is shown to have an asymptotically normal
distribution.
\abs{Edit Gombay}
{Department of Mathematical Sciences, University of Alberta,
Edmonton, Alberta~~T6G 2G1, Canada}
{Correcting some limit theorems about the likelihood ratio}
To put the new results into perspective, first, we review some aspects
of the history of the likelihood ratio since its definition by Neyman
and Pearson in 1928.
Strong approximation of the maximum likelihood ratio statistic by a
diffusion process under the null hypothesis is given. This allows one
to develop statistics using different weight functions. Sequential
tests will be defined and the precision of the approximation is
examined.
The asymptotic distribution of the likelihood ratio under noncontiguous
alternatives is shown to be normal for the exponential family of
distributions. The rate of convergence of the parameters to the
hypothetical value is specified where the asymptotic noncentral
chi-square distribution no longer holds. It is only a little slower
than $O(n-1/2)$. The result provides compact power approximation
formulae and is shown to work reasonably well even for moderate sample
sizes.
In conclusion, we briefly consider the consequences of these theorems
in sequential testing theory.
\abs{Lajos Horv\'ath}
{Department of Mathematics, University of Utah, Salt Lake
City,
Utah~~84112, USA}
{Best approximations for bootstrapped processes with
applications}
We study the asymptotic properties of bootstrapped empirical
processes
based on ``naive'' and weighted bootstrap. We show that the order of
the best possible approximation for the bootstrapped processes cannot
be better than $O_P(n^{-1/2}\log n)$ and we construct sequences of
Brownian bridges where this rate is achieved. We discuss applications
of the main results to density estimation and change-point detection.
\abs{Gail Ivanoff}
{Department of Mathematics and Statistics, University of
Ottawa, Ottawa, Ontario~~K1N 6N5, Canada}
{Set-indexed Martingales}
We give an appropriate framework and definitions for the theory of
set-indexed martingales. Compensators and quadratic variation
processes will be defined, and used to develop set-indexed analogues
of
the classical martingale characterizations of the Poisson process and
Brownian motion. Sufficient conditions will be given in terms of
compensators for a Poisson convergence theorem and in terms of
quadratic variation processes for a martingale central limit theorem.
Applications to set-indexed empirical processes will be discussed.
\abs{Michael Kouritzin}
{University of Alberta, Edmonton, Alberta~~T6G 2G1, Canada}
{Parabolic equations with random coefficients}
Questions related to the asymptotic behavior (as $\epsilon \rightarrow
0$) of systems of random ordinary differential equations
\[
\dot{X}^{\epsilon }(t)=F\Bigl(X^{\epsilon }(t),\frac{t}{\epsilon
},\omega
\Bigr),\quad \epsilon >0,\ X^{\epsilon }(0)=x_{0},
\]
where $\{F(x,{s}),s\geq 0\}$ is a random process for each $x\in \Re
^{d}$, have attracted a multitude of investigations due to
applications
in such diverse areas as celestial mechanics, oscillation theory,
adaptive filtering, recursive identification, and stochastic adaptive
control.
A natural question that is important to filtering theory and
stochastic
control is whether these convergence results continue to hold for the
parabolic partial differential equations
\[
\partial _{t}u^{\epsilon }(x,t)=\sum_{|k|\leq 2p}A_{k}\Bigl(
x,\frac{t}{\epsilon},\omega\Bigr)\partial _{x}^{k}u^{\epsilon }(x,t),
\quad \epsilon >0,\ u^{\epsilon}(x,0)=\phi (x).
\]
For second order parabolic equations with various technical and
simplifying assumptions, earlier results indicate that laws of large
numbers and fluctuation results continue to hold, provided one resorts
to spaces of generalized functions for the fluctuation results. In
this talk, we will discuss general convergence and rate of convergence
results for $u^{\epsilon}$. In particular, we will only assume that
the coefficients themselves satisfy natural convergence or fluctuation
results and we will prove our fluctuation results on a natural Hilbert
space. Finally, our setting is general enough to allow for long-range
dependence and/or heavy-tail distributions within our work on
fluctuations.
\abs{Reg Kulperger}
{Department of Statistical and Actuarial Sciences, University
of Western Ontario, London, Ontario~~N6A 5B7, Canada}
{Empirical processes and tests of independence}
Empirical processes and empirical distributions loose certain
information. For example the usual empirical distribution is invariant
under a permutation of the data. However some information about the
dependence structure can be retained either by the use of random
weights or blocks. This talk will consider empirical processes and
their use in tests of randomness.
\abs{Brenda MacGibbon}
{D\'epartement de math\'ematiques, Universit\'e du
Qu\'ebec \`a Montr\'eal, Montr\'eal, Qu\'ebec~~H3C 3P8, Canada}
{On statistical minimax estimation and principal
eigenfunctions
of the Laplacian}
In many parametric statistical estimation problems, there is definite
prior information concerning the values of the parameter vector. There
may be bounds on the individual components or on a particular
functional of the whole vector. Many computationally feasible
estimation methods have been developed to capitalize on such
information, but theoretical results have lagged behind. One approach
is the worst case analysis: given some error measure, compute the
maximum expected error over the restricted parameter space. The study
of the resulting best or minimax risk is related here to the study of
the classical Dirichlet problem and of principal eigenfunctions of the
Laplacian.
\abs{Neal Madras}
{Department of Mathematics and Statistics, York University,
Toronto, Ontario~~M3J 1P3, Canada}
{In search of faster simulations}
Markov Chain Monte Carlo has been one of the most active topics in
applied probability in recent years. The main idea is that one can
generate random samples from a given complicated distribution by first
inventing a Markov chain whose equilibrium distribution is the
distribution of interest, and then simulating the chain on a computer.
Unfortunately, the chain may converge to equilibrium very slowly; for
example, this can be the case when the equilibrium distribution is
strongly multimodal. One class of remedies involves creating a
sequence of overlapping distributions that interpolate between the
distribution of interest and some ``easy'' distribution (whose
associated Markov chain equilibriates rapidly). These methods have
been successful in practice, but the mathematical theory behind them
is
not yet complete.
\abs{Don L. McLeish}
{University of Waterloo, Waterloo, Ontario~~N2L 3G1, Canada}
{Estimating parameters of financial time series using highs
and lows}
Observations on security prices, currency exchange rates interest rates
and other financial time series usually include not only open and
closing values over periods of time but also highs and lows for the
same periods. This information on highs and lows is often disregarded,
in part because the joint distributions are very complicated in fact
most are unknown). However, these observations are highly informative
about the volatily, the correlations among securities, how to hedge one
using another, testing for a change in drift, and the valuation of
certain options lookback, barrier, {\it etc.}) that depend on upper and
lower barriers. We suggest some semi-parametric methods for using this
information efficiently, illustrating the methods using the stock
prices of the major Canadian Banks.
\abs{Majid Mojirsheibani}
{School of Mathematics and Statistics, Carleton University,
Ottawa, Ontario~~K1S 5B6, Canada}
{Combined estimation and probabilistic classification}
There has recently been a growing interest in combining different
classifiers in order to develop more effective classification rules
with higher predictive power. The individual classifiers could be, for
example, tree classifiers, partitioning rules, Fisher's linear
discriminant function, or nearest neighbour classifiers, to name a
few. Combining classifiers may also be viewed as a partial answer to
the question of: Given a few classification rules, which one should
the user choose if his/her main concern is to have a low error rate?
Quite often, in a given situation, one classifier performs better than
another; the reason can be directly related to the nature of the
underlying parent distributions of the classes involved.
In this talk we will propose a new combining procedure which is quite
simple to use in practice. We will also show that under certain
conditions, the proposed combined classifier is asymptotically,
strongly, at least as good as any one of the individual classifiers.
\abs{Bruno Remillard}
{Universit\'e du Qu\'ebec \`a Trois-Rivi\`eres}
{Empirical processes based on pseudo-observations}
Pseudo-observations can be described as estimations or predictions of
non observable random variables. In practice, they are used to estimate
the distribution function of the non observable random variable. For
example, regression residuals are pseudo-observations.
In this talk, I will give some conditions in order that the empirical
process based on pseudo-observations converges in law to a continuous
stochastic process. Many examples will illustrate the results.
\abs{P\'al R\'ev\'esz}
{Mathematical Institute of Hungary, Academy of Science,
Budapest,
Hungary}
{Critical branching Wiener process in the $d$-dimensional
Eucledean space {\rm(3 talks)}}
Consider a particle which executes a critical branching Wiener
process.
Assume that this particle is living till time $T$. Then we investigate
the empirical distribution function defined by the locations of the
particles at time $T$ as well as some functionals of this
distribution. A related question is the behaviour of the probability
that at least one particle is located in a given ball. We also
consider the properties of a critical branching random field. That is
we have a Poisson point process in time $0$ and the points of this
field execute independent branching Wiener processes. A typical
question is to investigate the radius of the smallest ball around the
origin which consists of at least one particle at time $T$.
\abs{Jeffrey Rosenthal}
{Department of Mathematics, University of Toronto, Toronto,
Ontario~~M5S 3G3, Canada}
{The mathematics of Markov chain Monte Carlo algorithms}
Markov chain Monte Carlo ($\MCMC$) algorithms, such as the Gibbs
sampler and the Metropolis-Hastings algorithm, are now widely used in
statistics, computer science, physics, and chemistry to understand
complicated probability distributions. While the implementation of
these algorithms is often routine, many fundamental questions---such
as
convergence rates---are much more difficult. In this talk, we will
review some of these issues and the partial progress that has been
made
towards resolving them.
\abs{Thomas S. Salisbury}
{Department of Mathematics and Statistics, York University,
Toronto, Ontario~~M3J 1P3, Canada}
{The complement of the planar Brownian path}
Take a planar Brownian path, run until it exits the unit ball. The
complement of the path consists of many components, and we can ask
about their general shape. The talk will describe joint work with
Yuval
Peres, in which we show that the components are round, in the sense
that their areas are comparable to the square of their diameters. More
formally, we show that for every $\beta>0$, $\sum A_i^\beta<\infty$ if
and only if $\sum R_i^{2\beta}<\infty$, where $A_i$ is the area of the
$i$-th component, and $R_i$ is its diameter.
\abs{Byron Schmuland}
{University of Alberta, Edmonton, Alberta~~T6G 2G1, Canada}
{Rademacher's theorem on configuration space}
Rademacher's theorem says that a Lipschitz function on $R^d$ is
differentiable almost everywhere. Recently, R\"ockner and Schied have
proved a version of this result for functions on the configuration
space $\Gamma_{R^d}$ of locally finite subsets of $R^d$. In this
talk,
we will explain how this result can be used in the construction and
quasi-sure analysis of a diffusion process living on $\Gamma_{R^d}$.
\abs{Qi-Man Shao}
{Department of Mathematics, University of Oregon,
Eugene, Oregon~~97403, USA}
{Gaussian correlation conjecture and small ball probabilities}
The Gaussian correlation conjecture states that for any two symmetric
convex sets in $n$-dimensional space and for any centered Gaussian
measure on that space, the measure of the intersection is greater than
or equal to the product of the measures. It is known that the
conjecture is true for some special cases, which has been found a very
useful tool in the study of the so called small ball probabilities for
Gaussian processes. This talk will review recent progress on the
conjecture. A new Gaussian correlation inequality as well as its
application to the existence of small ball constant for fractional
Brownian motion will be discussed.
\abs{Zhan Shi}
{Laboratoire de Probabilit\'es, Universit\'e Paris VI,
4, Place Jussieu 75252~~Paris cedex 05, France}
{The maximum of the uniform empirical process}
Some problems---with or without an answer---concerning the maximum of
the uniform empirical process will be discussed. This is joint work
with Endre Cs\'aki.
\abs{Gordon Slade}
{Department of Mathematics and Statistics, McMaster
University,
Hamilton, Ontario~~L8S 4K1, Canada}
{The scaling limit of the incipient infinite cluster in
high-dimensional percolation}
We describe recent joint work with Takashi Hara proving that, for
independent bond percolation in high dimensions, the scaling limits of
the incipient infinite cluster's two-point and three-point functions
are those of integrated super-Brownian excursion ($\ISE$).
\abs{Christopher G.~Small}
{Pure Mathematics Department, University of Waterloo,
Waterloo,
Ontario~~N2L 3G1, Canada}
{The analysis of random shapes}
The shape of an object or image, regarded as a subset of $\bbd R^n$,
can be defined as the total of all properties which are invariant under
similarity transformations of the object in $\bbd R^n$. In practice,
the shape of an object is encoded from the configuration of a finite
set of points called {\it landmarks\/} chosen at important locations on
the object. In this talk, we shall survey two of the main approaches
to shape analysis due to F.~L.~Bookstein and D.~G.~Kendall. Small
(1996) provided an extension of the Bookstein representation by
representing the shapes of $p$-simplexes on manifolds. These manifolds
are quite distinct from those proposed by D.~G.~Kendall based upon
Procrutes distances. A formula for geodesic distance in simplex shape
space permits the implementation of multidimensional scaling methods
for the statistical shape analysis of $2$- and $3$-dimensional
objects.
We apply this representation of simplex shapes to a shape analysis of
Iron Age brooches from modern day Switzerland. The brooches are then
grouped into five classes based upon chronological ordering. It is
shown that these five classes form relatively distinct groups when the
first two principal coordinates of shape variation are displayed. This
supports the assumptions used in archaeological seriation where
artifacts are given a rough chronological ordering based upon stylistic
features.
\abs{Barbara Szyszkowicz}
{School of Mathematics and Statistics, Carleton University,
Ottawa, Ontario~~K1S 5B6, Canada}
{An interplay of weighted approximations and change-point
analysis}
When studying change-point problems, weighted partial sum-type
processes frequently appear as the natural outcome of some
nonparametric as well as parametric considerations. We present how
approximation methods can lead to obtaining convergence results for
such processes and to some ``unexpected'' results for their
sup-functionals. Results and methods of strong and weak
approximations
have become an integral part of the theory and applications of
probability and statistics in the last 40 or so years. Recent
contributions in this area are mainly concerned with weighted
approximations of stochastic processes based on $\iid$ observations
({\it cf.}, {\it e.g.}, M.~Cs\"org\H{o} and L.~Horv\'ath, Weighted
Approximations in Probability and Statistics, Wiley 1993). We
construct tools for use in weighted approximations of additive
processes in various metrics under $\iid$ sampling and combine these
techniques with Le Cam's theory of contiguous measures. An
appropriate
parametrization of contiguity enables us to quantize a possible change
from $\iid$ sampling to small disturbances afterwards in large sets of
chronologically ordered data. Constructed new tools allow us to
obtain results under the most stringent conditions.
\abs{Keith Worsley}
{Department of Mathematics and Statistics, McGill University,
Montreal, Quebec~~H3A 2K6, Canada}
{The geometry of correlation fields, with an application
to functional connectivity of the brain}
Repeated 3D images of brain function, obtained by Positron Emission
Tomography (PET) or functional Magnetic Resonance Imaging (fMRI), are
now being used to detect pairs of points in 3D that show high
functional connectivity. This is defined as the usual sample
correlation coefficient measured at the two points in the images, which
generates a 6D correlation random field. To detect pairs of regions
that are highly correlated, we find tail probabilities of local maxima
of the correlation field, and the size of the largest set of connected
points in 6D where the correlation field is above a fixed high
threshold. The main tool used is the expectation of the Euler
characteristic of the excursion set. Results are applied to an
experiment to determine which brain regions are functionally connected
during an attention task. This is joint work with Jin Cao (Bell Labs,
Lucent Technologies).
\abs{Hao Yu}
{Department of Statistics and Actuarial Sciences, University
of Western Ontario, London, Ontario~~N6A 5B7, Canada}
{Weighted Kolmogorov-Smirnov test of stock return
distribution}
In this talk we demonstrate how Chibisov-O'Reilly theorem be used to
do
goodness-of-fit test for stock return distributions. The problem
related to find the critical values for the test is discussed. The
result shows this test is quite effective to deal with heavy tail
distributions typically presented in stock return data.
\abs{Ri\v cardas Zitikis}
{Carleton University, Ottawa, Ontario~~K1S 5B6, Canada}
{The Vervaat, Lorenz and some other related processes of
probability and mathematical statistics in weighted metrics}
We shall discuss various probabilistic properties of the general
Vervaat process (that appears in: M.~Cs\"org\H o and R.~Zitikis, {\it
Strassen's $\LIL$ for the Lorenz curve}. J.~Multivariate Anal. (1)
{\bf
59}(1996), 1--12) along with its crucial role when analyzing the
Lorenz, Bonferroni, Goldie and other related processes of statistical
relevance. We shall discuss the importance of investigating the
convergence of these processes in weighted supremum metrics, and under
minimal assumptions as well.
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{\bf Topology---4 sub-sessions / Topologie---4 sous-sessions}}
\org{E.~Campbell, Organizer}
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{\it 1)~~Differential Geometry and Global Analysis / G\'eom\'etrie
diff\'erentielle et analyse globale}}
\org{Muang Min-Oo and McKenzie Wang, Organizers}
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\abs{Igor Belegradek}
{Department of Mathematics and Statistics, McMaster University,
Hamilton, Ontario~~L8S 4K1, Canada}
{Pinching and Pontrjagin classes}
We proof some pinching theorems for the class ${\cal M}_{a,b,\pi, n}$
of $n$-manifolds that have fundamental groups isomorphic to $\pi$ and
that can be given complete Riemannian metrics of sectional curvatures
within $[a,b]$ where $a\le b<0$. For example, given a word-hyperbolic
group $\pi$ and an integer $n$ there exist $\epsilon=\epsilon(n,\pi)>0$
such that the tangent bundle of any manifold in the class ${\cal
M}_{-1-\epsilon, -1, \pi, n}$ has has zero rational Pontrjagin
classes.
\abs{Christoph B\"ohm}
{Department of Mathematics and Statistics, McMaster
University,
Hamilton, Ontario~~L8S 4K1, Canada}
{Inhomogeneous Einstein metrics on spheres}
The standard metric on the $n$-dimensional sphere $S^n$ is Einstein,
{\it i.e.} its Ricci tensor is a multiple of itself. The first
non-trivial example of an Einstein metric on $S^n$ was given by Jensen
for $n=4m+3$ in 1973. Five years later Bourguignon and Karcher
described a further Einstein metric on $S^{15}$. Up to now these
metrics were the only known examples of Einstein metrics on spheres.
All of them are homogeneous and Ziller proved that there exists no
further homogeneous Einstein metric on $S^n$. We describe infinitely
many inhomogeneous Einstein metrics with positive scalar curvature on
$S^5$, $S^6$, $S^7$, $S^8$, $S^9$. The resulting sequence of Einstein
metrics converges to an explicit known limit metric which is smooth
outside two submanifolds. Hence we get new examples of manifolds
where
the Palais-Smale condition for the total scalar curvature functional
is not fulfiled.
\abs{Jeffrey Boland}
{Department of Mathematics and Statistics, McMaster
University,
Hamilton, Ontario~~L8S 4K1, Canada}
{Magnetic fields on negatively curved manifolds}
Closed $2$-forms $\Omega$ on a Riemannian manifold $M$ induce magnetic
field flows in the unit tangent bundle $SM$ which model the motion of
a
charged particle under the influence of the ``magnetic field''
$\Omega$. When $M$ is negatively curved and the magnetic field is
weak, these flows are hyperbolic and have many interesting properties.
We discuss in particular their entropy, the regularity of their
hyperbolic splitting, and the question of when they are isomorphic to
(Finsler or Riemannian) geodesic flows. Particular attention will be
given to the case when $M$ is a locally symmetric space.
\abs{Virginie Charette}
{University of Maryland, College Park, Maryland~~20742, USA}
{Properly discontinuous actions of free groups on Minkowski
space}
Margulis showed that free, non-abelian groups can act freely and
properly discontinuously on three-dimensional Minkowski space. These
examples of ``affine Schottky groups'' came as a surprise.
The talk will focus on quotients of Minkowski space by such groups of
isometries that can be factored as products of inversions. Certain
preferred fundamental polyhedra for their action will be introduced and
facts about recurring geodesics will be presented.
\abs{Jingyi Chen}
{Department of Mathematics, Massachusetts Institute of
Technology, Cambridge, Massachusetts~~02139, USA}
{Triholomorphic curves and complex $\ASD$ connections}
We study the adiabatic limit of the complex anti-self-dual connections
over a product of two Calabi-Yau surfaces and the triholomorphic
curves
between kyperk\"ahler manifolds.
\abs{Ailana Fraser}
{Courant Institute, New~York, New~York~~10012, USA}
{On the free boundary variational problem for minimal disks}
We will discuss the problem of extremizing the energy (equivalently
area) for maps from the unit disk $D$ into a Riemannian manifold $N$
having boundary lying on a specified embedded submanifold $M$. The
critical points of this geometric variational problem are minimal
surfaces which meet the submanifold orthogonally along the boundary.
We
derive a partial Morse theory for this problem in arbitrary
dimensions. In addition, we use the geometry to obtain certain lower
bounds on the Morse index for such disks.
\abs{Michael Gage}
{University of Rochester, Rochester, New~York~~14627, USA}
{Remarks on B.~S\"ussmann's proof of the Banchoff-Pohl
inequality}
This is an expository talk describing Bernd Suessmann's use of the
curve shortening flow to prove the Banchoff-Pohl isoperimetric
inequality for non-simple closed curves on simply connected surfaces
with Gauss curvature bounded above by a non-positive constant $K_0$.
The inequality is
$$
L^2 - 4\pi\int_M w(x)^2\, dA(x)+K_0\Bigl(\int_M |w(x)|\,dA(x)\Bigr)^2
\ge 0
$$
were $L$ is the length of the curve $\gamma$ and $w(x)$ is the winding
number of $\gamma$ about the point $x$. The idea of the proof is to
show that the left hand side cannot be increased under the curve
shortening flow. This is sufficient because the curve shortening flow
deforms an arbitrary closed curve to a ``circular'' point (the curve
may temporarily develop cusps during process) and the left hand side
is non-negative for curves near these points.
The inequalities S\"ussmann derives in order to prove that this
quantity decreases under the curve shortening flow are interesting and
probably more powerful than the final result.
\abs{Miroslav Lovri\'c}
{Department of Mathematics and Statistics, McMaster
University,
Hamilton, Ontario~~L8S 4K1, Canada}
{Multivariate normal distributions parametrized as a
Riemannian symmetric space}
The construction of a distance function between probability
distributions is of importance in mathematical statistics and its
applications. Distance function based on the Fisher information metric
has been studied by a number of statisticians, especially in the case
of the multivariate normal distribution (Gaussian) on ${\bf R}^n$. It
turns out that, except in the case $n=1$, where the Fisher metric
describes the hyperbolic plane, it is difficult to obtain an exact
formula for the distance function (although this can be achieved for
special families with fixed mean or fixed covariance). We propose to
study a slightly different metric on the space of multivariate normal
distributions on ${\bf R}^n$. Our metric is based on the fundamental
idea of parametrizing this space as the Riemannian symmetric space
$\SL(n+1)/\SO(n+1)$. Symmetric spaces are well understood in
Riemannian
geometry, allowing us to compute distance functions and other relevant
geometric data.
\abs{Mohan Ramachandran}
{SUNY at Buffalo, Buffalo, New~York~~14214-3093, USA}
{To be announced}
\abs{Patrick Ryan}
{McMaster University, Hamilton, Ontario~~L8S 4K1, Canada}
{Real hypersurfaces in complex space forms}
We consider hypersurfaces immersed in a complex projective space ${\bf
C}P^n$ or complex hyperbolic space ${\bf C}H^n$---the space forms of
nonzero constant holomorphic sectional curvature. We are interested
in
characterizing certain standard examples ({\it e.g.} the homogeneous
ones) in terms of intrinsic and extrinsic geometrical properties.
Recent results and open problems in this area will be discussed.
\abs{Patrice Sawyer}
{Department of Mathematics and Computer Science, Laurentian
University, Sudbury, Ontario~~P3E 2C6, Canada}
{Ghostly symmetric spaces}
Various objects associated with symmetric spaces have specific ranges.
We are thinking in particular of root multiplicities in the root system
(nonnegative integers) and of the Cartan subalgebra (${\bf R}^l$). We
will see that by extending these ranges and thus going beyond the
``physical'' symmetric spaces (hence the title), we can solve problems
concerning ordinary symmetric spaces.
\abs{Alina Stancu}
{Courant Institute of Mathematical Sciences, New~York,
New~York~~10012, USA}
{Asymptotic behavior of a crystalline evolution}
Motion by crystalline curvature is viewed as a typical example of
geometric evolution by a nonsmooth boundary energy. Assume that a
planar curve is endowed with an energy density defined on a finite set
of normal directions. It is natural then to consider the restricted
class of piecewise linear curves with just this ordered set of
normals. These curves do not have a motion by curvature in the
conventional geometric sense, but, following M.~Gurtin and J.~Taylor,
one can still define the so-called crystalline curvature flow which is
analogous to the motion by weighted curvature for smooth planar
curves.
We consider Gurtin's defintion, with no driving term, for closed
convex
curves, so that the inward normal velocity of each segment of an
admissible polygonal curve as above is inversely proportional to the
length of the segment, where the proportionality factor is only
required to be positive. Our results show that, if the curve has more
than four sides, it will shrink to a point while approaching the shape
of a homothetic solution to the flow. This implies the existence of
at
least one self similar solution for any flow associated to an energy
density defined on more than four unitary directions. The number of
homothetic solutions will be discussed based on the properties of the
energy density.
\abs{John Toth}
{McGill University, Montreal, Quebec~~H3A 2K6, Canada}
{To be announced}
%\vskip .75cm
%\hrule
%\vskip .3cm
\session{
{\it 2)~~Homotopy Theory / Th\'eorie de l'homotopie}}
\org{Lisa Langsetmo and Jim Shank, Organizers}
%\hrule
%\vskip .4cm
\abs{Dan Christensen}
{Johns Hopkins University, Baltimore, Maryland~~21218, USA}
{Phantom maps: all or nothing}
The general theme is to determine conditions on spectra $X$ and $Y$
under which either every map from $X$ to $Y$ is phantom, or no
non-zero
maps are. I will also address the question of whether such all or
nothing behaviour is preserved when $X$ is smashed with a finite
spectrum $W$. It turns out that there are close connections with the
divisibility and rationality of the group $[X,Y]$, and with
Brown-Comenetz duality.
If there is time, I will discuss $n$-phantom maps, a chromatic analog
of phantom maps, and use this concept to give another explanation of
the results on phantom maps.
This is joint work with Mark Hovey and Sharon Hollander.
\abs{F. R. Cohen}
{Department of Mathematics, University of Rochester,
Rochester, New~York~~14627, USA}
{On stunted projective spaces}
F.~Sergeraert and V.~Smirnov have recently studied the homology of the
loop spaces for $RP^{\infty}/RP^n = X_n$. It is the purpose of this
note to record the homotopy type of these spaces after looping
(sometimes more than once). The following is joint work with R.~Levi.
\lproclaim{Theorem}{
(1)~~$\Omega(X_1)$ is homotopy equivalent to $S^1\times\Omega S^3$.
\endgraf
(2)~~$\Omega(X_2)$ is the homotopy theoretic fibre of a map
$S^3\rightarrow Z$ where $Z$ denotes the $6$-skeleton of the Lie
group $G_2$. Thus $\Omega^4_0(X_2)$ is homotopy equivalent to
$\Omega^3_0S^3 \times \Omega^4_0(Z)$.
\endgraf
(3)~~$\Omega(X_3)$ is homotopy equivalent to
$S^3 \times \Omega\bigl(S^7 \vee P^6(2)\bigr)$.
\endgraf
(4)~~$\Omega(X_n)$ is the homotopy theoretic fibre of a map from a
finite complex to the Lie group $\Spin(n)$.}
Further information concerning the map $X_n \rightarrow \Sigma{RP^n}$
is given. For example, the theorem above for $X_2$ gives a different
proof of a theorem of Jie Wu concerning a splitting of the homotopy
groups for $\Sigma RP^2 $.
\abs{Gustavo Granja}
{Department of Mathematics, MIT, Cambridge,
Massachusetts~~02139, USA}
{On self maps of $\HP^n$}
In 1975, Feder and Gitler computed the action of the Adams operations
on the $K$-theory of $\HP^n$ and used this to get restrictions on the
possible degrees of self maps. They conjectured that all the integers
satisfying these restrictions are degrees of actual self maps. The
conjecture is known to be true for $n=2,3$ and $\infty$. We show
that
it also holds for $n=4,5$ by using computations by Curtis and Mahowald
of the unstable Adams spectral sequence for $S^3$ and explain why this
gives qualitatively stronger evidence for the conjecture.
\abs{Steve Halperin}
{Department of Mathematics, University of Toronto, Toronto,
Ontario~~M5S 3G3, Canada}
{The homotopy Lie algebra of a finite complex}
Sullivan models provide an effective means of using finiteness
hypotheses on homology to deduce non-trivial properties of the
homotopy
Lie algebra. This talk presents a survey, including some new results
joint with Y.~Felix and J.-C.~Thomas.
\abs{Philip R. Heath}
{Department of Mathematics, Memorial University of Newfoundland,
St. John's, Newfoundland~~A1C 5S7, Canada}
{Fibre techniques in Nielsen periodic point theory}
Fibre techniques have been used in ordinary Nielsen fixed point theory
since the late sixties. In this talk I indicate how recent developments
allow fibre techniques to be use in Nielsen periodic point theory.
Applications to nilmanifolds and solvmanifolds (including the Klein
Bottle) are forthcoming.
\abs{Barry Jessup}
{Department of Mathematics and Statistics, University of
Ottawa, Ottawa, Ontario~~K1N 6N5, Canada}
{Estimating the rational $\LS$-category of elliptic spaces}
An {\it elliptic\/} space is one whose rational homotopy and rational
cohomology are both finite dimensional. Using the equality of
rational
$\LS$ category and Toomer's invariant for elliptic spaces, established
by F\'elix, Halperin and Lemaire, we provide new lower bounds for the
rational category of elliptic spaces. We do this by proving two
special cases of an even dimensional analogue of a theorem which
improved the estimate of the Mapping theorem in some settings. Several
interesting examples are presented to illustrate the results, which
are
joint work with Sonia Ghorbal.
\abs{Brenda Johnson}
{Department of Mathematics, Union College, Schenectady,
New~York~~12308, USA}
{Constructing and characterizing degree $n$ functors}
Let $F$ be a functor from a basepointed category with finite
coproducts
to a category of chain complexes over an abelian category. Such a
functor is homologically degree $n$ if its $n+1$-st cross effect (in
the
sense of Eilenberg and Mac~Lane) has trivial homology. We describe a
method for constructing, by means of cotriples associated to the cross
effects of $F$, a universal tower under $F$,
$$
\cdots \rightarrow P_{n+1}F\rightarrow P_nF\rightarrow
P_{n-1}F\rightarrow
\cdots \rightarrow P_1F\rightarrow P_0F=F(\ast),
$$
in which each functor $P_nF$ is homologically degree $n$. This
construction arose from the study of Goodwillie's Taylor tower in the
case of functors of modules over a ring. Using this model, we will
characterize homologically degree $n$ functors in terms of modules
over
a certain $\DGA$, and discuss some related constructions and examples
due to Eilenberg-Mac~Lane, and Dold-Puppe. This is joint work with
Randy McCarthy.
\abs{Keith Johnson}
{Department of Mathematics, Statistics, and Computing Science,
Dalhousie University, Halifax, Nova Scotia~~B3H 3J5, Canada}
{Elliptic homology cooperations}
In this talk I will survey three approaches to describing the hopf
algebroid $\Ell_\ast \Ell$ of stable cooperations for elliptic
homology
and some of the information about the $E_2$ term of the Adams spectral
sequence based on $\Ell$ which results.
\abs{Sadok Kallel}
{Department of Mathematics, University of British Columbia,
Vancouver, British Columbia~~V6T 1Z2, Canada}
{The homology structure of free loop spaces}
We present recent results on the structure of fibrations with section
(joint with D.~Sjerve), and give some applications to the theory of
free loop spaces. We show how higher differentials on spherical classes
in the Serre spectral sequence for the evaluation fibration associated
to $\Map(S^k,M)$ are given in terms of certain homology operations in
the fiber. As an illustration of our methods, we discuss the example of
free loops on configuration spaces (this is a space of fundamental
interest in non-linear analysis).
\abs{Kathryn Lesh}
{University of Toledo, Toledo, Ohio~~43606, USA}
{Progress toward a partial splitting of $E_{2}$
in the UASS for $\SO$}
In the 1970s, Bousfield conjectured a splitting for the $E_{2}$ term
of
the unstable Adams spectral sequence (UASS) for $\SO$. We discuss
recent progress towards a proof of the first stages of Bousfield's
conjecture.
\abs{L. Gaunce Lewis, Jr.}
{Mathematics Department, Syracuse University, Syracuse,
New~York~~13244-1150, USA, current address: Mathematics Department,
MIT, Cambridge, Massachusetts~~02139, USA (on leave for 1998--99)}
{Recent results on Mackey functors for a compact Lie group}
Mackey functors were first introduced as a tool for proving induction
theorems in representation theory. They have, however, become an
important tool in equivariant homotopy theory because any reasonable
equivariant cohomology theory is implicitly Mackey functor valued.
For
applications in equivariant homotopy theory, it would be very nice to
have a well-behaved extension of the notion of a Mackey functor from
its original context of finite groups to the context of compact Lie
groups. Unfortunately, various technical difficulties have so far
severely limited the utility of the available extensions. However,
the
new approach to Mackey functors for a finite group taken recently by
Florian Luca turns out to extend nicely to compact Lie groups. The
effectiveness of this extension of his methods to compact Lie groups
will be illustrated in this talk by presenting its implications for
the
structure of the spectrum of a commutative Mackey functor ring.
\abs{John Martino}
{Western Michigan University, Kalamazoo, Michigan~~49008, USA}
{A Minami-Webb formula for compact Lie groups}
The Minami-Webb formula expresses the classifying space of a finite
group $G$ as the linear combination,
$$
BG = \sum_H \alpha_H H,
$$
of $\mod$-$p$ cyclic groups ({\it i.e.}, extensions of $p$-groups by
cyclic $p'$-groups). The formula and its variants have proven very
useful in calculating group cohomology. We generalize the Minami-Webb
formula to compact Lie groups.
\abs{Joseph Neisendorfer}
{University of Rochester}
{James-Hopf invariants, Anick's spaces, and decompositions
of the double loops on a Moore space}
Using spaces introduced by Anick, we construct a decompostion into
indecomposable factors of the double loop spaces of odd primary Moore
spaces when the powers of the primes are greater than the first power.
If $n$ is greater than $1$, this implies that the odd primary part of
all the homotopy groups of the $2n+1$ dimensional sphere lifts to a
$\mod p^r$ Moore space.
\abs{Stewart Priddy}
{Department of Mathematics, Northwestern University, Evanston,
Illinois~~60208, USA}
{Decomposing products of classifying spaces}
Let $G = H \times K$ be the product of two finite groups. The problem
of determining the stable type of the classifying space $BG=BH \times
BK$, localized at a prime $p$, is often quite difficult due to the
complicated nature of the subgroups of $G$ which generally do not
relate well to those of $H$ and $K$. Of course, the usual stable
decomposition of a product of spaces $X \times Y \cong X \vee Y \vee (X
\wedge Y)$ gives a decomposition of $BG$ but these summands may be
decomposable, even in elementary cases such as $B({\bf Z}/2\times {\bf
Z}/2)$. In some situations however, the stable type of $BG$ is closely
related to that of $BH$ and $BK$. We say $BG$ is {\it smash
decomposable\/} if the indecomposable summands of $BG_{+}$ are given by
simply smashing together those of $BH_{+}$ and $BK_{+}$, that is, a
complete splitting for $BG_{+}$ localized at $p$ has the form
$$
BG_{+}= BH_{+}\wedge BK_{+} \cong \bigvee_{i,j} X_i \wedge Y_i
$$
where $BH_{+}\cong \bigvee_i X_i$ , $BK_{+} \cong \bigvee_j Y_j$ are
complete splittings into indecomposable summands. Here $BG_{+}$ is
$BG$ with a disjoint basepoint. This is convenient for converting from
Cartesian products to smash products since $(X \times Y)_{+}=
X_{+}\wedge Y_{+}$. Further $BG_{+} = BG \vee S^0$. Thus $BG$ is smash
decomposable when the $X_i\wedge Y_j$ are indecomposable. In this talk
we study groups for which $BG$ is smash decomposable.
(Joint work with John Martino and Jason Douma)
\abs{Charles Rezk}
{Northwestern University, Evanston, Illinois~~60208, USA}
{A model for the homotopy theory of homotopy theory}
We describe a category, the objects of which may be viewed as models
for homotopy theories. We show that for such models, ``functors
between two homotopy theories form a homotopy theory'', or more
precisely that the category of such models has a well-behaved internal
hom-object. This set-up has applications to problems of realizing
diagrams up to homotopy by ``honest'' diagrams.
\abs{Laura Scull}
{Department of Mathematics, University of Chicago,
Chicago, Illinois~~60637, USA}
{Rational $S^1$-equivariant homotopy theory}
I will discuss an algebraicization of rational $S^1$-equivariant
homotopy theory. There is an algebraic category of ``$T$-systems''
which is equivalent to the homotopy category of rational $S^1$-simply
connected $S^1$-spaces. There is also a theory of ``minimal models''
for $T$-systems, analogous to Sullivan's minimal algebras. Each
$S^1$-space has an associated minimal $T$-system which encodes all of
its rational homotopy information, including its rational equivariant
cohomology and Postnikov decomposition.
\abs{Paul Selick}
{Department of Mathematics, University of Toronto, Toronto,
Ontario~~M5S 3G3, Canada}
{Natural decompositions of loop suspensions and tensor
algebras}
(Joint work with Jie Wu).
Consider the full subcategory of pointed topological spaces whose
objects are simply connected suspensions of finite type. For $X$ in
the above category we examine natural decompositions of $\Omega\Sigma
X$ localized at a prime $p$ as a product (up to homotopy) of other
spaces. Since as a Hopf algebra, $H_\ast(\Omega\Sigma X;\bbd Z/p\bbd
Z)$ is isomorphic to the tensor algebra $T(V)$, where $V={\tilde
H}_\ast(X;\bbd Z/p\bbd Z)$, any such decomposition yields a natural
coalgebra decomposition of $T(V)$ (which need not be a Hopf algebra
decomposition since we have not required our decomposition to respect
the $H$-space structure on $\Omega\Sigma X$). We have shown that the
converse is true: every natural decomposition of $T(V)$ can be
geometrically realized as a natural decomposition of the space
$\Omega\Sigma X$. Having thus translated the problem to algebra, we
next consider the algebraic problem of finding natural coalgebra
decompositions of tensor algebras. We show that there is a natural
coalgebra decomposition of $T(V)$ (natural with respect to the vector
space $V$) $T(V) = A^{\min}(V)\otimes B_2(V)\otimes B_3(V)\cdots$
where $A^{\min}(V)$ contains $V$ itself and is minimal in the sense
that it is (up to isomorphism) a retract of any coalgebra containing
$V$ which is a natural retract of $T(V)$. The coalgebra $B_n(V)$ is
the smallest natural coalgebra retract containing a certain submodule
$L^{\max}_n(V)$ described below. This decomposition generalizes that
given by the Poincar\'e-Birkhoff-Witt Theorem, except it is natural
with respect to maps of vector spaces, whereas $\PBK$ is natural only
with respect to maps of ordered vector spaces.
Some properties of $A^{\min}(V)$ and of the product $B(V)=B_2(V)\otimes
B_3(V)\cdots$ of all the other factors are as follows. $B(V)$ is a
sub-Hopf-algebra of $T(V)$ which is a retract as a coalgebra. We show
that, as conjectured by Cohen, the only primitives in $A(V)$ occur in
weights of the form $p^t$. Also, $A(V)$ has a filtration where each of
the filtration quotients is a polynomial algebra. A description of the
generators for these polynomial algebras is given for the first $p^2-1$
filtration quotients, computation of the others remaining beyond our
present capabilities.
One important aspect of this work is its relationship to the $\mod p$
representation theory of the symmetric group $\Sigma_n$. It provides
some information about the important $\Sigma_n$-module $\Lie(n)$
described below which has arisen in many contexts and appears in
current work of Cohen, Dwyer, Arone, and others. To define $\Lie(n)$
consider the vector space $V$ with basis $\{v_1,\ldots, v_n\}$. There
is an action of $\Sigma_n$ on $(V)$ (and thus on $T(V)$) given by
$v_j\mapsto v_{\sigma(j)}$ for $\sigma\in\Sigma_n$. Let $\gamma(V)$
be
the subspace of $V^{\otimes n}$ spanned by $\{v_{\sigma(1)}\cdots
v_{\sigma(n)} \}$. Let $L_n(V)$ be the primitives of ``weight'' $n$
in
$T(V)$ which are indecomposable ({\it i.e.} not $p$-th powers).
Explicitly $L_n(V)$ consists of commutators of length $n$ in the
elements of $V$. Let $L^{\max}_n(V)=L_n(V)\cap B$. Let
$\Lie(n)=L_n(V)\cap\gamma$ and let $\Lie^{\max}(n)=
L^{\max}_n(V)\cap\gamma$. We show that $\Lie^{\max}(n)$ is a
projective $\Sigma_n$-submodule of $\Lie(n)$ and that any projective
$\Sigma_n$-submodule of $\Lie(n)$ is a retract (up to isomorphism) of
$\Lie^{\max}(n)$. If $n$ is invertible modulo $p$ then it is well
known that $\Lie(n)$ is itself projective and easy to see that
$L^{\max}_n(V)=L_n(V)$. In particularly, in characteristic $0$,
$L^{\max}_n(V)=L_n(V)$ for all $n$.
\abs{Stephen D. Theriault}
{Department of Mathematics, MIT, Cambridge,
Massachusetts~~02139, USA}
{Homotopy exponents for certain $\mod$-$2^{r}$ Moore spaces}
For a prime $p$, the $\mod$-$p^{r}$ Moore space $P^{m}(p^{r})$ is the
cofiber of the degree $p^{r}$ map on $S^{m}$. A special case of
Barratt's conjecture is that if $p^{r}\neq 2$ then the $p$-primary
torsion in $\pi_{\ast}\bigl(P^{m}(p^{r})\bigr)$ is annihilated by
$p^{r+1}$. It is well known that this is true when $p$ is an odd
prime. We investigate to what extent it is true when $p=2$ and $r\geq
2$.
%\vskip .75cm
%\hrule
%\vskip .3cm
\session{
{\it 3)~~Set Theoretic Topology / Topologie des ensembles}}
\org{Juris Steprans and Steve Watson, Organizers}
%\hrule
%\vskip .4cm
\abs{Murray Bell}
{Department of Mathematics, University of Manitoba, Winnipeg,
Manitoba~~R3T 2N2, Canada}
{Cardinal functions of centered spaces}
For a collection of sets $S$, give $\cen(S) = \{T \subset S : T$ is
centered$\}$ the compact Hausdorff topology that it inherits as a
subspace of $2S$. The spaces $\cen(S)$ are exactly the Adequate
Compact spaces of Talagrand. They have served topologists well as a
rich source of examples. This talk is concerned with spaces which are
by definition those Hausdorff spaces that are continuous images of some
$\cen(S)$. This is a good generalization of the dyadic spaces. Our
focus will be on the relationships between the 9 popular cardinal
functions of $w$, $\pi$, $t$, $\chi$, $d$, $c$, $\pi\chi$, $d\chi$ and
$d\pi\chi$ for this class of spaces.
\abs{Maxim R. Burke}
{University of Prince Edward Island}
{Continuous functions which take a somewhere dense set of
values on every open set}
We study the class of Tychonoff spaces that can be mapped continuously
into the real line in such a way that the preimage of every nowhere
dense set is nowhere dense. We show that every metric space without
isolated points is in this class. We also give examples of spaces which
have nowhere constant continuous maps into the real line and are not in
this class.
\abs{Krzysztof Ciesielski}
{West Virginia University}
{Each Polish space cocompactly quasimetrizable}
On April of 1998 Ralph Kopperman and Bob Flagg asked me whether every
for Polish space there exists a countable collection $C$ of closed
subsets of $X$ such that:
(1)~~each subset of $C$ with the finite intersection property has
nonempty intersection,
(2)~~for every open set $T$ and $x$ from $T$ there exists a $C$ in $C$
such that $C$ is a subset of $T$ and $x$ belongs to the interior of
$C$, and
(3)~~for every $C$ from $C$ and $x$ from the complement of $C$ there
exists a finite subcollection $G$ of $C$ such that $C$ is contained in
the interior of the union $U$ of $G$ and $x$ is still in the
complement
of $U$.
I was able to answer this question positively. In fact, the
constructed family satisfies condition (3) with the singleton families
$G$. This fact implies, in particular, that the following properties
are equivalent for every topological space $X$.
(A)~~$X$ is a Polish space.
(B)~~$X$ is cocompactly quasimetrizable, that is, $X$ is a Lindelöf
space arising from a quasimetric whose dual yields a compact (not
necessarily $T_2$) topology.
(C)~~$X$ has a bounded complete approximation (computational) model,
that is (loosely), the points can be encoded and approximated by sets
containing them in a computer program.
\abs{W. W. Comfort}
{Wesleyan University}
{Continuous cross sections on abelian groups equipped with the
Bohr topology}
All groups here are Abelian. A closed subgroup $H$ of a topological
group $G$ is a {\it $\ccs$-subgroup\/} if there is a continuous cross
section from $G/H$ to $G$---that is, a continuous function $\Gamma$
such that $\pi\circ\Gamma=\id|_{G/H}$ (with $\pi\colon G\rightarrow
G/H$ the natural homomorphism).
The symbol $G^{\#}$ denotes $G$ with its Bohr topology, {\it i.e.},
the topology induced by $\Hom (G,{\Bbb T})$.
A topological group $H$ is {\it an absolute $\ccs$-group\/}
$({\#})$ [resp., {\it an absolute retract\/} $({\#})$] if $H$
is a $\ccs$-subgroup [resp., is a retract] in every group of the form
$G^{\#}$ containing $H$ as a (necessarily closed) subgroup. One then
writes $H\in\ACCS ({\#})$ [resp., $H\in\AR({\#})$].
\lproclaim{Theorem 1}{ Every $\ccs$-subgroup $H$ of a group of the form
$G^{\#}$ is a retract of $G^{\#}$ (and $G^{\#}$ is homeomorphic to
$(G/H)^{\#}\times H^{\#}$); hence $\ACCS ({\#}) \subseteq \AR
({\#})$.}
\lproclaim{Theorem 2}{ $H^{\#}\in\ACCS ({\#})$ [resp.,
$H^{\#}\in\AR({\#})$] if{}f $H^{\#}$ is a $\ccs$-subgroup of its
divisible hull $\bigl(\rmdiv(H)\bigr)^{\#}$ [resp., $H^{\#}$ is a
retract of $\bigl(\rmdiv(H)\bigr)^{\#}$].}
\lproclaim{Theorem 3}{ (a)~~Every cyclic group is in $\ACCS({\#})$.
\endgraf
(b)~~The classes $\ACCS ({\#})$ and $\AR({\#})$ are closed
under finite products.}
\lproclaim{Theorem 4}{ Not every Abelian group is in $\ACCS
({\#})$.}
Question [van~Douwen, 1990]. Is every Abelian group in
$\AR({\#})$?
$^{*}$~Presented in Kingston by this co-author.
\abs{Ilijas Farah}
{York University, Toronto, Ontario~~M3J 1P3, Canada}
{$\Exp(N^\ast)$ need not be an $N^\ast$-image}
We prove, using an idea due to M.~Bell, that the \v Cech-Stone
remarinder of the integers does not map onto its exponential space.
This result uses the Open Coloring Axiom, $\OCA$, and it is another
instance of the effect of $\OCA$ to structures closely related to real
numbers which is opposite to the effect of Continuum Hypothesis (see
[1]).
{\centering
{\bf References}
\par}
{\bf 1.}~~I.~Farah, {\it Completely additive liftings}. Bull. Symb.
Logic
{\bf 4}(1998), 37--54.
\abs{Douglas L. Grant}
{University of College of Cape Breton}
{Alternative universes: the role of set theory in topological
algebra}
This survey will explore some of the theorems of topological algebra
in
which significantly more powerful results are available than in the
analagous case for Hausdorff spaces. In addition, some cases will be
considered where the construction of counterexamples has required more
powerful set-theoretic assumptions than is the case in general
topology. In most of those instances, solutions without set-theoretic
axioms beyond $\ZFC$ remain open problems.
\abs{Gary Gruenhage}
{Auburn University, Auburn, Alabama~~36849-5310, USA}
{More on $a$-Toronto spaces}
A Toronto space is an uncountable non-discrete Hausdorff space which
is
homeomorphic to every one of it's uncountable subspaces. It is known
that a Toronto space, if one exixts, is scattered of Cantor-Bendixon
$\rank w_1$ with each level countable. J.~Steprans defines an
$a$-Toronto space, where a is an ordinal, to be a scattered space of
rank a which is homeomorphic to each subspace of the same rank. We
will
discuss results related to our recent proof that, consistently, there
are countable $a$-Toronto spaces for any $a < w_1$; for example, we
show that the proof can be modified to obtain, for any cardinal $k$, a
$k$-Toronto space in which each level has cardinality $k$.
\abs{Melvin Henriksen}
{Harvey Mudd College, Claremont, California~~91711, USA}
{Embedding a ring of continuous functions in a regular ring:
preliminary report}
$C(X)$ denotes the ring of continuous real-valued functions on a
Tychonoff space $X$. If $f \in C(X)$, let $f'(x) = \frac1f(x)$ when
$f(x) = 0$, and let $f(x) = 0$ otherwise. Let $G(X)$ denote the
subalgebra of $\bbd R_X$ generated by $C(X)$ and $\{f':f \in C(X)\}$.
Then $G(X)$ is the smallest von Neumann regular sublagebra of $\bbd
R_X$ containing $C(X)$ and is closed under inversion, {\it i.e.}, any
element of $G(X)$ that never vanishes has an inverse in $G(X)$. Let
$Ba_1(X)$ denote that family of pointwise limits and of sequences of
elements of $C(X)$, $G^u(X)$ the family of uniform limits of sequences
of elements of $G(X)$, and $X\delta$ the topological space obtained by
taking the zerosets of $C(X)$ as a base for a topology on $X$. Then
$C(X) \subset G(X) \subset G^u(X) \subset Ba_1(X) \subset C(X\delta)$.
Each of these inclusions can be proper, each of these families are
vector lattices under the usual pointwise operations, and each of them
are algebras with the possible exception of $G^u(X)$. Each $f \in G(X)$
is continuous on an open dense subspace of $X$, and if $X$ is a Baire
space then each $f \in G^u(X)$ is continuous on a dense $G\delta$ of
$X$. The bounded elements of $G^u(X)$ are closed under multiplication,
each $f \geq 1$ in $G^u(X)$ is invertible, and $G^u(X)$ is an algebra if{}f
$f^2 \vee 1 \in G^u(X)$ whenever $f \in G^u(X)$. It is not known whether
$G^u(Q)$ (where $Q$ is the space of rational numbers) is an algebra, but
this latter is not closed under inversion. The relationship between
some of these vector lattices and the complete ring of quotients and
the epimorphic hull of $C(X)$ is studied by making use of results on
the latter due to R.~Raphael and R.~G.~Woods that are not as yet
published. Indeed, the present research is joint work with these two
authors.
\abs{James Hirschorn}
{University of Toronto, Toronto, Ontario~~M5S 3G3, Canada}
{Towers of measurable functions}
We give mathematical reformulations of the cardinals $\frak p$ and
$\frak t$ in terms of families of measurable functions, proving that
both of these cardinals are invariant under the addition of one random
real. We also describe the cardinal $\frak f$ after adding one random
real.
\abs{Valery Miskin}
{Kemerovo State University}
{Set ideals everywhere}
Although set ideals are objects of algebraic nature they penetrate
like
comets many branches of mathematics from Set Theory and Combinatorics
to Topology, Analysis, Measure, Ergodic and Number Theory. They
appears
naturally in these settings and adorn the tree of Mathematics just as
the lights of Christmas lighting strings connecting old and new
mathematical concepts and problems. As to the ideals arising in Set
Theory and Infinite Combinatorics I concentrate on the classical
Commutative Algebra concept of quotient of two set ideals (which is
not
yet a well known instrument in Set Theory and Topology) and related
cardinal invariants of set ideals. This concept proved to be crucial
in
Boolean ring contexts while solving the problem of isomorphism of
symmetry groups of set ideals and the problem of description of set
ideals with complete or maximal symmetry groups (problem of
H.~Macpherson and P.~Neumann) and in general while studding the
automorphisms and normal subgroup lattice of their symmetry groups. I
discuss the solutions to these problems and related open problems. As
to the properties of set ideals of topological origin I discuss the
concepts related to Stone-ech compactification of $w$ and state a
generalization of a Banach-Kuratowski theorem in terms of
imprimitivity
domains and provide some nice properties of the automorphism group of
an absolutely homogeneous countably separated Borel space. I mention
some resently posed open problems particularly related to meager and
null sets on the real line.
\abs{Justin Tatch Moore}
{University of Toronto, Toronto, Ontario~~M5S 3G3, Canada}
{A linearly fibered Souslinean space Under Martin's axiom}
We will sketch the construction, under $MA + \neg CH$, of a c.c.c.
nonseparable compact space which maps continuously into the Real line
with linear fibers. Such a space can not, for instance, map onto
$[0,1]^{\omega_1}$.
\abs{Alexander Shibakov}
{Tenessee Technological University}
{Controlling sequential order in topological vector spaces}
We discuss methods of constructing topological vector spaces with
prescribed convergence properties. A possible strategy for making the
examples locally convex is also considered. At the end, some open
questions will be mentioned.
\abs{Slawomir Solecki}
{Indiana University}
{Polish group actions and measures}
The talk will be about a connection between continuous actions of
Polish groups and the structure of the equivalence relation of mutual
absolute continuity among Borel probability measures defined on a
Polish space. We will show that this equivalence relation is induced
by a special kind of a continuous action of a Polish group.
Background
and consequences of this result will be presented. For example, a
theorem of Kechris and Sofronidis (that it is not possible to
classify,
by ``simple objects'' and in Baire measurable fashion, Borel
probability measures up to mutual absolute continuity) will be
deduced.
\abs{Paul J. Szeptycki}
{Ohio University}
{Normality and property (a)}
A space $X$ is said to have property (a) if for each open cover $U$ and
each dense $D \subseteq X$, there is a closed discrete $E \subseteq D$
such that $st(E,U) = \bigcup \{u \in U:u \cap E \neg \emptyset\} = X$.
This property was recently introduced by M.~Matveev and is of
particular interest when $X$ is countably compact. Although not a
priori obvious, property (a) is closely related to normality. I will
discuss recent theorems, examples and open questions.
\abs{Andrzej Szymanski}
{Slippery Rock University of Pennsylvania}
{On a class of special Namioka spaces}
If $f\colon X\times Y \rightarrow M$ is a separately continuous
function on the product of a compact space $Y$ and a strongly countably
complete space $X$ into a metric space $M$, then there exists a dense
$G\delta \subset A$ of $X$ such that $f$ is jointly continuous at each
point of $A\times Y$.
We introduce a game-theoretic description of a class of topological
spaces that is substantially wider than, for example, the class of
strongly countably complete spaces or the class of metric Baire
spaces, yet the conclusion of the Namioka theorem holds for spaces
from this class. The class itself is also closed under perfect
preimages. We apply our results to the theory of semitopological
groups.
\abs{Frank Tall}
{University of Toronto, Toronto, Ontario~~M5S 3G3, Canada}
{The topology of elementary submodels}
Results and problems concerning the topology of elementary submodels
will be discussed.
\abs{Murat Tuncali}
{Nipissing University}
{On generalizations of the Hahn-Mazurkiewicz theorem}
The Hahn-Mazurkiewicz theorem characterizes the Hausdorff continuous
images of $[0,1]$ as the class of locally connected metric continua
(Peano continua). A theorem of Alexandroff gives a characterization of
the Hausdorff continuous images of the Cantor ternary set as the class
of compact metric spaces. Following these theorems, it was natural to
ask whether one could obtain generalizations of these results in the
category of Hausdorff spaces.
Nikiel (1988) obtained a characterization of locally connected
continuous images of compact ordered spaces. Bula and Turzanski (1986)
gave a characterization of conitnuous images of compact ordered
spaces. Since these characterizations were obtained, there has been a
considerable amount of development in the study of continuous images of
ordered continua (arcs) and compact ordered spaces. In this talk, we
will give a survey of the results concerning the classes of spaces that
are continuous images of ordered continua and compact ordered spaces,
and related classes.
\abs{E. D. Tymchatyn}
{University of Saskatchewan, Saskatoon, Saskatchewan~~S7N 5E6,
Canada}
{Measures and topological dynamics on Menger manifolds}
Joint research with H.~Kato, K.~Kawamura and M.~Tuncali.
We study non-atomic, locally positive, Lebesgue-Stieltjes measures on
compact, connected, Menger manifolds. We show that each such manifold
$X$ admits an essentially unique, normalized, non-atomic, locally
positive, Lebesgue-Stieltjes measure. The set of ergodic
homeomorphisms
on $X$ forms a dense $G_d$ in the space of all measure preserving
autohomeomorphisms of $X$ in the compact open topology. In
particular,
there exists a topologically transitive homeomorphism on $X$. We also
prove the existence of chaotic homeomorphisms on $X$.
\abs{Grant Woods}
{Department of Mathematics, University of Manitoba, Winnipeg,
Manitoba~~R3T 2N2, Canada}
{Recent developments in rings of continuous functions}
In 1960 Gillman and Jerison's text ``Rings of Continuous Functions'',
one of the most influential books on general topology ever written, was
published. Thirty-eight years later researchers are still investigating
the structure of the ring $C(X)$ of continuous real-valued functions
defined on a Tychonoff topological space.
In this talk I review recent work by myself and others on three
different topics concerning $C(X)$:
(1)~~The weak topology on the set $X \times Y$ induced by the family of
separately continuous functions on the product $X \times Y$ of
Tychonoff spaces $X$ and $Y$.
(2)~~Regular rings of quotients of $C(X)$, particularly its epimorphic
hull.
(3)~~The relationship among the cardinal numbers $|C(X)|$, $|C(Y)|$,
and $|C(X \times Y)|$.
\abs{Eduardo Santillan}
{York University, Toronto, Ontario~~M3J 1P3, Canada and
Cinvestav, Mexico DF}
{Topological properties of removable singularities for
analytic
functions}
A closed set $X$ of a complex manifold $M$ is said to be a {\it
removable singularity\/} if every analytic function defined on $M$-$X$
has got an analytic extension to $M$. In his doctoral tesis, Shiffman
proved a partial characterisation of the removable singularities by
using the Hausdorff measure. Moreover, in 1994, professors Chirca,
Stout and Lupacciolu showed more results which are completely
independet of those of Shiffman; they used the cover dimension. The
actual challenge is to improve that results to characterise the
removable singularities by using just topological properties.
%\vskip .75cm
%\hrule
%\vskip .3cm
\session{
{\it 4)~~Symplectic/Low Dimensional Topology / Topologie en basses
dimensions/Topologie symplectique}}
\org{Steve Boyer, Jacques Hurtubise and Francis Lalonde, Organizers}
%\hrule
%\vskip .4cm
\abs{Steven Boyer}
{D\'epartement de math\'ematiques, Universit\'e du Qu\'ebec
\`a Montr\'eal, Montr\'eal, Qu\'ebec~~H3C 3P8, Canada}
{Norm duality and hyperbolic $3$-manifolds}
The $\SL_2(C)$-character variety of a $1$-cusped hyperbolic
$3$-manifold has given rise to two convex, balanced plane
polygons---the unit ball of the Culler-Shalen norm and the Newton
polygon of the (two-variable) $A$-polynomial of the manifold. In
joint
work with Xingru Zhang, we show that these polygons are dual in the
sense that the line segments joining antipodal vertices of one are
parallel to the the sides of the other. Indeed, we show that the
Newton polygon can be thought of as a ball of the norm dual to the
Culler-Shalen norm. This duality was one of the key ideas in our
recent proof of the finite surgery conjecture.
\abs{Jacques Hurtubise}
{McGill University, Montreal, Quebec~~H3A 2K6,
Canada}
{To be announced}
\abs{Francois Lalonde}
{University of Quebec at Montreal, Montreal, Quebec~~H3C 3P8,
Canada}
{Applications of Quantum cohomology to some fundamental
problems
of dynamics}
I will first present briefly the origins and the construction of
Quantum cohomology, and I will then indicate how the fundamental group
of Hamiltonian diffeomorphisms acts on that cohomology. This leads to
new invariants attached to groups of diffeomorphisms, and has
immediate
applications to the topology of Hamiltonian dynamical systems.
\abs{Yuriy Shkolnikov}
{Department of Mathematics and Statistics, University of
Calgary, Calgary, Alberta~~T2N 1N4, Canada}
{A generalisation of Whitney's trick in dimension $4$,
borromeanism and related questions}
It is well known that the classical Whitney's trick for
$2$-submanifolds $X$ and $Y$ of a $1$-connected $\PL$ (or smooth)
$4$-manifold $M$ does not work. It has not been noticed that the
obstruction to it lies in the topology of Borromean-like links in
$S^3$. This gives a rise to an idea of simultaneous performing of the
Whitney-like trick with $2$ or more Whitney pairs of intersection
points of $X$ and $Y$. Under certain conditions such a trick (called a
Whitney's multitrick) turns out to be successful. The distinctive
feature of a multitrick is that it can be performed only collectively
which means that for each separate pair of Whitney's points the
classical trick does not work.
\abs{Jennifer Slimowitz}
{Universit\'e du Qu\'ebec \`a Montr\'eal, Montr\'eal,
Qu\'ebec~~H3C 3P8, Canada}
{Length minimizing geodesics in the group of Hamiltonian
diffeomorphisms}
To any symplectic manifold $(M, \omega)$, one can associate the group
$\Ham^c(M)$ of compactly supported Hamiltonian diffeomorphisms of $M$.
Hofer has constructed a norm on this group which can be used to define
the notion of a length minimizing path. A new class of examples of
length minimizing paths in $\Ham^c(M)$ for $M$ of dimension $2$ or $4$
will be presented. The proofs rely on a technique described by Lalonde
and McDuff using a new estimate on the Hofer-Zenhder capacity of
certain manifolds.
\abs{Xingru Zhang}
{Mathematics Department, SUNY at Buffalo, Buffalo,
New~York~~14214-3093, USA}
{On simple points of the character variety of a cusped
hyperbolic $3$-manifold}
In recent years, the study of the $\SL(2,C)$-character varieties of
$3$-manifolds has brought great progress in underlying the topology and
geometry of 3-manifolds. Yet many fundamental questions concerning
these varieties remain unanswered. In this talk I will discuss one
aspect of the character variety of a cusped hyperbolic $3$-manifold,
namely to determine which points of the variety are simple in the sense
of algebraic geometry. This is a joint work with Steve Boyer.
%\vskip 1.0cm
%\hrule
%\vskip .3cm
\session{
{\bf Universal Algebra and Multiple-Valued Logic / Alg\`ebre
universelle et logique multi-valu\'ee}}
\org{L.~Haddad, Organizer}
%\hrule
%\vskip .4cm
\abs{Clifford Bergman}
{Department of Mathematics, Iowa State University, Ames,
Iowa~~50011, USA}
{Complexity of some problems in universal algebra}
Given two finite algebras $A$ and $B$, there are several natural
questions one might ask about the relationship between $A$ and $B$.
Among them: Do $A$ and $B$ generate the same (quasi)variety? And, are
$A$ and $B$ term-equivalent? Although these problems are well-known
to
be decidable, the precise computational complexity seems to be
nontrivial.
In this talk, we will survey some old and new results on these
questions, as well as pose some open problems.
This is joint work with Giora Slutzki (Iowa State University).
\abs{Jie Fang}
{Department of Mathematics, Simon Fraser University, Burnaby,
British Columbia~~V5A 1S6, Canada}
{Ockham algebras with pseudocomplementation}
The variety ${\bf pO}$ consists of those algebras $(L, \land, \lor,
f,
\star, 0, 1)$ of type $\langle 2, 2, 1, 1, 0, 0\rangle$, where $(L,
\land, \lor, f, 0, 1)$ is an Ockham algebra, $(L, \land, \lor, \star,
0, 1)$ is a $p$-algebra, and the unary operations $f$ and $\star$
commute. We describe the structure of the subdirectly irreducible
algebras that belong to the subclass ${\bf pK}_{1,1}$ characterised by
the property $f^3=f$, and give a description of the lattice of
subvarieties of ${\bf pK}_{1,1}$.
\abs{Ibrahim Garro}
{Institute for the History and Philosophy of Science and
Technology, University of Toronto, Toronto, Ontario~~M5S 3G3, Canada}
{An application of non-wellfounded sets to infinite valued
infinitary propositional calculus}
The present article is a continuation of my work [1] demonstrating
natural applications of non-wellfounded sets in logic.
A stream is given as a domain of interpretation for infinitary
propositional logic. This stream is the set of ordered pairs
$V=\Bigl(0,\bigl(1,(2,\dots )\bigr)\Bigr)$. $0$ is the empty set and
the second component of every pair is taken as its complement in $V$.
The logical operations are interpreted in $V$ in the manner of post
algebras. Negation is interpreted as complement. `or` is interpreted
as max. and `and` as min. (with special care when one of the terms is
infinite expression {\it i.e.}, a negation, and for infinitely long
expressions.)
We then show that we get as a special case the generalized post
algebra
$Pw$ described in Urquhart [2] p.~91. This opens the road to
applications of the methods used in Post algebras to streams and
vice-versa.
We then study the extension of these results to predicate logic and
look at its implications on the complexity of the decision problem for
propositional logic by comparing the underlying set theories.
{\centering
{\bf References}
\par}
{\bf 1.}~~I.~Garro, {\it Resolving paradoxes of self reference using
the
theory of non-wellfounded sets}. Submitted to the ASL meeting May
22--25,
Toronto, 1988.
{\bf 2.}~~A.~Urquhart, {\it Many valued logic: Handbook of
Philosophical
Logic}. Vol III, Dordrecht, 1994.
\abs{George Gr\"atzer}
{Department of Mathematics, University of Manitoba, Winnipeg,
Manitoba~~R3T 2N2, Canada}
{Independence Theorems for automorphism groups and
congruence lattices of lattices}
In my book, {\it General Lattice Theory}, in 1978, I raised the
following problem:
Let $L$ be a lattice and let $G$ be a group. Does there exist a
lattice $K$ such that $K$ and $L$ have isomorphic congruence lattices
and the automorphism group of $K$ is isomorphic to $G$?
Several papers have been published providing technical tools that
could
be used to attack this problem, in particular,
$\bullet$~~earlier papers by G.~Birkhoff and R.~Frucht;
$\bullet$~~more recent papers by V.~A.~Baranski\u\i,
M.~Tischendorf, and A.~Urquhart;
$\bullet$~~a number of relevant categorical results by the Prague
group
(reported, in part, in a book of A.~Pultr and V.~Trnkov\'a);
$\bullet$~~a number of papers by G.~Gr\"atzer and E.~T.~Schmidt on
congruence-preserving extensions;
$\bullet$~~earlier papers on the topic of tensor products of
lattices with zero by J.~Anderson and N.~Kimura, G.~A.~Fraser, and
G.~Gr\"atzer, H.~Lakser, and R.~W.~Quackenbush and a series of very
recent papers on the same topic (and on some generalizations) by
G.~Gr\"atzer and F.~Wehrung.
Based on these contributions, F.~Wehrung and I have succeeded in
solving this problem.
To state the new results, we need two definitions.
Let $L$ be a lattice. A lattice $K$ is a {\it congruence-preserving
extension\/} of $L$, if $K$ is an extension of $L$ and every
congruence
of $L$ extends to exactly one congruence of $K$. Of course, then the
congruence lattice of $L$ is isomorphic to the congruence lattice of
$K$.
A lattice $K$ is an {\it automorphism-preserving extension\/} of $L$,
if $K$ is an extension of $L$ and every automorphism of $L$ has
exactly
one extension to $K$, and in addition, every automorphism of $K$ is
the
extension of an automorphism of $L$. Of course, then the automorphism
group of $L$ is isomorphic to the automorphism group of $K$.
\smallskip
{\bf The Strong Independence Theorem for Lattices with Zero.}
Let $L_{\rm A}$ and $L_{\rm C}$ be lattices with zero, let $L_{\rm C}$
have more than one element. Then there exists a lattice $K$ that is a
\hbox{$\{0\}$-preserving} extension of both $L_{\rm A}$ and $L_{\rm
C}$, an
automorphism-preserving extension of $L_{\rm A}$, and a
congruence-preserving extension of $L_{\rm C}$.
\smallskip
{\bf The Strong Independence Theorem for Lattices.}
Let $L_{\rm A}$ and $L_{\rm C}$ be lattices, let $L_{\rm C}$ have more
than one element. Then there exists a lattice $K$ that is an
automorphism-preserving extension of $L_{\rm A}$ and a
congruence-preserving extension of $L_{\rm C}$.
\abs{Jennifer Hyndman}
{Mathematics and Computer Science, University of Northern
British Columbia, Prince George, British Columbia~~V2N 4Z9, Canada}
{Dualizable is not the same as fully dualizable}
I will present a bi-unary algebra that is dualizable but not fully
dualizable in the sense of natural duality. In fact the algebra is not
fully dualizable by any alter ego with only a set of relations,
operations or partial operations in its signature. This is joint work
with R.~Willard (University of Waterloo).
\abs{Benoit Larose}
{Champlain Regional College, Quebec, Canada}
{Projective graphs and Hedetnyemi's conjecture}
Let $\bf G$ be a finite simple graph, which for our purposes we define
as a pair $\bf G = \langle G,\theta \rangle$ where $\theta$ is a
symmetric and areflexive binary relation on the finite set $G$. We say
that $\bf G$ is {\it $n$-projective\/} if the only $n$-ary idempotent
operations in the clone $\Pol \theta$ are the projections, and $\bf G$
is {\it $n$-quasiprojective\/} if all the $n$-ary idempotent operations
in the clone $\Pol \theta$ preserve every subset of $G$. We prove that
for all $n \geq 2$, $n$-projectivity, $n$-quasiprojectivity and
$2$-quasiprojectivity are equivalent. (the result also holds for
infinite graphs). This can be used to show that complete graphs, odd
cycles and Kneser graphs are $n$-projective for every $n \geq 2$.
These results are related to Hedetnyemi's conjecture, which states that
the chromatic number of the product of two graphs is the minimum of the
chromatic numbers of the factors; indeed, this conjecture can be
reformulated as follows: every complete graph $K_n$ is {\it
multiplicative}, {\it i.e.} if $K_n$ is a retract of a product $G
\times H$ then it is a retract of $G$ or $H$. We show that under
certain conditions, a projective graph $\bf P$ is {\it weakly
multiplicative}, {\it i.e.} that if $\bf P$ is a retract of a product
of {\it connected\/} graphs then it is a retract of one of the
factors. In particular, complete graphs and Kneser graphs are shown to
be weakly multiplicative. (Note that there exist Kneser graphs which
are not multiplicative (Tardif, Zhou)).
This is joint work with C.~Tardif.
\abs{Jonathan Leech}
{Department of Mathematics, Westmont College, Santa Barbara,
California~~93108-1099, USA}
{Noncommutative lattices: foundational issues and recent
results}
Since the 1940s, a number of individuals have considered algebras
which, while noncommutative, nonetheless resemble lattices. Typically,
such algebras include a pair of binary operations which are assumed to
be associative and to satisfy certain absorption identities, but are
not necessarily commutative. Because of the different types of
absorption possible in a noncommutative context, there has been
considerable variation in the classes of algebras introduced. The talk
will focus on issues such as: What should one expect a noncommutative
lattice to look like? What is the role of lattice theory in framing
such an expectation? Likewise, what is the role of semigroup theory?
These and other issues will be discussed in the light of recent
developments, some as yet unpublished.
\abs{Robert W. Quackenbush}
{Department of Mathematics, University of Manitoba, Winnipeg,
Manitoba~~R3T 2N2, Canada}
{Varieties of binary linear codes}
A binary linear code is a vector space $V$ over $\GF(2)$ with an added
unary operation, $'$, satisfying $0' = 0$; $x'' = x'$, and $(x' + y )'
= x' + y'$. This last law exactly expresses the facts that the set of
codewords $C :=\{x' \mid x \in V\}$ is a subspace and that $'$ on any
coset of $C$ is a translation by a fixed element. I will discuss the
lattice of subvarieties and discuss some connections to classical
linear codes, {\it e.g.}, Hamming codes are closely related to the
$n$-generated free codes in the variety.
\abs{I. G. Rosenberg}
{D\'epartement de math\'ematiques et statistique,
Universit\'e de Montr\'eal, Montr\'eal, Quebec~~H3C 3J7, Canada}
{Completeness for uniformly delayed circuits}
We survey algebraic results on combinatorial circuits constructed from
many-valued uniformly delayed gates. These involve the composition
ofniformly delayed operations, the lattice of uniform clones, the
corresponding relational theory, uniform completeness and the search
for an effective completeness criterion. This is joint work with
T.~Hikita, Dept of Computer Science, Meiji University, Japan.
\abs{Ross Willard}
{Pure Mathematics Department, University of Waterloo,
Waterloo, Ontario~~N2L 3G1, Canada}
{Independence of the linear commutator}
Years ago, Lampe discovered some unexpected interactions between the
congruence lattice of an algebra (as an abstract lattice) and the
term-condition commutator operation on this lattice. We claim that no
result of this kind is possible for the linear commutator. In
particular, no lattice-theoretic property of the congruence lattice of
an algebra can force the algebra to be quasi-affine. This is joint
work with K.~A.~Kearnes (University of Louisville) and W.~A.~Lampe
(University of Hawaii).
%\vskip 1.0cm
%\hrule
%\vskip .3cm
\session{
{\bf Education I/ \'Education I\\
Identifying and Overcoming Barriers to \\
Teaching and Learning Mathematics at University / \\
Identifier et vaincre les obstacles \`a \\
l'enseignement et l'apprentissage des math\'ematiques\\
\`a l'universit\'e}}
\org{Morris and Grace Orzech, Organizers}
%\hrule
%\vskip .4cm
\abs{Edward Barbeau}
{Department of Mathematics, University of Toronto, Toronto,
Ontario~~M5S 3G3, Canada}
{The teacher as coach}
Effective learning by students depends in part on their perception of
the nature of mathematics and how it is to be mastered, as well as on
their understanding of their goals, abilities and learning styles.
Such issues need to be raised explicitly to avoid a conflict of
expectation between student and teacher and to allow students to
marshall their resources most effectively. This talk will discuss how
failure to consider such factors can impede progress.
\abs{Bill Beyers}
{Department of Mathematics and Statistics, Concordia
University,
Montreal, Quebec~~H4B 1R6, Canada}
{Revealing the inner mathematician}
Teachers of university level mathematics often present the subject to
their students in a way which is very different from the way they
themselves understand it. The student is expected to see through the
surface of the presentation (often rigorous, axiomatic) and somehow
``get it''. How you go about ``getting it'' and what it is precisely
that you are supposed to ``get'' is usually unclear to the student.
Part of the problem is the logical structure of mathematics. When
should rigor be introduced? In what context? And to which students?
Bill Thurston wrote that ``what we [should be]$\dots$doing [as
mathematicians and teachers] is finding ways for {\it people\/} to
understand and think about mathematics''. As teachers of mathematics,
we must show the students how to think by revealing honestly how we
think about mathematics.
These questions will be discussed in the context of various curricular
reforms that we have undertaken at Concordia.
\abs{Morris Orzech}
{Department of Mathematics and Statistics, Queen's University,
Kingston, Ontario~~K7L 3N6, Canada}
{Addressing student difficulties specific to linear algebra}
At my university, first year students who plan a mathematics
concentration as part of their degree programme take courses in linear
algebra and in calculus. Each course offers its challenges, but
linear
algebra seems a more uncomfortable experience for both teachers and
students. My outlook on this phenomenon is that linear algebra seems
to be more difficult for students despite it being taught with as much
care, and as much attention to content, as is calculus. This
viewpoint
has led me to shift my attention from an almost exclusive focus on
choosing material and presenting it clearly to an increased concern
with aligning the way material is offered to the students' abilities
to
learn it. Some of my attempts in this direction depart from my
traditional teaching practice by looking for guidance to
learning-theory based ideas of people like Harel, Hillel and
Sierpinska. Some are more ad-hoc, but are unconventional in other
ways: in the name of having students better learn and appreciate
mathematics they challenge some shibboleths that our conventional
teaching practice seeks to inculcate.
\abs{Thomas W. Rishel}
{Cornell University, Ithaca, New~York~~14853-7901, USA}
{Teaching and job initiatives for graduate programs in
mathematics}
In the past few years, a number of new initiatives have developed for
preparation of graduate students for the professoriate. Among these
are TA training programs, college teaching courses, ``professors for
the future'' programs, job search and placement services, peer
mentoring, and teaching portfolios.
Rishel, who has been involved in many of the above for the mathematics
department at Cornell, will outline the design, implementation and
rationale for such initiatives.. He will also provide sample materials
from his program.
\abs{Martha Siegel}
{Towson University, Towson, Maryland~~21252-7097, USA}
{To be announced}
\abs{Keith F. Taylor}
{University of Saskatchewan, Saskatoon, Saskatchewan~~S7N 5E6,
Canada}
{The Math readiness project at the University of Saskatchewan}
For several years, we have been developing and conducting a
preuniversity preparation programme in mathematics at the University of
Saskatchewan. Among the components in the programme are a two week
summer camp of intensive review, evening versions of the course, a
distance delivery version that uses the internet, and an internet based
course on basic mathematics for mature students who have forgotten
everything. I will describe all aspects of Math Readiness including
our successes and failures and how we fund the project. If we have the
right technology available, I will give a demonstration of MRC (the
Math Readiness Course on the internet).
%\vskip .75cm
%\hrule
%\vskip .3cm
\session{
{\bf Education II/ \'Education II\\
Teaching Mathematics---Why We do What We do in the Classroom? / \\
L'enseignement des math\'ematiques---pourquoi fait-on ce qu'on fait
en classe?}}
\org{William Higginson, Grace and Morris Orzech, Organizers}
%\hrule
%\vskip .4cm
\abs{George Gadanidis}
{Durham District School Board, 400 Taunton Road East, Whitby,
Ontario~~L1R 2K6, Canada}
{If Piaget was a math teacher---Theoretical images, classroom
practice and the power of students' minds}
This talk will address theory-practice links at the classroom level.
Subject to the availability of time and equipment we will:
- discuss the theory-practice gap/overlap at the classroom level;
- illustrate with video clips how theoretical images may be translated
into practice;
- discuss how we may work with teachers so that they develop and
experiment with personal theoretical images.
\abs{William Higginson}
{Faculty of Education, Queen's University, Kingston,
Ontario~~K7L 3N6, Canada}
{Having, knowing, and being: some fundamental questions
about mathematics teaching}
[Abstract not available]
\abs{Lynn McAlpine and Cynthia Weston}
{Centre for University Teaching and Learning, McGill
University, Montreal, Quebec~~H3A 2K6, Canada}
{How six outstanding math professors use reflection to
improve their teaching}
An increased value is being placed on quality teaching in higher
education. An important step in developing approaches to better
instruction is understanding how those who are successful at teaching
go about improving their own teaching. Although reflection has
frequently been referred to as a mechanism for accomplishing this
improvement process (in both literatures on university and public
school teaching), there had been no research which documented it as a
cognitive process. So, a number of years ago, we undertook a program
of research in which the concept of ``reflection'' provided the frame
of reference. We wanted to see if and how it improved practice.
We chose to look at outstanding professors since the literature
suggests that experts tend to reflect more. And, we chose mathematics
as the subject area since there appears to be a general public concern
about the quality of students' learning in this area. We investigated
in detail the reflective processes of six professors each teaching an
undergraduate course in mathematics or mathematics education. We
tracked their day-to-day planning, instructing and evaluating of
learners. We documented what these professors were monitoring during
and after teaching, what changes they made as a result, and what
knowledge they drew upon in making decisions. Of particular interest
was their attention to feedback from students in making decisions about
how to adjust their teaching.
What we will do in the presentation is describe how these professors
used reflection to improve their teaching, and present the model of
reflection that resulted from the analysis. We have found that this
model provides a language for describing reflection and therefore a way
to be intentional in using reflection.
\abs{Pat Rogers}
{Department of Mathematics and Statistics, York University,
Toronto, Ontario~~M3J 1P3, Canada}
{The importance of why: encouraging students to reflect}
[Abstract not available]
\abs{Nathalie Sinclair}
{Queen's University, Kingston, Ontario~~K7L 3N6, Canada}
{Romancing powerful mathematical ideas}
Technology can provide students with interactive tools for
visualizing,
transforming and simulating mathematical objects; this enables them to
explore powerful mathematical ideas, find connections between them and
construct their knowledge of them by creating their own microworlds.
Classroom experiences in trying to achieve this potential with middle
school students will be presented, as well as a description of the
JavaBean-based technologies developed and used.
%\vskip .75cm
%\hrule
%\vskip .3cm
\session{
{\bf Mathematics on the Internet / \\
Math\'ematiques sur internet}}
\org{June Lester, Nathalie Sinclair and Malgorzata Dubiel, Organizers}
%\hrule
%\vskip .4cm
\abs{William Casselman}
{Department of Mathematics, University of British Columbia,
Vancouver, British Columbia~~V6T 1Z2, Canada}
{Colour, animation, interaction---the next generation of
electronic journals}
So far, electronic journals have only acted as cheaper versions of
paper journals. In the near future we can expect to see more
imaginative productions, but they will require much more work by both
authors and staff. How can we prepare for this? What will be the
rewards?
\abs{Stan Devitt}
{Waterloo Maple Inc., Waterloo, Ontario, Canada}
{Extracting mathematical meaning from MathML notation---an
essential step towards live math on the Web}
The meaningful representation of mathematics on the web has had a
checkered past. Approaches have fallen into essentially three
categories: Pictures of formulae, prepared using other software such
as Maple, Expressionist, TeX or Mathtype, embedded special encodings
processed by plugins and applets such as techexplorer or WebEQ, and
pictures of entire pages formatted using page layout software such as
PageMaker, \TeX or FrameMaker.
While there are extensive problems deriving from scaling and alignment
of the pictorial fragments, especially after rescaling of the page,
each of these approaches has a much more fundamental flaw. Their heavy
reliance on pictures of formulae and page layout based encodings
prevents the encoding the mathematical meaning of the original
formulae. Without a mechanism for the author to specify the intended
semantics, there no way to automatically process formulae that are
embedded on web pages.
W3C's MathML 1.0 specification addresses the semantics issues by
providing encodings for both the visual/aural encoding and the
mathematical semantics. The approach taken allows authors to actually
define and reference their notation. This talk explores how MathML
allows you to work with the mathematical meaning of expressions,
especially when interacting with computer algebra systems such as
Maple.
\abs{Stan Devitt}
{Waterloo Maple Inc., Waterloo, Ontario, Canada}
{The Relationship between OpenMath and MathML}
The primary purpose of OpenMath is to facilitate reliable communication
of mathematical objects between mathematical applications. To
accomplish this the actual meanings of the objects must be available.
To see why consider that in the absence of such definitions we would be
unable to determine if $D^2 y$ represents two applications of the
differential operator $D$ to the function $y$, or is simply a monomial
corresponding to the square of the symbol $D$ multiplied by the the
symbol $y$, where $D$ and $y$ are elements in some algebra chosen by the
author. In this talk we review the current status of the OpenMath
standard, how it is being used, and how this relates to similar
requirements for the communication of mathematical meaning in MathML.
\abs{Nicholas Jackiw}
{KCP Technologies, 1150 65th Street, Emeryville,
California~~94608, USA}
{Taking dynamic geometry on-line}
This session will investigate educational and technical issues
surrounding JavaSketchpad, a prototype web-based version of The
Geometer's Sketchpad which emerged from research into pen-based
mathematical modeling software. The software and some examples of its
use shall be demonstrated, and the author will discuss both the
specific technical difficulties and mathematical questions posed by its
``dynamic geometry'' paradigm; and more generally, the Internet's
current position within the technology landscape of secondary
mathematics education.
\abs{Loki J\"orgenson}
{Centre for Experimental and Constructive Mathematics, Simon
Fraser University, 8888 University Drive, Burnaby Mountain Campus,
Burnaby, British Columbia~~V5A 1S6, Canada}
{The CMS Journals On-Line: A Study in Digital Publishing}
A guided tour of the Canadian Journal of Mathematics and the Canadian
Mathematics Bulletin On-line will be presented with an opportunity for
the audience to ask questions and make suggestions for their future.
Plans for their development will be sketched out with reference to some
of the new Web technologies presented at the conference. Predictions
for the future of digital publishing in mathematics will be offered,
some disquieting, some controversial and others exciting, but all
interesting to those concerned about scholarly communications.
\abs{Robert Miner}
{Geometry Technologies, Inc., 400 Sibley Street, St. Paul,
Minnesota~~55101, USA}
{Putting math on the Web}
Nearly eight years after the invention of the World Wide Web, it is
still a difficult medium to use for the communication of mathematical
and scientific material in spite of its phenomenal success in other
areas. In this talk, we will briefly survey the status of current
technologies for putting math on the Web, and in particular the
development and implementation of the World Wide Web Consortium's
Mathematical Markup Language (MathML).
%\vskip 1.0cm
%\hrule
%\vskip .3cm
\session{
{\bf Contributed Papers Session / Communications libres}}
\org{L. Haddad, Organizer}
%\hrule
%\vskip .4cm
%%%HOLD%%%%
\abs{Alexandru Gica}
{Faculty of Mathematics, University of Bucharest,
RO-70109 Bucharest 1, Romania}
{A conjecture which implies the theorem of
Gauss-Heegner-Stark-Baker}
In 1801 Gauss conjectured that the ring of integers for a quadratic
imaginary field $K= {\bf Q} (\sqrt d)$ is principal only for a finite
number of $d$. Heegner, Stark and Baker proved that only for
$$
-d=1,2,3,7,11,19,43,67,163
$$
the ring of integers for ${\bf Q} (\sqrt d)$ (where $d\in {\bf Z}$,
$d<0, d$ squarefree) is principal. We prove two result which are
equivalent with this theorem and we pose a conjecture which implies
the
Gauss-Heegner-Stark-Baker theorem (we will abbreviate: the G.H.S.B.
theorem). Finally we pose a ``weaker'' conjecture which seems to be
more approachable than the other one.
%%%%HOLD%%%%
\abs{I. Nikolaev}
{CRM, Universit\'e de Montr\'eal, Montr\'eal, Qu\'ebec~~H3C
3J7,
Canada}
{$3$-manifolds, foliations and $K_0\bigl(C(X)\rtimes Z\bigr)$}
J.~F.~Plante [1] classified $3$-dimensional manifolds $M$ such that
every foliation $\cal F$ on $M$ has a compact leaf. Plante's method is
based on estimation of `growth rate' of the fundamental group
$\pi_1(M)$.
We show that calculation of a `dimension function' of $K_0$-group of
the crossed product $C^\ast$-algebra generated by $\cal F$ leads to
series of results covering Plante's classification.
{\centering
{\bf References}
\par}
{\bf 1.}~~J.~F.~Plante, {\it Foliations with measure preserving
holonomy}.
Ann. of Math. {\bf 102}(1975), 327--361.
\abs{Dieter Ruoff}
{Department of Mathematics and Statistics, University of
Regina, Regina, Saskatchewan~~S4S 0A2, Canada}
{Solution of a non-Euclidean convexity problem}
The curve that will be investigated is made up from the vertices of the
angles which have size $\alpha$, share the common chord $\AB$ and lie
in one and the same halfplane with respect to the line through $\AB$.
It is a well-known fact that in Euclidean geometry these points form an
arc of a circle. Also, it is clear that any one of the given angles
can be moved into any other by a rotation. In non-Euclidean geometry
all this does not hold. A sketch of a convexity proof for this latter
setting will be provided.
\abs{Andrew Toms}
{Fields Institute, University of Toronto, Toronto, Ontario,
Canada}
{Perforated $K_0-$groups of $C^\ast-$algebras}
Every weakly unperforated ordered abelian group arises as the
$K_0-$group of some simple, separable $C^\ast-$algebra. What of
perforated abelian groups? In this talk we outline methods for
constructing $C^\ast-$algebras having a maximal free cyclic subgroup
of
$K_0$, say $H$, with the property that $H \cap K_0^+ \simeq \langle
k,l
\rangle$, $k$ and $l$ relatively prime positive integers. The methods
used also provide many other such algebras with subgroups whose
intersection with the positive cone of $K_0$ further extend the
preceding result, though these are less succinctly described.
%\vskip 1.0cm
%\hrule
%\vskip .3cm
\session{
{\bf Graduate Student Seminar / S\'eminaire pour \'etudiants
dipl\^om\'es}}
\org{David Gregory, Organizer}
%\hrule
%\vskip .4cm
\abs{Leo Butler}
{Department of Mathematics and Statistics, Queen's University,
Kingston, Ontario~~K7L 3N6, Canada.}
{A New Class of Homogeneous Manifolds with
Liouville-Integrable Geodesic Flows}
A family of nilmanifolds possessing Riemannian metrics whose geodesic
flow is Liouville-integrable is demonstrated. These homogeneous spaces
are of the form $D \backslash H$, where $H$ is a connected,
simply-connected and two-step nilpotent Lie group and $D$ is a
discrete, cocompact subgroup of $H$. The metric on these homogeneous
spaces is obtained from a left-invariant metric on $H$. These
nilmanifolds provide the first example of manifolds whose fundamental
group possesses no commutative subgroup of finite index, yet they admit
a Liouville-integrable geodesic flow. The conclusions of
Ta\u{\i}manov's theorem do not obtain in the category of
Liouville-integrable geodesic flows with smooth first integrals.
\abs{Leo Creedon}
{Department of Mathematical Sciences, University of Alberta, Edmonton,
Alberta~~T6G 2G1, Canada}
{Constructing free groups in modular group algebras}
We investigate the existence (and construction) of free pairs of units
in the unit group of a (modular) group algebra $KG$. We generalise a
result of Goncalves/Passman to do this, and use the programming package
$\GAP$ to investigate the units of $\bbd F_2(t)A_5$, where $t$ is an
element transcendental over $\bbd F_2$.
\abs{Mark V. DeFazio}
{Department of Mathematics and Statistics, York University,
Toronto, Ontario~~M3J 1P3, Canada}
{The behaviour of the complex zeroes of the Laguerre
polynomial}
When $\alpha > -1$, the behaviour of the zeroes of $L_n^{(\alpha)}(x)$
is well understood. There are $n$ positive distinct real zeroes and
they are continuously increasing functions of $\alpha$. When $\alpha <
-1$, these zeroes can be complex and their behaviour as $\alpha$ varies
is not well known. I will present results on the location of the
complex zeroes of $L_n^{(\alpha)}(z)$ and describe their behaviour as
$\alpha < -1$ varies.
\abs{Jody Esmonde}
{Department of Mathematics and Statistics, McGill University,
Montr\'{e}al, Qu\'{e}bec~~H3A 2K6, Canada}
{Parametric solutions to the generalized Fermat equation}
We will discuss parametric solutions $a(t), b(t), c(t) \in F[t]$ to the
equation $x^p+y^q=z^r$, proving that such solutions exist if and only
if $1/p+1/q+1/r>1$.
\abs{Shaun Fallat}
{Department of Mathematics, College of William and Mary,
Williamsburg, Virginia~~23187, USA}
{Multiplicative principal minor inequalities for totally
nonnegative matrices}
An $m$-by-$n$ matrix $A$ is said to be {\it totally nonnegative\/} if
the determinant of every square submatrix ({\it i.e.}, {\it minor}) of
$A$ is nonnegative. Our main interest lay in characterizing all the
inequalities that exist among products of principal minors (minors of
$A$ based on the same row and column index sets) of nonsingular
totally nonnegative matrices. We give a complete characterization of
all such inequalities for $n$-by-$n$ totally nonnegative matrices with
$n \le 5$. We also illustrate certain key steps in the proof of this
characterization, which offer interesting applications of symbolic
computation, graph theory, convexity, and linear programming. Other
results are presented including general conditions which guarantee
when
the product of two principal minors is less the product of two other
principal minors, and many others.
This is joint work with Professors M.~Gekhtman and C.~Johnson.
\abs{Andrei V. Gagarin}
{Department of Mathematics, University of Manitoba, Winnipeg,
Manitoba~~R3T 2N2, Canada}
{Characterizations of $(\alpha, \beta)$-polar graphs by
forbidden induced subgraphs}
Let $\alpha$ and $\beta$ be positive integers or infinity. A graph
$G$
is called {\it $(\alpha, \beta)$-polar\/} if there is a partition of
its vertex set $V(G)=A\cup B$, $A \cap B=\varnothing$, such that the
induced subgraph $G(B)$ is a union of disjoint cliques of order at
most $\beta$ and $G(A)$ is the complement to a union of disjoint
cliques of order at most $\alpha$.
In this talk, we will consider the problem of characterizing $(\alpha,
\beta)$-polar classes in terms of a finite list of forbidden induced
subgraphs.
\abs{Malcolm Harper}
{Department of Mathematics and Statistics, McGill University,
Montreal, Quebec~~H3A 2K6, Canada}
{A family of Euclidean rings containing $Z [\sqrt{14}]$}
Let $K$ be an algebraic number field with ring of integers ${\cal
O}_{K}$ and suppose that ${\cal O}_{K}$ has an infinite unit group.
Assumming a suitable generalized Riemann hypothesis, ${\cal O}_{K}$ is
a Euclidean ring (in the sense of Samuel, 1971) if and only if $K$ has
class number $1$ (Weinberger, 1973). $Z[\sqrt{14}]$ has an infinite
unit group and is the ring of integers in $K=Q (\sqrt{14})$ which has
class number $1$. Cardon (1997) showed that the fundamental
obstruction to the norm acting as a Euclidean algorithm in
$Z[\sqrt{14}]$ lies at one of the residue classes modulo $2$ and thus
$Z[\sqrt{14}] [1/2]$ is Euclidean. Using the sieve techniques of
Gupta, Murty and Murty (1987), Clark (1992) and Clark and Murty (1995)
we show that inverting any non-unit in $Z [\sqrt{14}]$ yields a
Euclidean ring. That is, $Z [\sqrt{14}] [1/a]$ is Euclidean for any
$a$ in $Z[\sqrt{14}]$ not a unit.
\abs{Yu-Ru Liu}
{Department of Mathematics, Harvard University, Cambridge,
Massachusetts~~02138-2901, USA}
{The Tur\'{a}n sieve and probabilistic Galois theory}
We introduce the Tur\'{a}n sieve and apply it to the probabilistic
Galois theory problem in both the rational and the functional cases.
More precisely, we estimate the number of polynomials of degree $n$
and
height $\le N$ whose Galois group is a proper subgroup of $S_n$.
\abs{David McKinnon}
{Department of Mathematics, University of California,
Berkeley,
California~~94618, USA}
{An arithmetic B\'ezout theorem}
In this paper, we prove two versions of an arithmetic analogue of
B\'ezout's theorem, subject to some technical restrictions. The
basic formula proven is $h(X\cap Y)=h(X)\deg(Y) + h(Y)\deg(X) + O(1)$.
The theorems are inspired by the arithmetic B\'ezout theorem of Bost,
Gillet, and Soul\'e, but improve upon them in two ways. First, we
obtain an equality up to $O(1)$ as the intersecting cycles vary in
projective families. Second, we generalise this result to
intersections of divisors on any regular, generically smooth,
projective arithmetic variety. We present an application of these
results to proving an analogue of Hilbert's Irreducibility Theorem for
intersections of curves in ${\bf P}^2$.
\abs{Satya Mohit}
{Department of Mathematics and Statistics, Queen's University,
Kingston, Ontario~~K7L 3N6, Canada}
{The $ABC$-Conjecture and bounds for the order of the
Tate-Shafarevich group}
We will discuss various equivalent formulations of the $ABC$-Conjecture
in terms of bounds for the order of invariant quantities associated to
elliptic curves, particularly, the Tate-Shafarevich group. We will
show that, in order to prove the $ABC$-Conjecture, it suffices to prove
the following fact:
{\it For every elliptic curve $E$, there is some `small' quadratic
twist $E_D$ whose Tate-Shafarevich group is bounded by
$DN^{1/2+\epsilon}$}.
\abs{Daniel Pich\'e}
{Department of Applied Mathematics, University of Waterloo,
Waterloo, Ontario~~N2L 3G1, Canada}
{Wavelet compression on fractal tilings}
Over the last decade, wavelets have become increasingly useful for
studying the behaviour of functions, and for compression. Though much
remains to be investigated in this field, certain types of wavelets are
fairly easy to construct, namely Haar wavelets. This paper ties
together the theory of these wavelets with that of complex bases. An
algorithm is proposed, for doing wavelet analysis, with wavelets
arising in this fashion. This will enable further study of the
properties of these wavelets.
\abs{Filip Saidak}
{Department of Mathematics and Statistics, Queen's University,
Kingston, Ontario~~K7L 3N6, Canada}
{On zero-free regions of the zeta function}
A powerful analytic method of Tur\' an brought several advances in
understanding the local behaviour of zeros of zeta-like functions. We
discuss certain simplifications and extensions of these classical
results, highlighting the main unsolved problems and conjectures of the
theory.
\abs{Gregory S. Smith}
{Department of Mathematics, University of California at
Berkeley, Berkeley, California~~94720-3840, USA}
{Computing global extension modules}
Let $X$ be a projective scheme; let ${\cal M}$ and ${\cal N}$ be two
coherent ${\cal O}_{X}$-modules. Given an integer $m$, we present an
algorithm for computing the global extension module $\Ext^{m}(X;{\cal
M},{\cal N})$. In particular, this allows one to calculate the sheaf
cohomology $H^{m}(X,{\cal N})$ and to construct the sheaf corresponding
to an element of the module $\Ext^{1}(X;{\cal M},{\cal N})$. This
algorithm can be implemented using only the computation of Gr\"{o}bner
bases and syzygies, and it has been implemented in the computer algebra
system {\it Macaulay2}.
\abs{Sarah Sumner}
{Department of Mathematics and Statistics, Queen's University,
Kingston, Ontario~~K7L 3N6, Canada}
{Investigating transcendence in the field of $p$-adic numbers}
In the field of $p$-adic numbers, whether the number
$\sum_{n=1}^{\infty}\, n!$ is transcendental or even irrational is an
unsolved problem. However, it can be shown that numbers of the form
$\sum_{n=1}^{\infty} n^k \cdot n!$ can be written in the form $v_k -
u_k\sum_{n=1}^{\infty}\,n!$ where $u_k$ and $v_k$ are integers.
Interesting congruences involving the series $\{u_k\}$ are developed
and conditions that will produce a $p$-adic transcendental number are
discussed.
This is joint work with Professor Ram Murty.
\abs{Drew Vandeth}
{Macquarie University, Centre for Number Theory Research,
Sydney, Australia}
{Transcendence of the values of generalized Mahler functions}
In this talk we examine conditions under which generalized Mahler
functions, of one or several variables, assume transcendental values at
given algebraic points. The results are achieved through the injection
of the Bombieri-Vaaler Siegel Lemma into the work of P-G.~Becker and
the work of F.~Gramain, M.~Mignotte and M.~Waldschmidt.
\end{document}
\end{document}