Public Lecture / Conférence Publique
Ed Barbeau (University of Toronto)
Delicious mathematics

Perhaps one per cent of the population have the ability for seminal achievements in music and a similar proportion are incapable of any appreciation of music at all. In between lies a whole spectrum of talent and interest. A range of activity from choral and chamber groups to jazz and rock bands exists to provide pleasure and satisfaction to almost everyone of whatever level of sophistication and willingness to participate.

Surely, mathematics is similar. Yet who would know this from a school experience which many pupils see as drudgery and a few see as an elite undertaking? Yes, it is useful, but it is much much more. Mathematics, like music, is an artifact of human invention, some of it decidedly from the ``folk'' culture. There are gems to delight even those with no particular background in the discipline - puzzles, games, demonstrations and models. Come and behold!

Jeffery-Williams Lecture / Conférence Jeffery-Williams
John Friedlander (University of Toronto)
Prime Values of Polynomials

Consider an irreducible polynomial with integer coefficients. It is an old and interesting problem to try to prove whether or not the polynomial is a prime number for infinitely many integer values of the variable(s). One expects that, apart from certain trivial counter-examples, this will always be the case. Nevertheless, despite the long history of the question, relatively few cases have been settled in the affirmative. In this talk we survey some of that history and then go on to describe a number of the ideas behind recent joint work with Henryk Iwaniec leading to further progress.

Krieger Nelson Lecture / Conférence Krieger Nelson
Nicole Tomczak-Jaegermann (University of Alberta)
Asymptotic Aspects in Geometric Functional Analysis

Asymptotic finite- and infinite-dimensional approaches in Geometric Functional Analysis will be discussed on examples of solutions of the homogeneous Banach space problems, in finite and infinite dimensions.

Plenary Speakers / Conférenciers principaux
Ed Barbeau (Cambridge)
Men as trees walking

We know how beautiful our subject is. But for much of the past, mathematicians have accepted that a love of mathematics is a minority taste and that it is enough for most of the population to have some level of operational competence. However, recent attempts to revolutionize education have been predicated on the worthy proposition that understanding and enjoyment of mathematics can be made more generally accessible. In particular, a lot of emphasis has been put on problem solving. Among other things, this has spawned a lot of competition activity. Unfortunately, like many revolutions, this one has been executed unevenly. New fissures have developed in the educational community and new inequities have replaced the old ones.

Michel Van den Bergh (Limburg)
Non commutative algebraic geometry

The classifications of curves and surfaces are classical subjects in algebraic geometry. In recent years an understanding has grown that parts of this classification admit a non-commutative analogue. In fact for curves this has more or less been settled whereas for surfaces the main results are still conjectural. In the talk I will explain in what framework these non-commutive analogues are formulated. I will also explain in what sense non-commutative curves have been classified and finally I will show how several classical constructions from algebraic geometry have been extended to the non-commutative case. In this way it is possible to construct non-commutative projective planes, quadrics and cubics, and more generally, non-commutative Del-Pezzo surfaces.

Ranee Brylinski (Pennsylvania State)
Star products and quantization of manifolds with symmetry

A star product on an algebra of (smooth or algebraic) functions is a non-commutative product which deforms the ordinary multiplication by adding "correction terms" involving a formal parameter t. I will explain some basic notions about star products equivariant under a symmetry group. Such star products are in general given by quotients of differential operators (and so fall outside the theories of Fedosov, Kontsevich, etc.). I will discuss examples occuring in algebraic geometry and representation theory: star products on polynomial functions on orbits of complex nilpotent matrices (more generally, complex nilpotent coadjoint orbits).

Tom Korner (Cambridge)
Does Order Matter?

In Fourier analysis we split a signal into components and then reassemble them. Does it matter in which order we use the components? If it does, will natural methods always work? Are there better approaches to the whole problem?

Doug Stinson (Waterloo)
Orthogonal arrays: variations on a theme, and applications

An orthogonal array OA(N,t,k,v) is an N by k array of symbols chosen from a v-set such that, within any t columns of the array, every list of t symbols appears in the same number of rows. Orthogonal arrays were introduced over 50 years ago for use in certain types of experimental design. Mutually orthogonal Latin squares, which are equivalent to orthogonal arrays with t = 2 and N = v², have been studied studied for over 200 years. In recent years there have been numerous situations where various structures that are generalizations of orthogonal arrays have been used for interesting applications in cryptography. We give several illustrative examples in this talk, including resilient functions, zigzag functions, (t,m,s)-nets, ramp schemes and threshold schemes, and all-or-nothing transforms.

Luc Vinet (CRM)
Integrable Systems, Special Functions and Algebraic Combinatorics

I shall discuss examples that illustrate aspects of the fruitful interplay that has developed recently between these three areas.

Perspectives in Ring Theory / Perspectives de la théorie des anneaux
Yuri Bahturin (Moscow)
Algebras with actions of Hopf algebras

Let (H,m,u,D,e,S) be a Hopf algebra over a field k, A a k-algebra, not necessarily associative. We say that A is an H-algebra if A is a left H-module and

h·(ab) = å (h)  (h(1)·a)(h(2)·b)

for any a,b Î A and h Î H with D h = å_(h)h₍₁₎h₍₂₎, and also h·1_A = e(h)1_A.

This definition absorbs many classical situations such as graded algebras, automorphism groups and derivation Lie algebras of algebras. The idea of the approach via Hopf algebras is to give unified solutions to similar problems in seemingly different situations. Our examples are mainly some theorems about algebras with polynomial identities. In this talk we stress the situation of non-associative algebras.

Here we formulate just one relevant result, from a joint paper with Susan Montgomery.

Theorem. Let L be a finite-dimensional colour Lie superalgebra (here the dual of a group algebra of a finite group G acts) over a field of characteristic 0. Then any associative PI-envelope A of L is finite-dimensional if and only if the even part L₊ of L is a perfect algebra (i.e. with [L₊,L₊] = L₊).

Previously such result was known only in the case of ordinary Lie algebras.

Anthony Bak (Bielefeld)
Multiplicative Spectra for Rings and Automorphism Groups of Modules and Modules with Forms

This talk introduces a concept of spectrum for arbitrary associative rings with identity and uses it to extend significantly classical results on the normal subgroup structure of automorphism groups of modules over arbitrary rings and of modules with a quadratic or not necessarily even Hermitian form over arbitrary rings with involution.

The concept of spectrum above is based on denominator sets rather than prime ideals and is considerably more flexible to apply than its predecessors. The flexibility is used to define the important notion of local stable rank lsr(A) of a ring A. This notion is smaller than that of Bass stable rank sr. In fact, it is usually much smaller and the class of rings whose lsr is finite, but whose sr is infinite, is huge. Some explicit properties of lsr are that a commutative or module finite ring has lsr  1 and an arbitrary ring A has lsr(A) £ sup {sr (A  /m) | m primitive in A}.

The results on the automorphism groups of modules (resp. modules with forms) are proved under the assumption on the module that its unimodular (resp. Witt) index ³lsr (A) + 2. The results show that the elementary subgroup of the automorphism group is normal and establish a sandwich classification of all subgroups of the automorphism group normalized by its elementary subgroup. Results of this kind for modules (resp. modules with quadratic forms) are known if lsr is replaced by sr, thanks to classical results of H. Bass (resp. the author), and if A is module finite thanks to results of A. Suslin (resp. several others). Normality results for not necessarily even Hermitian forms were not known even in the classical setting of finite stable rank or module finite rings.

Further applications of multiplicative spectra to extend classical results over commutative or module finite rings, such as splitting and cancellation theorems, to arbitrary rings will be listed, but not discussed in detail.

Margaret Beattie (Mount Allison)
Counting pointed Hopf algebras with a given coradical

Necessary and sufficient conditions for two Ore extension Hopf algebras to be isomorphic are given. As an application we derive formulas for the number of isomorphism classes of Ore extension Hopf algebras with coradical of dimension 4 and with skew-primitives whose square lies in the coradical.

Howard Bell (Brock)
Setwise Commutativity Conditions in Rings

Motivated by recent results in group theory, we discuss commutativity or near-commutativity in rings satisfying finiteness and/or commutativity constraints on certain subsets of the ring. We shall pay particular attention to a condition introduced by Brailovsky: for each k-set K, |K²| £ k(k+1)/2

V. Bovdi (Hungary)
On the order of the unitary subgroup of a modular group algebra

Let KG be the group algebra of a locally-finite p-group G over a commutative ring K (with 1) and V(KG) be the group of normalized units. The anti-automorphism g® g^-1 extends linearly to an anti-automorphism a® a^* of KG. Let

V*(KG) = {  x Î V(KG)  |   xx* = 1  }

be the unitary subgroup of V(KG).

We proved the following theorems

Theorem 1. Let K be a finite field of characteristic 2.

• [(i)] If G is an extraspecial 2-group of order |G| = 2²ⁿ⁺¹ with n ³ 2, then

  |V*(KG)| = |K|2n-1(2n+2-2n+(-1)n)-1.

• [(ii)] If G is a central product of an extraspecial 2-group H of order |H| = 2²ⁿ⁺¹ with a cyclic group <d> of order 4, then

  |V*(KG)| = |K|3·22n-1.





Recall that a p-group is extraspecial if its centre, commutator subgroup and Frattini subgroup coincide and have order p.

Theorem 2. Let K be a finite field of characteristic 2 and G has an abelian subgroup A of index 2 and an element b which inverts every element of A.

• [(i)] If b has order 2, then

  |V*(KG)| = |K| (3|A|+|A[2]|-2)/2.

• [(ii)] If b has order 4, then

  |V*(KG)| = 2·|A2[2]|·|K||A|+0.5|A[2]|-1.

Corollary Let K be a finite field of characteristic 2.

• [(i)] If

  D_2ⁿ⁺¹ = <  a,b  |   a²ⁿ = b² = 1,   a^b = a^-1>   is a dihedral group of order 2ⁿ⁺¹, then

  |V*(KD2n+1)| = |K|3·2n-1.

• [(ii)] If

  Q_2ⁿ⁺¹ = <  a,b  |   a²ⁿ = 1, a^2n-1 = b²,   a^b = a^-1>   is a quaternion group of order 2ⁿ⁺¹, then

  |V*(KQ2n+1)| = 4·|K|2n.

Hans Brungs (Alberta)
Valuation trees

There exists for every rooted tree T a finite dimensional division algebra with T as its valuation tree. This result can be extended to a particular class of labelled trees.

Stefaan Caenepeel (Brussel)
Separable functors applied to Doi-Hopf modules

Let H be a Hopf algebra, A a right H-comodule algebra, and C a right H-module coalgebra. A Doi-Hopf module is at once a right A-module and a right C-comodule, satisfying a certain compatibility condition. Special cases are graded modules, Sweedler Hopf modules, relative Hopf modules and Yetter-Drinfel'd modules. One can also give pairs of adjoint functors between categories of Doi-Hopf modules, and these include many existing functors (forgetful functor, induction functor) as special cases. The starting point of this lecture is the following problem. When do we have a Maschke type Theorem for such a functor? Several attempts have been given in the literature, under various conditions. Here we give a general answer, based on the theory of separable functors. This leads to a generalized notion of separability idempotent, and a generalized notion of integral.

Gerald Cliff (Alberta)
Hecke Algebras for GL(n,R) where R is Z mod a^p

The Hecke algebra for G=GL(n,k), where k is a finite field, is the endomorphism ring of the permutation module of G/B where B is the subgroup of upper triangular matrices. This is closely related to the group ring of the symmetric group. We replace the field k by R=Z mod a^p, and study the resulting analog of the Hecke algebra. The applications are to representations of GL(n,R).

Jairo Gonçalves (S#&227o Paulo)
Free objects in algebras

Let D be a division ring with center k, and let D* be its multiplicative group. We investigate the existence of free(noncyclic) subgroups in D*. In many instances, as in quaternion algebras, field of fractions of skew polynomial rings and symbol algebras, we are able to exhibit them explicitly. These results also apply to produce symmetric free groups in group rings of finite groups in arbitrary characteristic.

C. K. Gupta (Manitoba)
Primitive sets and automorphisms

Let R be the algebra generated by two generic 2×2 matrices x and y over a field K of characteristic 0 and let T be the algebra generated by R and the traces of the matrices from R. We construct new automorphisms of T and R. Our automorphisms of R are wild. We do not know if the new automorphisms of the polynomial algebra K[x₁,x₂,x₃,x₄,x₅] are tame. We study the Lie automorphisms of R and we prove that the subgroup generated by \textGL₂(K) and the automorphisms defined by

fkl(x) = x,   fkl(y) = >y+y \textad2kx \textad2l+1[y,x], >    k ³ 1, l ³ 0,

is dense in the group of all Lie automorphisms of R with respect to the formal power series topology. In other words, for any Lie automorphism f of R and for any positive integer n, there exists an automorphism y_n of our subgroup such that the polynomials f(x)-y_n(x) and f(y)-y_n(y) have no terms of degree less than n.

This is joint work with V. Drensky.

Klaus Hoechsmann (UBC)
Units in commutative integral p-group rings

Given a finite abelian group A, there is an explicitly constructible group C(A) in the integral group ring ZA. It contains the classical Bass units and is contained in the group K(A) of units which are mapped to cyclotomic ones by all characters of A. For p-groups A, C(A) equals K(A) if and only if p is a regular prime. For any p, the the index [K(A):C(A)] equals the order of a canonical group of ideal classes.

Colin Ingalls (Washington)
Quantizable Orders over Surfaces

Infinitesimal deformations of maximal orders over smooth algebraic surfaces are studied. We classify which maximal orders may admit deformations that are not finite over their centres. The classification is in terms of the algebraic surface that corresponds to the centre of the order and the ramification divisor of the order.

Eric Jespers (Brussel) and Stanley Juriaans* (São Paulo)
Isomorphisms of integral group rings of infinite groups

This talk will deal with the isomorphism problem for integral group rings of infinite groups. We shall answer a question of Mazur by giving conditions for the isomorphism problem to be true for integral group rings of groups that are a direct product of a finite group and a finitely generated free abelian group. It will be shown that the isomorphism problem for infinite groups is strongly related to the normalizer conjecture and that the automorphism conjecture holds for infinite finitely generated abelian groups G if and only if ZG has only trivial units. A partially answer is also given to a problem of Sehgal. It is shown that the class of a finitely generated nilpotent group G is determined by its integral group ring provided G has only odd torsion. When G has nilpotency class two then the finitely generated restriction is not needed. This, together with a result of Ritter and Sehgal, settles the isomorphism problem for finitely generated nilpotency class two groups. A link will be pointed out between this problem and the dimension subgroup problem.

Thomas Kucera (Manitoba)
Explicit descriptions of the indecomposable injective modules over Jategaonkar's rings

In 1968-1969, A. V. Jategaonkar [1], [2] published his famous constructions of left but not right Noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. Indeed, in remarking about the difficulties associated with one-sided Noetherian rings, Jategaonkar [3] says ``Our decision to work with Noetherian rings rather than right Noetherian ones is often dictated by the exigencies of the situation under consideration. We note though that, after Jategaonkar (69) [that is, [2] below], an attempt to stay with right Noetherian rings at all costs is generally regarded as futile.'' The family of examples developed in the paper cited are important and fascinating; they illustrate how strongly the left ideal structure of a ring can be disconnected from the right ideal structure of the ring. Nonetheless, they do not provide examples of indecomposable injectives that are difficult to understand or impossible to describe. It is the purpose of this paper not only to describe the indecomposable injectives over these badly behaved rings, but to convince the reader that the structure of the indecomposables is easily and naturally deduced from elementary facts about injective modules combined with the description of the rings themselves.

In this talk I will discuss the principal results of my paper. I will outline how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational representations of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables.

[1]

Jategaonkar, A. V., "Left principal ideal domains", J. Algebra 8 (1968), 148-155.

[2]

Jategaonkar, A. V., "A counter-example in ring theory and homological algebra", J. Algebra 12 (1969), 418-440.

[3]

Jategaonkar, A. V., "Localization in Noetherian Rings", London Math. Soc. Lecture Notes 98, Cambridge University Press, 1986.

Janos Kurdics (Hungary)
Nilpotency class in group algebras

Let F be a field of prime characteristic p and G a group with a nontrivial p-Sylow subgroup. By Khripta's result which goes back more than two decades, it is well-known that the group of units U(FG) is nilpotent if and only if G is nilpotent and its commutator subgroup G¢ is of p-power order. Since then remarkable developments have been achieved concerning the study of the nilpotency class cl(U(FG)) of the group of units, although there is no complete description known yet.

We start from two lemmas which may be of independent interest. The first deals with the Lie structure of an associative ring; the second one assigns a `weak complement' to a subgroup in a finite abelian p-group.

By means of these lemmas our main results are the following.


Theorem 1. Let F be a field of prime characteristic p, G a nilpotent group such that its commutator subgroup G¢ is of p-power order and g₃(G) Í G¢^p. Put I = I(G¢). Then

1. FG^(k) = I^k-1 (k ³ 2) and t^L(G) = t(G¢)+1;
2. if G¢ is cyclic of order pⁿ then t_L(G) = t^L(G) = pⁿ+1.


Theorem 2. Let F be a field of prime characteristic p and G a nilpotent group such that the commutator subgroup G¢ is a finite abelian p-group with invariants (p^m₁,p^m₂,¼,p^m_s). Then the following statements hold:

1. t_L(G) ³ t(G¢)+1 = 2+å_{i = 1}^s(p^m_i-1);
2. t_L(G) = t^L(G) = t(G¢)+1 if g₃(G) Í G¢^p;
3. cl(U(FG)) = t(G¢) if G is a p-group and g₃(G) Í G¢^p.


Theorem 3. Let F be a field of prime characteristic p ¹ 3 and G a group nilpotent of class greater than 2 such that the commutator subgroup G¢ is a finite abelian p-group

G¢ = áa1ñ×áa2ñ×¼×áasñ,  |ai| = pmi,     m1 ³ m2 ³ ¼ ³ ms,

and |g₃(G)G¢^p/G¢^p| = p^r, 0 < r < s.

Furthermore, let A = áa_u1ñ×áa_u2ñ×¼×áa_us-rñ, u₁ < u₂ < ¼ < u_s-r be the weak complement of g₃(G) in G¢ relative to the basis {a_i}, and let {1,2,¼,s} = {u₁,u₂,¼,u_s-r,v₁,v₂,¼,v_r}, v₁ < v₂ < ¼ < v_r. Then

1. t_L(G) ³ t(G¢)+t(G¢/A) = 2+å_{i = 1}^s(p^m_i-1)+ å_{j = 1}^r (p^m_vj-1);
2. t_L(G) = t^L(G) = t(G¢)+t(G¢/A) if G is of class 3;
3. cl(U(FG)) = t(G¢)+t(G¢/A)-1 if G is a p-group of class 3.


References

1. [1.] B. Bódi and J. Kurdics, Lie Properties of the Group Algebra and the Nilpotency Class of the Group of Units, J. Algebra, 212 (1999), 28-64.

Greg Lee (Alberta)
Symmetric elements in group rings

Let KG denote the group ring of a group G over a field K. Then KG admits a natural involution * which sends each group element to its inverse. The symmetric elements of KG are those which are fixed under *. We will examine the extent to which the symmetric elements determine the structure of the group ring. Also, we will look at the symmetric units, and the extent to which they determine the structure of the unit group of the group ring.

Yuanlin Li (MUN)
The normalizer of a metabelian group in its integral group ring

In this paper, we first give a necessary and sufficient condition for which the normalizer property holds for the integral group ring of a finite metabelian group. We then discuss two important situations for which all $C$-automorphisms are inner and therefore the normalizer property holds for those cases. Our results generalize Marciniak and Roggenkamp's result, Proposition 12.3). As an application of our theorems, we prove the following interesting result: the normalizer property holds for the integral group ring of a split finite metabelian group with a dihedral Sylow 2-subgroup.

César Polcino Milies (São Paulo)
Group Rings of Frobenius Groups

We shall give a result on the orders of torsion units in integral group ring of Frobenius groups. Then, we shall show some applications to the Zassenhaus conjecture ZC3 and related questions. Also, this result implies that the normalizer conjecture has a positive solution for Frobenius groups.

Olaf Neisse (Augsburg)
On bicyclic units

Bicyclic units have an important role in the theory of units in the integral group ring of a finite group. For example, for a huge class of groups it is known that the group generated by bicyclic units and Bass-Milnor units is of finite index in the group of all units. The first part of the talk describes some relations between generalised bicyclic units and how to use them to descend partly in some cases. The second part is about explicit matrix representations for some p-groups and consequences for the torsion part of the group generated by bicyclic units.

W. K. Nicholson (Calgary)
Strongly clean rings and Fitting's lemma

A ring is called strongly clean if every element is the sum of an idempotent and a unit which commute. These rings are shown to be a natural generalization of the strongly pi-regular rings, and several properties of strongly pi-regular rings are extended, including their relationship to Fitting's Lemma.

Jan Okninski (Warsaw)
Subspace semigroups

This is a joint work with Mohan S.Putcha. For a finite dimensional algebra A over a field K, by the subspace semigroup S(A) we mean the set of all K- subspaces of A endowed with the operation V*W = lin _K (VW). While the structure of S(A) is of interest on its own, one of our motivations is to create a new context for developing geometric tools in representation theory of finite dimensional algebras. On the other hand, dealing with S(M_n(F_q)), one can focus on combinatorial aspects and linear representations for a new promising class of finite monoids, analogous to the case of M_n(F_q) and, more generally, of the class of finite reductive monoids. Notice also that S(M_n(K)) has arisen recently in the context of discrete dynamical systems.

We describe the structure of S = S(M_n(K)), and relate it to the structure of M_n(K). As the key intermediate step we first study the closed set semigroup C = C(M), for a connected linear monoid M defined over an algebraically closed field K, which consists of all closed irreducible subsets of M subject to the operation X·Y = Zcl(XY), where Zcl(Z) denotes the Zariski closure of a subset Z. Structural invariants of C(M) are characterized in terms of certain connected algebraic groups associated to M. This is applied to the case where M = A is a finite dimensional K- algebra. The main properties of S(A) are derived via the natural onto homomorphism h: C(A)® S(A) that maps every X Î C(A) to its linear span lin _K (X). In this context, the role of basic algebras and connections with the Morita equivalence are explained. The main ingredients of our structure theorem for S(A) are then extended to any finite dimensional algebra A over an arbitrary field K.

Donald Passman (Wisconsin)
Simple Lie algebras of Cartan type

Let K be a field, let A be an associative, commutative K-algebra and let D be a nonzero K-vector space of commuting K-derivations of A. Then, with a rather natural definition, W(A,D) = AÄ_KD = AD becomes a Lie algebra, a Witt type algebra. Combined with earlier work of David Jordan, we were able to obtain necessary and sufficient conditions for this Lie algebra to be simple. Indeed, with one minor exception in characteristic 2, simplicity occurs if and only if A is D-simple and A^D ÄD = A^DD acts faithfully as derivations on A. In addition, there is a map div: W(A,D)® A called the divergence and its kernel S = S(A,D) is a Lie subalgebra, a special type algebra. In a recent joint paper with Jeff Bergen, we studied S from a ring theoretic point of view, and obtained sufficient conditions for the Lie simplicity of [S,S]. While the main result was somewhat cumbersome to state, it did handle a number of examples in a fairly efficient manner. Furthermore, some of the preliminary lemmas were of interest in their own right and may, in time, lead to a more satisfactory answer.

Mohan Putcha (North Carolina State)
Monoids and Hecke Algebras

Representation theory of finite semigroups is markedly different from group representation theory since the complex representations of a finite semigroup need not be completely reducible. We use Hecke algebra constructions for finite semigroups to answer complete reducibility questions about representations of finite semigroups.

If time permits we will also show how semigroup representation theory can be used to obtain a new concept of weights for group representations.

Lex Renner (UWO)
Blocks of algebraic monoids

A block can be thought of as an equivalence class of irreducible representations. The equivalence relation in this setup is generated by declaring the irreducible representations (r,V) and (f,W) in the same block if there exists an indecomposable representation (y,U) such that both (r,V) and (f,W) occur as factors in a composition series of (y,U).

If G is an algebraic group (defined over an algebraically closed field K), the blocks of G may be infinite and somewhat out of control. However, much more can be said if we consider the blocks of algebraic monoids. We shall discuss the following contrasting classes of algebraic monoids M.

(a) M solvable with 0.
(b) M reductive with 0 and char(K) = p > 0; (e.g., M = M_n(K)).
Case (a) is based on my own work (J. Algebra 188 (1997), 272-291). Case (b) was investigated by J. A. Green and S. Donkin for M = M_n(K) and, more recently, by S. Doty in the general case.

Akhar Rhemtulla (Alberta)
Braid Groups and their group rings

It has been shown recently that Braid Groups B_n can be right ordered and the Pure Braid Groups P_n are two-sided orderable. The interesting point is that none of the right orders of B_n are "lexicographic" (or of Conrad type). Moreover, the restriction of every right order on B_n to P_n is also non-lexicographic. In particular, no two-sided order on P_n can be lifted to a right order on B_n. A natural question arises: can B_n be embedded in the multiplicative group of a division ring?

Sudarshan K. Sehgal (Alberta)
Torsion in Matrices over ZG

I shall report on the problem: When is a matrix of finite multiplicative order with entries in an integral group ring conjugate to a diagonal matrix with group elements as entries?

Qiang Wang (MUN)
Hereditary semigroup rings

In his book on semigroup algebras, Okninski posed the following problem (Question 37): characterize hereditary semigroup algebras. In this talk I report on some recent joint work with E. Jespers. First we give a description of (prime contracted) semigroup algebras K[S] that are hereditary and Noetherian when S is either a Malcev nilpotent monoid, a cancellative monoid or a monoid extension of a finite non-null Rees matrix semigroup. Second, for the class of monoids which have an ideal series with factors that are non-null Rees matrix semigroups, an upper bound is given for the global dimension of its contracted semigroup algebra.

Paul Wauters (Diepenbeek)
Primitive localisations of group algebras of polycyclic-by-finite groups

Let G be a polycyclic-by-finite group such that D(G) is torsion-free abelian and K is a field. Denote by S a multiplicatively closed set of non-zero central elements of the group algebra K[G]. The main result states necessary and sufficient conditions such that K[G]_S, the localization of K[G] with respect to S, is a primitive ring. Some special cases and examples will be considered.

Yiqiang Zhou (MUN)
Two non-classical finiteness conditions of modules

Two modules are orthogonal if they have no nonzero isomorphic submodules. A nonzero module is called an atomic module if any two nonzero submodules of it are not orthogonal. If there exist pairwise orthogonal atomic submodules A₁,¼,A_n of a module M such that A₁żÅA_n £ _eM, then such a number n is uniquely determined by M in the sense that if N₁żÅN_m £ _eM such that 0 ¹ N_i (i = 1,¼,m) and N_i, N_j (i ¹ j) are orthogonal, then m £ n. In this case, we call n the type dimension of M and write t.dim(M) = n. If such an n does not exist, we say the type dimension of M is ¥ and write t.dim(M) = ¥. If M = 0 we write t.dim(M) = 0. The type dimension is a `type' analogue of the Goldie dimension. In this talk, we discuss how the type dimension can be used to study the `type' analogues of the right Noetherian ring and the right QFD-ring, i.e., the ring whose cyclics are of finite Goldie dimension. Part of this is a joint work with John Dauns.

Harmonic Analysis / Analyse harmonique
J. Benedetto (Maryland)
The role of tiling in sampling and wavelet theory

The notion of tiling is used to prove multidimensional decomposition theorems in the setting of Fourier frames as well as for the case of orthonormal wavelets. The former results depend on Beurling's theory of balayage, and give rise to a constructive irregular sampling algorithm having applications in MRI problems. The latter results, which in art go back to ideas of da Vinci and Escher, provide a general construction in Euclidean space of single dyadic orthonormal wavelets. Part of this construction is reminiscent of the Littlewood-Paley theory, and there are computer generated pictures of the wavelets. In light of multiresolution analysis, such constructions were once thought to be unlikely.

T. Chen (UWO)
Generalized Hardy Operators and Normalizing Measures

We study the boundedness between weighted Lebesgue spaces of positive integral operators with variable limits of integration. Normalizing measures are introduced to provide an integral form for the resulting necessary and sufficient conditions.

C. Finet (Belgium)
Transfer principles in Orlicz spaces

We extend to Orlicz spaces a transfer principle of R. Coifman and G. Weiss, concerning L_p inequalities on some convolution operators.

B. Forrest (Waterloo)
Operator spaces in noncommutative harmonic analysis

As preduals of von Neumann algebras the Fourier and Fourier-Stieltjes algebras of a locally compact group inherit a natural operator space structure. In this talk, we will show that for noncommutative locally compact groups these operator space structures can provide additional information about the nature of the underlying group that cannot be deduced from the norm structure alone.

J-P. Gabardo (McMaster)
Determinacy in truncated trigonometric moment problems and the extension property

Given a finite set S Ì Zⁿ which is symmetric, i.e. 0 Î S and - k Î S whenever k Î S, we say that a positive Borel measure on Tⁿ is S-determinate if the Fourier coefficients m(k), k Î S, characterize the measure m uniquely. We give necessary and sufficient conditions for S-determinacy and show that, when S is a finite set, the validity of this property can be verified algebraically by an algorithmic procedure involving trigonometric polynomials with spectrum in S vanishing on the support of m and satisfying certain positivity properties. We also give necessary and sufficient conditions for a measure to be S-determinate in the case where S is a difference set S = K-K, where K is a finite subset of Z² satisfying the so-called ``extension property".

E. Granirer (UBC)
Some functional analytic properties of quotients of the Fourier algebra as reflected by some subsets of the real line

Let E be a closed subset of the real line. Properties such as Arens regularity, amenability and local nonergodicity of algebras related to A(E) will be discussed.

H. Heinig (McMaster)
Modular inequalities for the Calderón operator

If P,Q:R⁺ ® R⁺ are increasing functions satisfying Q(0+) = P(0+) = 0, and T the Calderón operator defined on decreasing functions, then optimal modular inequalities (*) ò P(Tf) £ cò Q(f) are proved. If P = Q the conditions on P are necessary and sufficient for (*). In addition interpolation theorems for modular spaces are given.

Zhiguo Hu (Windsor)
Isomorphism and homomorphism results on the von Neumann algebra VN(G)

It is well-known that if G is a locally compact abelian group with dual group G, by means of the Fourier transformation, L¹(G) is isomorphic to an algebra A(G) of functions on G. For a general locally compact group G, Eymard defined the Fourier algebra A(G) directly on G and showed that the Banach dual space of A(G) can be identified isometrically with the von Neumann algebra VN(G). In this talk, we will present results which construct isomorphism and homomorphism between VN(G) and l^¥(X), where G is any non-discrete locally compact group and X is an infinite set closely related to G. Applications of such mappings on the structure of subspaces and quotient spaces of VN(G) will also be discussed.

R. Kerman (Brock)
Weighted inequalities for semigroups of operators and the norm convergence of the Abel means of certain eigenfunction expansions

We give a condition sufficient to guarantee uniform weighted norm inequalities for semigroups of operators and hence for the weighted mean convergence of the Abel means of various eigenfunction expansions. The condition turns out to be necessary in the case of the Gauss-Weierstrass semigroup. This semigroup is then used to treat special Hermite and Hermite expansions. To deal with Laguerre expansions we use the Bessel semigroup. The above is joint work with S. Thangavelu.

T. Lau (Alberta)
On the centre of some Banach algebras associated to a locally compact group

Let A be a commutative Banach algebra. Richard Arens shows that there is a natural extension of the multiplication on A to its second dual A^** such that A^** is also a Banach algebra. In general A^** is not commutative. In this talk, I shall discuss the problem in determining the centre of A^** when A is the Fourier algebra of a locally compact group, some related results and implications.

D. Oberlin (Florida)
Convolution with affine arclength measures in the plane

We discuss an L^p - L^q convolution estimate for affine arclength measure. The estimate is uniform over a certain class of curves in the plane.

J-O. Ronning (Skovde)
Generalized Perron trees - What and why

This talk will describe some new necessary conditions for subsets of the unit circle to give collections of rectangles (by means of orientations) which are density bases or give Hardy--Littlewood type maximal functions which are bounded on L^p, p>1. This is done by proving that a well known method, the Perron Tree Construction, can be applied to a larger collection of subsets of the unit circle than was earlier known. Two applications will be presented; a partial converse to a well known result of Nagel, Stein and Wainger regarding boundedness of maximal functions with respect to rectangles of lacunary directions, and a result regarding the cardinality of subsets of e.g. arithmetic progressions in sets of the type described above.

This talk describes a joint work with Professor K.E. Hare, Waterloo, Ontario.

G. Sinnamon (UWO)
From Nörlund matrices to Laplace representations

The existence of Laplace representations for functions in weighted Hardy spaces on the right half plane is established. The method uses an extension of an inequality involving Nörlund matrices and corresponding convolution operators on the line to prove Fourier inequalities on weighted Hardy spaces. Analogous inequalities are proved for power series representations of functions in weighted Hardy spaces on the disc.

Nicolaas Spronk (Waterloo)
Diagonal type conditions on group C$^*$-algebras

Let G be a locally compact group with C^*(G) and C^*_r(G) its enveloping and reduced C^*-algebras respectively. We show that if C^*(G) is residually finite dimensional, then G is maximally almost periodic, and C^*_r(G) is residually finite dimensional if and only if G is both amenable and maximally almost periodic. Letting l_G be the left regular representation of G, we show that a certain quasidiagonality condition on {l_G(s):s Î G} implies that G is amenable.

This is joint work with Peter Wood.

S. Wainger (Wisconsin)
Some discrete problems in harmonic analysis

Let S_l be the d - 1 dimensional sphere in R^d centered at the origin of radius Ö{l}, and denote by ds_l normalized surface area measure on S_l. For f defined on R^d set

Al f = f * dsl.

Then there is the following classical theorem.

Theorem A: If d ³ 2 and p > d/(d-1),

|| sup l  |Al f |   || Lp (Rd) £ C || f||Lp(Rd) ,

Theorem A is due to E. M. Stein if d ³ 3 and J.  Bourguin if d = 2.

We will discuss recent joint work with A. Magyar and E. M.  Stein on a discrete analogue of Theorem A. Let f be a function defined on Z^d (the points in R^d with integer coordinates). Denote by r(l) the number of lattice points (n₁, ¼, n_d) with n²₁ + ¼+ n²_d = l, and for m in Z^d, set

Bl f(m) = (r(l))-1 å [(n Î Zd), ( |n|2 = l)]  f(m-n).

Then we have the following recent theorem.

Theorem B: (A. Magyar, E. M. Stein, and S. Wainger) Suppose p > d/(d-2> and d ³ 5, then

|| sup l  |Bl f|   ||lp(Zd) £ C || f||lp(Zd) .

Surveys in Mathematics / Études en mathématiques
Jacques Hurtubise (McGill)
Integrable systems and algebraic geometry

Niky Kamran (McGill)
Recent progress in the study of the Dirac operator in black hole geometries

The rigorous analytical study by J.Smoller, A. Wasserman, S.T. Yau and collaborators of the gravitational collapse of matter into a black hole is at the heart of one of the most interesting current research programs in mathematical physics. One of the striking results in this line of work is a theorem due to F.Finster, J.Smoller and S.T.Yau, who have shown that the Dirac equation in a spherically symmetric black hole admits no time-periodic solutions which are L² on space-like hypersurfaces. This means that a Dirac particle will either "fall into" the black hole singularity, or escape to infinity, but that it cannot stay on a periodic orbit around the black hole. This result is quite surprising, because it is well-known that there exist closed time-like geodesics in spherically symmetric black hole geometries. We will give a motivated introduction to the main themes of this research program and present, if time permits, some of the results that we have recently obtained in collaboration with Finster, Smoller and Yau in the axisymmetric case. This can be thought of as the generic case in view of the black hole uniqueness theorems of Carter and Israel.

James Lewis (Alberta)
Algebraic cycles and Hodge theory

The subject of Algebraic Cycles has been influenced to a large extent by developments in Hodge theory and Algebraic K-theory. I would like to explain some of these developments, and in particular how the methods of Hodge theory have played a role in refining our understanding about some of the outstanding problems in this area. This talk will be aimed at a general audience.

Neal Madras (York)
Some simple models of phase transitions

The simplest example of a phase transition is percolation, which may be described as follows: Each edge of the graph Z^d is coloured black with probability p, independently of all other edges. Does the subgraph of black edges have an infinite black component? The answer depends sensitively on the value of p. More generally, we say that a system exhibits a phase transition if a small change in a ``microscopic'' parameter causes a ``macroscopic'' qualitative change in the systems (e.g., from many small components to one infinite component in percolation; from liquid to solid in water). In this talk I shall describe some current research and open questions concerning phase transitions in percolation and in some simple stochastic spatial population models.

Heydar Radjavi (Dalhousie)
Operator semigroups and reducibility questions

A family S of bounded operators on a Banach space X is said to be reducible if there is a non-trivial closed subspace of X invariant under every member of S. (Non-trivail means other than {0} and X.) If there is a maximal chain of such subspaces, then S is said to be (simultaneously) triangulaizable. If X has finite dimension, for example, this means a chain of invariant subspaces starting with dimension one and increasing one dimension at a time. Triangularizability is a generalization of commutativity, and is linked to various partial spectral mapping theorems and to properties of spectral radii and traces. The case where S is a semigroup, i.e., a family closed under multiplication, has received a great deal of attention in the last decade or so. The links to spectral conditions have been examined and, in particular, many classical finite-dimensional results have been maximally extended. Some of these development will be discused.

Damien Roy (Ottawa)
Diophantine approximation and transcendental numbers

The object of the talk is to review recent results in the area together with their applications: the subspace theorem of W. M. Schmidt and its generalizations, the theorem of the algebraic subgroup of M. Waldschmidt, the zero estimates of D. W. Masser, G. Wüstholtz and P. Philippon, the criterion of algebraic independence of P. Philippon, and the algebraic independence of p, e^p and G(1/4) by Yu. V. Nesterenko.

Dana Schlomiuk (Montreal)
Analytic, geometric and algebraic methods, intertwined in the study of planar vector fields

The study of planar vector fields was initiated by Poincare who introduced many of the basic notions ( limit cycle, return map, homoclinic connection, etc ), a new qualitative aproach to differential equations and founded the field of dynamical systems. In studying dynamical systems we are interested in the behaviour of all solutions for all future and past times, in a region of the phase space. This makes it natural to opt for a geometric viewpoint along with an analytic one. Problems on planar vecor fields are easy to state and very hard to solve. Thus three problems stated a century ago are still open in this area. The subject has advanced at a slow pace with many setbacks. Thus in the early eighties Il'yashenko discovered an error in a proof given in 1923 by Dulac, of the statement that any planar vector field defined by polynomials over the reals has a finite number of limit cycles. This statement was later proved indepenedently by Il'yashenko and Ecalle. There is now a lot of activity in this area and while some new results were obtained by analytic methods only, more and more an integrated approach of analytic, geometric and algebraic methods is used. In this lecture we shall survey some of these developments which were obtained by a truly interdisciplinary approach. Part of the newest and more exciting develpments are of this nature, some of these are based on beautiful geometric ideas of Khovansky or on differential algebraic methods (Mourtada's latest work).

Nicole Tomczak-Jaegermann (Alberta)
Infinite-dimensional Geometry of Banach Spaces

We shall present several purely infinite-dimensional geometric and structural phenomena in Banach spaces that have been discovered and developed in recent years.

Ravi Vakil (MIT)
Gromov-Witten theory and enumerative geometry

Kontsevich's moduli space of stable maps (and quantum cohomology and Gromov-Witten theory) has had far-reaching consequences in many fields, including algebraic geometry, mathematical physics, and symplectic topology. Some of the consequences of this point of view can already be seen in previously-unsuspected patterns in the enumerative geometry of plane curves and covers of the projective line, and in intersection theory on the moduli space of curves. Much recent work in this area has been motivated by various conjectures and results implying that this enumerative geometry and intersection theory is somehow related to modular forms on one hand, and the Virasoro algebra on the other. The conjectures and theorems (due to Witten, Kontsevich, Gottsche, etc.) and background will be sketched, mainly in the context of explicit examples.

Education / Éducation
Peter Crippin (Waterloo)
Creative Problems from Competitions

This presentation will consider creative aspects of competitions and discuss ways in which they encourage students to function as aspiring mathematicians.Specific problems from recent competitions will be considered, with discussion about surprising and fascinating originality required in their solutions. Also included will be suggestions for helping teachers make the best possible use of problems with their students.

Ron Dunkley (Waterloo)
Competitions: More than student identifiers

The role of competitions in identifying and challenging bright students is well recognized. This presentation will consider the further role of competitions in the development of students on a wider scale and in helping teachers develop problem solving activities in their classrooms. Included will be suggestions for materials they need. Also included will be comments on the role of multiple choice tests in comparison with papers requiring full solutions.

Tony Gardiner (Birmingham)
How national mathematics competitions support the mathematics community.

We shall present examples and data covering the last 10 years to illustrate how participants in the mathematics competition pyramid in the UK have fed through into undergraduate and postgraduate courses. We shall identify potential strengths, limitations and pitfalls of mathematics competitions and will pinpoint key principles which allow one to exploit the strengths while avoiding the pitfalls.

Sava Grozdev (Bulgaria)
The influence of the Mathematics competitions in Bulgaria to education

It is proposed a review of Mathematics competitions and olympiads in Bulgaria as well as of international ones with the participation of Bulgarian students. The preparation for above events is analized and the participation of University professors and scientists in it is examined. Basic role in the preparation is played by the special Mathematics schools which exist in all bigger Bulgarian cities. Conclusions are done concerning the rise of the level in Mathematics Education and the corresponding perspectives.

Rita Janes (NCTM Director)
The Newfoundland Senior High Mathematics League - A Cooperative Approach

This talk will outline the objectives of the League, and how for the past thrteen years these objectives have been met. There will be a sharing of the experiences, that not only has made this league an enrichment activity, but also has provided a friendly and motivating environment for students to do mathematics problem solving and for teacher coaches to meet and discuss issues of importance to them. The advantages and limitations of the mathematics league will also be discussed.

David Leeming (Victoria)
MATHCOUNTS - Challenges and Opportunities

MathCounts is a math program at the junior high school level implemented jointly (in the United States) by the National Society of Professional Engineers and the National Council of Teachers of Mathematics. The program has been very successful in British Columbia since 1989 but has not caught on in the other provinces. In this talk, I will describe the MathCounts program, how it operates in British Columbia, and its long-term benefits to the discipline. I will also describe the partnerships which need to be in place in order to implement a successful provincial MathCounts program.

John Grant McLoughlin (MUN)
Popular distractors: one avenue into the mathematical thinking underlying errors in math contests

The response patterns within questions in multiple choice mathematics contests offer more than a means of generating final results. Large percentages of blank answers (nonresponses) may point to a high level of difficulty with a particular question. Selections of distractors point to a different difficulty, namely, that of a mathematical error in the problem solving process. Different distractors tend to suggest varying interpretations or processing errors due to computational, algebraic, or definitional circumstances. The paper focuses attention on distractors that were preferred over the correct answer according to the summaries of answer patterns of the 1995-96 and 1996-97 Flanders Mathematics Olympiad. Consideration of these "popular distractors" within the context of the specific contest problems proves to be a valuable indicator of processing errors and pertinent content knowledge issues. The analysis may serve as a springboard for closer examination of the results of other contests.

Shannon Sullivan (MUN student)
Why I Hated Achilles

This talk will examine high school math competitions from the perspective of an individual who has been on both sides of the floor, first as a student participant and subsequently as a volunteer helping to run the competition. I will discuss how the competitions were viewed by students when I was in high school, and how my own opinion has evolved since that time. I will also offer my thoughts on how the competitions benefitted me, both in the short-term and in the long-term, and what direction they might take in the future.

Jim Totten (University College of the Cariboo)
BC Colleges High School Math Contest

This contest, which is effectively a provincial high school mathematics contest, grew out of three distinct regional contests and now covers most of the province. Before discussing the contest, what it is about, how it is organized, and what it seems to be accomplishing, we will do a little study of the demographics of British Columbia and talk about post-secondary education in the province. At least part of the basis for the contest is found there. We will then look at the type of students we are trying to reach, what we are trying to accomplish, how we got to where we are today, what the future may hold and whether we have been successful in our objectives. I am also hoping to get some ideas from the audience to take back to the rest of the contest organizers for future use.

The education session will also host a panel discussion titled "From High School to University Mathematics: A Smooth Transition or Horrible Discontinuity" is planned. Panelists include Dr. Richard Nowakowski (Dalhousie), Dr. Cathy Baker (Mount Allison) and Dr. Herb Gaskill (MUN).

Combinatorics and its applications / Combinatoire et applications
Lynn Batten (Manitoba)
On Chen's construction of minimum distance five codes

In 1991, C.L. Chen introduced a new construction of minimum distance five codes. We examine this construction in depth and show that no codes are obtained from it unless the fields used are GF(8) or GF(32). Using Magma, we prove that the binary [11,4,5] code and the binary [15,7,5] extension found by Chen are the only possible such codes over the former field. We show also that precisely three non-equivalent binary [47,36,5] codes arise from the latter field, each resulting in the same extension to a [63,51,5] code.

Frank Bennett (Mount Saint Vincent)
Steiner Pentagon Packing and Covering Designs

Let K_n denote the complete undirected graph on n vertices. A Steiner pentagon system (SPS) of order n is a pair (K_n, B), where B is a collection of edge-disjoint pentagons which partition K_n, and such that every pair of distinct vertices of K_n is joined by a path of length two in exactly one pentagon of B. A Steiner pentagon packing (covering) of order n is a pair (K_n,B) where B is a collection of pentagons from K_n such that any two vertices are joined by a path of length one in at most (at least) one pentagon of B, and also by a path of length two in at most (at least) one pentagon of B. If no other such packing (covering) has more (fewer) pentagons, the packing (covering) is said to be maximum (minimum) and the number of pentagons in a maximum packing (a minimum covering) is called the packing number (the covering number), denoted by p(n) (c(n)). The packing problem is to determine the values of p(n) for all n ³ 5. Similarly, the covering problem is to determine the value of c(n) for all n ³ 5. It is fairly well-known that the spectrum of SPSs is precisely the set of all n º 1 or 5 (mod 10), except n = 15, for which no such system exists, and the functions p(n) and c(n) have been determined for all such values of n. For other values of n, some progress has been made with regards to the determination of p(n) and c(n) and the current state of affairs will be presented.

Jeff Dinitz (Vermont)
Bicoloring Triple Systems

A coloring of the points in a block design is a bicoloring if the points in each block are from exactly two color classes. In this talk we discuss some necessary conditions for the existence of a bicoloring of a triple system and give some recursive constructions for bicolored triple systems.

Katherine Heinrich (SFU)
Uniform Coverings of 2-Paths

A uniform l -covering of the 2-paths of a graph G is a collection of isomorphic subgraphs of G which cover every 2-path of G l times. For example, a Steiner quadruple system on n points is a 1-covering of the 2-paths of K_n by subgraphs isomorphic to K₄. In this talk, which is joint work with Dennis Langdeau and Helen Verrall, we will present results on the existence of such coverings when G is a complete graph or a complete bipartite graph.

Vaclav Linek (Winnipeg)
Large odd periodic and generalized Skolem sequences

A generalized Skolem sequence is a sequence of positive integers and null symbols, called holes, such that if an integer j appears in the sequence then it appears exactly twice with j-1 symbols separating the two appearances. Thus 3_232411_4 is a generalized Skolem sequence, ("_" is the null symbol). In this talk we discuss the existence question for sequences with two holes in which the consecutive integers from 1 to n appear. Certain special sequences comprised mainly of odd integers, called odd periodic sequences, play an important role in this investigation.

Alan Ling
Difference Triangle Sets

In this talk, I will define and survey some results on difference triangle sets.

William Martin (Winnipeg)
From synchronization to integration: ordered Codes and ordered orthogonal arrays

There is a well-known and very fruitful connection between error-correcting codes and orthogonal arrays. A recent paper of Stinson and Martin establishes a surprising analogue relating ordered orthogonal arrays and ordered codes. This talk will outline this new theory, including the introduction of a new family of self-dual association schemes. We will mention several applications of these structures such as an application of ordered codes to binary channels subject to synchronization errors and an application of ordered orthogonal arrays to numerical integration.

Rolf Rees^* and Nabil Shalaby (MUN)
Constructing starters with `Holes'

A holey starter is a quadruple (n,R,D,P) where n is an odd integer, R is a set of residues in Z_n \{ 0 }, D is a set of non-zero differences in Z_n satisfying -D =D, and P is a partition of (Z_n \{ 0 })- R into pairs (a_i, b_i) such that {± (a_i - b_i ) : (a_i, b_i) Î P } = (Z_n \{ 0})- D. Note that for some 0 £ k £ ( n-1) /2 we have |R| = |D| = 2k and |P| = (n-1-2k)/2. We discuss how such objects arise and some methods for their construction.

Chris Rodger (Auburn)
Maximal Sets of Hamiltonian Cycles

If S is a set of edge-disjoint hamilton cycles in G with the property that G - {e|e is an edge in a cycle in S} contains no hamilton cycles, then S is said to be a maximal set of hamilton cycles in G. In this talk, {m|there exists a maximal set of m hamilton cycles in K_n,n} is found. Progress towards settling the companion problem for all complete multipartite graphs is also discussed. The proof uses the method of amalgamations.

Alex Rosa (McMaster)
Orthogonal double covers and symmetric graph designs

Symmetric graph designs were introduced recently as a common generalization of symmetric balanced incomplete block designs, and of orthogonal double covers. We discuss some recent results on various classes of symmetric graph designs, including symmetric graph designs on friendship graphs, and Peter Cameron's characterization of doubly transitive symmetric graph designs. Several open problems are outlined as well.

Ruizhong Wei (Waterloo)
Constructions of Balanced Sampling Plan Excluding Contiguous Units

Balanced sampling designs for the exclusion of contiguous units (or BSEC for short) were first introduced by Hedayat, Rao and Stufken in 1988. These designs can be used for survey sampling when the units are arranged in one-dimensional ordering and the contiguous units in this ordering provide similar information. This design can be described as a pair (X, B), where X is a set of v points in cyclic ordering and B is a collection of k-subsets of X called blocks such that any two contiguous points do not appear in any block while any noncontiguous points appear in exactly l blocks. The properties and constructions of these objects were discussed in several papers, however, the existence problem of these designs is still wide open. For example, no infinite class of such designs is known for fixed block size k and fixed l. In this paper, we discuss the constructions of these designs in view of combinatorial design theory. Among other constructions, we use Langford sequence to construct all the possible BSEC with block size 3 and l = 1, thus give the necessary and sufficient conditions of such designs. Then we generalize this concept to two-dimensional situation and give some constructions by using modified group divisible designs. Some equivalent objects and open problems are also mentioned.

Nonlinear Analysis / Analyse nonlinéaire
Giampietro Allasia (Torino)
Cardinal basis interpolation on multivariate scattered data

Some problems relating to theoretical properties and numerical computation of multivariate cardinal basis interpolants at scattered data are outlined. It is shown that cardinal basis interpolants achieve a noteworthy performance in computation, being particularly suited to parallel, multistage and recursive procedures. As a straightforward consequence, adaptive interpolation is made very easy.

Jonathan Borwein (SFU)
Generic behaviour of generalized gradients

I will give an overview of recently discovered methods for determining that a multifunction is the generalized gradient of some Lipschitz (or continuous) function. Particular applications of this result include that: (i) in every Banach space "most" Lipschitz functions have a Clarke subgradient which is a constant multiple of the unit ball at every point; (ii) in every smoothable Banach space this is true for the limiting subgradient of Ioffe, Mordukhovitch-Kruger et al.

This is joint work by J. Borwein, W. Moors and X. Wang.

Raffaele Chiappinelli (Siena)
Nonlinear stability of eigenvalues of compact self-adjoint operators

We consider the eigenvalue problem Tu+N(u) = lu in a real Hilbert space H, where T is a linear compact self-adjoint operator in H and N is a positively homogeneous, completely continuous gradient operator in H. We prove that given any eigenvalue l₀ ¹ 0 of T, then for any N as above with ||N || < d/2 there is an eigenvalue l of T+N with |l-l₀ | £ ||N ||. Here

||N || = sup {||N(u) ||: ||u || = 1} - as for linear operators - and

d = d(l₀) = dist (l₀, s(T) \ {l₀}) denotes the isolation distance of l₀ in the spectrum s(T) of T. To prove this, we make use of a Constrained Saddle Point Theorem for the "quadratic form" Q(u) = (Tu,u)+(N(u),u) on the unit sphere S = {u Î H : ||u || = 1 }. Applications are given to a class of semilinear elliptic problems in bounded domains of Rⁿ.

AMS 1991 Mathematics Subject Classification: 47H12, 58E05, 35P30, 35J65.

Hanuman P. Dikshit (Bhopal)
Birkhoff interpolation on some perturbed roots of unity

Recently, it has been observed that almost all the results of interpolation on the unit circle, deal with the case of equidistributed nodes which except for rotation, are equivalent to some roots of unity. Xie Siquing, de Bruin, Sharma and Szabados and some others have considered the regularity of the problem of Birkhoff interpolation on non equidistributed nodes. The object of this talk is to examine the regularity of (0,m) and (0,1,...,r-2,r) interpolation on the zeros of (zⁿ+1)(z-z). We are indeed able to determine necessary and sufficient condition on the choice of z to ensure the regularity. Some other problems of interpolation which in particular, include corresponding earlier results are also proved.

Massimo Furi (Florence)
A new extension of the Leray-Schauder degree theory

Given a Fredholm operator of index zero between vector spaces L:E ® F L:E ® F, we say that a linear operator A:E ® F is a corrector of L if its image is finite dimensional and L+A is an isomorphism. We show that the set of correctors of L can be partitioned in just two equivalence classes, each of them is called an orientation of L. We prove that in the context of Banach spaces this purely algebraic notion of orientation is stable; in the sense that an orientation of a bounded Fredholm operator of index zero induces a natural orientation on any sufficiently close operator. These concepts are used to develop an oriented degree theory for a class of maps between Banach manifolds which extends the celebrated Leray-Schauder theory, as well as the theory of Elworthy-Tromba.

(The material is taken from some joint works with P.L. Benevieri)

Paul Gauthier (Montreal)
Regular quasiregular mappings

The variational integral ò|Ñu |ⁿ is strongly related to qr-mappings. Extremals are solutions of the associated Euler-Lagrange equation, which, for dimension n > 2, is nonlinear. Although for n > 2 one does not usually pay much attention to regular (smooth) qr-mappings because they are rather rigid, this talk will in fact focus on highly regular (harmonic) qr-mappings.

Kazimierz Goebel (Lublin)
Ways to retract balls onto spheres in Banach spaces

Let X be an infinite dimensional Banach space with the unit ball B and the unit sphere S. In contrary to the finite dimensional case, S is the retract of B. It means that there exists a continuous mapping  R: B® S  such that  Rx = x  for  x Î S. Such retraction can even be lipschitzian. There is an intriguing problem of finding retractions with smallest possible Lipschitz constant.

In this presentation we discuss some recent evaluations and constructions related to this problem.

Jinlu Li (Shawnee State U.)
Ky Fan's Best Approximation in Banach spaces

In this paper, we will use the techniques of KKM mapping and Fan-KKM Theorem to prove a very general theorem (the main theorem of this paper) which is an extension of the Fan KKM Theorem and we will make some applications of this theorem on the study of variational inequalities, approximation theory and fixed point theory.

M. Zuhair Nashed (Delaware)
Nonlinear ill-posed problems with Monotone Operators

In this talk I will present some recent results on iterative and projection methods for nonlinear ill-posed problems involving monotone operators in Hilbert spaces. Some applications and open problems will be indicated briefly.

M. Aslam Noor (Dalhousie)
Auxiliary principle techniques and variational inequalities

The auxiliary principle technique is used to obtain an equivalent differentiable optimization problem for a class of variational-like inequalities. This equivalent formulation enables us to construct a merit(gap) function for variational-like inequalities, which allows us to develop a unified descent framework for solving various classes of variational inequalities. An open problem is solved. Our results represent an improvement of known results.

Luigi Rodino (Torino)
Local solvability for nonlinear partial differential equations

We study the problem of the local solvability for general nonlinear partial differential equations in the frame of the Gevrey classes, reporting on the results of [1]. We also address to equations with multiple characteristics in standard/ weighted Sobolev spaces; by combining fixed point techniques and a priori estimates, we obtain results of local solvability for semi- linear and fully nonlinear equations.

[1] T. Gramchev, L. Rodino "Gevrey solvability for semilinear partial differential equations with multiple characteristics", Boll. Un. Mat. It., Ser.VIII, 2-B (1999), 65-120.

J.N. Singh (Nigeria)
Interior point methods for $L_p$ and their consequences in nonlinear optimization

In this paper we discuss some recent interior point methods for linear programming and examine their relations with nonliner optimization.

S.N. Singh (South Africa)
Solutions of boundary value problems by double Laplace Transformations

In solutions of boundary value problems, previous researchers have applied Laplace transformations with respect to one variable only. In the present paper, we have applied Laplace Transformations with respect to both the variables in a partial differential equation involving functions of two variables and have obtained the solutions with greater ease.

Pramila Srivastava (Allahabad)
Fuzzy Dynamical Systems and Entropy

Papagudi Subrahmanyam (India)
Applications of fixed point theorems for the solution of some functional equations

In this talk, applications of fixed point theorems for the solution of some functional equations are discussed. In particular,existence of solutions to simultaneous functional equations, generalizing a result of De Rham(1956/57) is discussed.

Wataru Takahashi (Tokyo)
Approximating fixed points and applications

We first prove fixed point theorems for families of nonexpansive mappings in a Banach space. Next, we prove nonlinear ergodic theorems of Baillon's type for nonlinear semigroups of nonexpansive mappings. In particular, we give an answer to the open problem posed during the Second World Congress on Nonlinear Analysts, Athens, Greece, 1996, by extending Takahashi's result and Rode's result to a Banach space for an amenable semigroup of nonexpansive mappings. Further, we establish weak and strong convergence theorems of Mann's type and Halpern's type for families of nonexpansive mappings. Finally, using these results, we discuss the problem of image recovery by convex combinations of nonexpansive retractions and the convex minimization problem of findind a minimizer of a convex function.

E. Tarafdar (Queensland)
Some Fixed Point Theorems in Uniform Topological Spaces and in Partially Ordered Sets

Some fixed point theorems for contraction mappings on uniform topological spaces including the Cantor Intersection Theorems will be presented. A generalized Knaster-Tarski theorem will also be presented and its possible applications will be discussed. Some results may not be new.

Representation Theory / Théorie des représentations
Jonathan Brundan (Oregon at Eugene)
A modular analogue of Zelevinsky's branching rule for GL_n(F_q)

In his 1981 book, Zelevinsky proved a branching rule for the restriction of an irreducible character of the finite general linear group GL_n+1(F_q) to the affine general linear group AGL_n(F_q) (embedded in a natural way as an ``almost'' parabolic subgroup). The multiplicities appearing in his branching rule are ``the same'' as in the classical branching rule of an irreducible polynomial representation of GL_n+1(C) to the subgroup GL_n(C).

In recent work using the Dipper-James theory of cross-characteristic representation theory of GL_n(F_q), we have recently obtained a generalization of this fact. We show that the branching multiplicities in the restriction of an irreducible GL_n+1(F_q)-module to AGL_n(_q) are ``the same'' as the multiplicities in the branching rule of an irreducible polynomial representation of quantum GL_n+1 (over the cross-characteristic field and at root of unity q). The result motivates some curious combinatorics for computing a certain basis of the Hall algebra which is useful when calculating the Brauer character values of GL_n(F_q) at unipotent elements. For example, we are able to give a simple closed formula for the dimension of an irreducible modular representation of GL_n(F_q) in terms of the (unknown!) weight multiplicities of the quantum linear groups.

Ranee Brylinski (Pennsylvania State)
Quantization of nilpotent orbits associated to symmetric spaces

I will talk about the geometry and quantization of certain nilpotent coadjoint orbits associated to symmetric spaces.

Clifton Cunningham (Massachusetts)
Topological nilpotence and the Fourier transform on a p-adic Lie algebra

This talk introduces a family of sets, defined by way of Moy-Prasad filtrations, which are contained in the set of topologically nilpotent elements in a classical p-adic Lie algebra. Although their definition is mainly combinatorial, these sets reflect properties of certain affine Springer fibres; in particular, they often coincide with the open sets used over ten years ago by Kazhdan and Lusztig to show the importance of affine Springer fibres to p-adic representation theory. To illustrate the utility of our sets, we use them to produce results concerning the Fourier transform of distributions arising naturally in representation theory.

Sam Evens (Arizona)
Poisson geometry, a theorem of Kostant, and Schubert calculus

I will explain some joint work with Jiang-Hua Lu where we exploit a Poisson structure on the flag manifold whose symplectic leaves are the Schubert cells to interpret and simplify some work of Kostant on n-cohomology. We also use Poisson geometry to simplify work of Kostant and Kumar, and as a result recover formulas for the structure constants for the Schubert basis of cohomology of the flag manifold without using BGG operators. The argument is partly based on regarding this Poisson structure as a deformation of a family of symplectic structures. This deformation occurs inside the compactification of a symmetric space of a complex semisimple Lie group, which we will explain if time permits.

Aloysius G. Helminck (North Carolina)
On orbit closures of symmetric subgroups in flag varieties

We study K-orbits in G/P where G is a complex connected reductive group, P Í G is a parabolic subgroup, and K Í G is the fixed point subgroup of an involutive automorphism q. Generalizing work of Springer, we parametrize the (finite) orbit set K\G/P and we determine the isotropy groups. As a consequence, we describe the closed (resp. affine) orbits in terms of q-stable (resp. q-split) parabolic subgroups. We also describe the decomposition of any (K,P)-double coset in G into (K,B)-double cosets, where B Í P is a Borel subgroup. Finally, for certain K-orbit closures X Í G/B, and for any homogeneous line bundle L on G/B having nonzero global sections, we show that the restriction map res_X:H⁰(G/B,L)® H⁰(X,L) is surjective and that Hⁱ(X,L) = 0 for i ³ 1. Moreover, we describe the K-module H⁰(X,L). This gives information on the restriction to K of the simple G-module H⁰(G/B,L). Our construction is a geometric analogue of Vogan and Sepanski's approach to extremal K-types.

This is joint work with M. Brion.

Marcus Hunziker (Brandeis)
Equivariant D-modules, Weyl group representations, and nilpotent orbits

There is a remarkable correspondence between nilpotent orbits in a semisimple Lie algebra and Weyl group representations, the so-called Springer correspondence. Using recent results on invariant differential operators, we will explain a new approach to this correspondence from a geometric/invariant theoretic point of view.

Alexander S. Kleshchev (Oregon at Eugene)
Irreducible subgroups of the symmetric group

Let F be an algebraically closed field of characteristic p ³ 0, and S_n be the symmetric group on n letters. We present the solution of the following problem for the case p ¹ 2,3.

Problem. Classify the pairs (G,D) where G < S_n is a subgroup, and D is a simple FS_n-module with the restriction D¯_G irreducible.

The case p = 0 has been settled by Saxl in 1987. The modular version of the problem has applications to maximal subgroups in finite classical groups. In the special case G = S_n-1 the problem is equivalent to a conjecture of Jantzen and Seitz, and the special case G = A_n requires the Mullineux conjecture.

Friedrich Knop (Rutgers)
On multiplicity free spaces

Let G be a connected complex reductive group. A finite dimensional representation V is called multiplicity free if every simple G-module appears in the algebra P of polynomial functions with multiplicity at most one. Multiplicity free representations form a very special cases of so-called spherical varieties.

One characterization of (affine) spherical varieties is that their ring PD^G of invariant differential operators is a polynomial ring. More precisely, there is a Harish-Chandra isomorphism h: PD^G® C[a^*]^W₀ which associates to each operator its ``radial part''. Here, W₀ is a certain finite reflection group acting on the vector space a^*.

Due to the additive structure of a multiplicity free representation, its ring of invariant differential operators is equipped with a distinguished basis, the Capelli operators. Their radial parts form therefore a basis of the W₀-invariants.

It turns out that for the ``classical'' multiplicity free representations, these radial parts are connected with Jack polynomials. We will report on recent progress on two series of non-classical multiplicity free representations. Both consist of matrix-vector pairs. There is an explicit system of difference operators of which the radial part polynomials are eigenfunctions. This implies many nice properties like Pieri formulas, binomial formulas, triangularity, duality etc.

Venkatramani Lakshmibai (Northeastern)
Some affine varieties related to Schubert varieties

Let G be a semi-simple group, T a maximal torus in G, and B a Borel subgroup, B É T. Let W( = N(T)/T) be the Weyl group of G. For w Î W, let X(w) = [`BwB](mod B) be the Schubert variety in G/B associated to w. Let B^- be the Borel subgroup opposite to B, and let O^- be the B^- orbit B^-e_id, where e_id is the point in G/B corresponding to the coset B. Let Y(w) = X(w)Ç O^-. We conclude the geometric properties - normality and Cohen-Macaulayness - for two classes of affine varieties, namely, the ladder determinantal varieties and the Quiver varieties, by identyfying them with suitable Y(w)'s.

George McNinch (Notre Dame)
Semisimplicity of Exterior Powers of Semisimple Representations of Groups

This talk will discuss the author's solution of a problem posed by Jean-Pierre Serre: the author has shown that if V is a semisimple finite dimensional representation of a group G over a field K of characteristic p > 0, and m(dim_K V -m) < p, then the mth exterior power of V is again a semisimple representation of G.

The proof involves a reduction to the setting of rational representations of simple algebraic groups in characteristic p; some of the techniques will be discussed in the talk.

Fiona Murnaghan (Toronto)
Germs of characters of admissible representations

Let F be a non-archimedean local field of characteristic 0. In this paper we study a correspondence between representations of symplectic groups Sp(n, F), and special even-orthogonal split groups SO(2r, F), where r ³ 2. Let w_n,r be the Weil representation of Sp(2nr,F) attached to a non-trivial additive character y_F of F. We show that the correspondence arising by restricting the Weil representation w_n,r to Sp(n,F) ×SO(2r,F) is functorial for generic square integrable representations.

This is joint work with Goran Muic.

Arturo Pianzola (Alberta)
Line bundles and conjugacy theorems

We explore a connection between line bundles of $Spec R$ and Cartan subalgebras of Lie algebras of the form $R\otimes g.$ Examples of this scenerario are the simply-laced Kac-Moody Lie algebras, as well as their many generalizations (toroidal, extended affines etc.)

Yasmine Sanderson (Rutgers)
On the connection between Macdonald polynomials and Demazure characters

We show that a certain specialization of nonsymmetric Macdonald polynomials gives ,up to multiplication by a simple factor, characters of Demazure modules for affine sl(n). This connection nishes Lie-theoretic proofs of the nonnegativity and monotonicity of Kostka polynomials.

Gordan Savin (Utah)
Symplectic-orthogonal theta lifts of generic discrete series

Let F be a non-archimedean local field of characteristic 0. In this paper we study a correspondence between representations of symplectic groups Sp(n, F), and special even-orthogonal split groups SO(2r, F), where r ³ 2. Let w_n,r be the Weil representation of Sp(2nr,F) attached to a non-trivial additive character y_F of F. We show that the correspondence arising by restricting the Weil representation w_n,r to Sp(n,F) ×SO(2r,F) is functorial for generic square integrable representations.

Eric Sommers (Harvard)
Functions on Nilpotent Orbits

For a nilpotent orbit in a complex semisimple Lie algebra, we survey some known and conjectured properties of the structure of the ring of (algebraic) functions on the orbit. We will discuss connections with the Springer theory for Weyl group representations and unitary representations of the corresponding complex Lie group.

Peter Trapa (Princeton)
Genuine characters of the metaplectic group

A fundamental symmetry in the representation theory of linear Lie groups is Vogan's character multiplicity duality theorem. The duality, which depends crucially on the hypothesis of linearity, can be thought of as a symmetry of the Kazhdan-Lusztig-Vogan algorithm for computing composition series of standard Harish-Chandra modules --- for instance, in the highest weight category, it manifests itself as a well-known relationship between the coefficients of the Kazhdan-Lusztig polynomials P_x,y and P_wox,woy. Here we establish an analogous duality on the set of genuine representations of the (nonlinear) metaplectic group (subject to some restrictions on infinitesimal character). The main tool is a sheaf-theoretic understanding of certain exotic translation functors.

This is joint work with David Renard.

Graduate session/ Séance pour étudiants diplômés
Peter Arpin (Manitoba)
Minimal Representations of Finite Distributive Lattices

For every finite distributive lattice D there is a finite lattice L whose congruence lattice, denoted by Con(L), is congruent to D. For a finite distributive lattice D we let M(D) = min {|L|:L a lattice and Con(L) is congruent to D}. We will consider M(C) where C is a chain.

Benoit Charbonneau (Université du Québec à Montréal)
The famous Riemann-Roch theorem

Since 1854, approaches to the Riemann-Roch theorem have greatly evolved. Many different versions of the theorem and of the proof exist and this theorem is the key to many mathematical problems. We will see how new mathematical methods have given us new insight into the theorem.

Michelle Davidson (Manitoba)
Extensions of BCH Codes

In 1991, Chen published a paper giving an extension of distance 5 BCH codes. I will discuss this construction and talk about attempting to generalize these results to extensions of distance 7 codes.

Ronald Haynes (SFU)
Invariant Manifolds for Dynamical Systems: theory and computation

The study of complex mathematical models of physical systems has prompted increased interest in the theory and techniques of dynamical systems. Many systems of Ordinary Differential Equations (ODEs) have the property that after some transient time solution trajectories are attracted to a low-dimensional invariant manifold. Simple examples are steady-state or equilibrium solutions, or periodic solutions. After introducing some key concepts in dynamical systems we present a brief survey of various computational approaches for approximating these invariant manifolds. One such technique pioneered by DiecI, Lorenz, and Russell involves the computation of an invariant torus as the solution of an associated hyperbolic partial differential equation (PDE). We discretize this PDE with a pseudospectral method and linearize with Newton's method. The linear system of algebraic equations for the Newton step can become quite large and is fairly sparse and structured. It is solved by various iterative methods. Numerical results are presented for the forced van der Pol equation.

Dave Morgan (MUN)
Graceful labelings of the complete k-ary tree

Given a graph G, a graceful labeling of G is an injective mapping f:V(G)-->{0,1,...,|E(G)|} such that the associated mapping f':E(G)-->{1,2,...,|E(G)|} defined by f'(uv)=|f(u)-f(v)| is a bijection. A graph that has a graceful labeling is called graceful.

The conjecture by Ringel and Kotzig that all trees are graceful has stood for over 30 years. Several families trees have been shown to be graceful, and in particular we will look at the gracefulness of the complete k-ary tree. An inductive approach will incorporate the graphting techniques introduced by Stanton and Zarnke which create graceful trees from smaller graceful trees. As well, we will discuss the relation of graceful labeling trees to the cyclic decomposition of complete graphs.

Mike Newman (Manitoba)
Graph isomorphism and refinement

Most current graph isomorphism algorithms are based on the method of partition refinement to produce a certificate for a given graph. We will consider a modification of this algorithm based on dividing the graph and applying refinements seperately.

This is a joint project with Professor Kocay.

Jianhua Pan (MUN)
Simplification and error analysis for moving Finite element

The finite element method involves the establishment a mesh at which one finds an approximate solution to a given PDE. A global approximation to the original problem can then be derived using piecewise polynomials. However, many practical time-dependent PDE's have moving steep fronts or large gradients making classical finite elements with fixed mesh inefficient. The idea of using a moving mesh is one way of dealing with the problem. Such a moving finite element scheme (MFE) gives rise to nonlinear ordinary differential equations for the mesh coupled to equations from the approximation of the original problem.

This paper will present a simplified MFE scheme, where we managed to decouple the ODE system for the mesh from the rest of the problem. Complications arising from inclusion of various penalty terms required to regularize mesh topology are also examined. Overall, we produce schemes whose computational cost is considerably reduced. Finally, a priori and a postpriori error estimate are derived.

Wade Parsons (MUN)
Waveform Relaxation Methods for Volterra Integro-differential Equations

This talk gives a brief outline of the Ph.D. thesis I submitted in February of this year. It concerns the solution of systems of Volterra integro-differential equations by the application of waveform relaxation methods. This is a timely topic since such methods can often be implemented efficiently on parallel architectures. I give convergence results for both the regular kernel and the weakly singular kernel cases, and although my primary concern is with numerical methods, both analytic and numerical solutions are considered.

Contributed papers / Communications libres
John Baker (Waterloo)
The Dirichlet Problem for Ellipsoids

This talk concerns a relatively simple (and perhaps somewhat novel) proof of the well-known theorem: Given an open ellipsoid, E, in Rⁿ and a continuous f: ¶E ® R, there is a unique continuous u:$\bar{E}$ ® R such that u is C² on E,

n å k = 1  ¶2u / ¶xk2 = 0    on     E    and    u(x) = f(x)    for all     x Î ¶E.

The proof involves two steps. In the first step the Theorem is proved in the case of polynomial f by invoking the weak maximum principle and a simple, elegant (but apparently not well-known) result due to Ernst Fischer - of Riesz-Fischer fame. The general case is then settled with the aid of the Weierstrass approximation theorem and the mean-value property of harmonic functions.

Line Baribeau* and Thomas Ransford (Laval)
Non-linear spectrum-preserving maps

We denote by M_n the set of complex n×n matrices, and say that a map F defined on M_n is spectrum-preserving if x and F(x) have the same spectrum (counting multiplicities) for all x. Marcus and Moyls showed that a spectrum-preserving linear map from M_n to itself must be of one of the forms x® uxu^-1 or x® ux^tu^-1 for a fixed matrix u. In this work, we replace the linearity assumption by a differentiability assumption. Our result implies in particular that if F is a spectrum-preserving C¹-diffeomorphism of some open subset U of M_n, then F(x) is conjugate to x for all x Î U.

Peter Booth (MUN)
Topology in Everyday Life

Topology is usually introduced as a rather abstract subject, related to the foundations of analysis and far removed from real life events. I will argue that topology is a part of everybody's day to day experience, and that a study of topology can develop on that basis.

Andrew Foster (MUN)
A Dynamical Systems Approach to Numerical Methods for Flows

Numerical methods for solving differential equations generally involve approximating the (continuous) flow by a (discrete) map. Dynamical systems theory is a natural choice for studying any behavioral differences caused by this type of transformation. In this talk, the usual local bifurcations of scalar flows will be transformed under a class of numerical methods known as linearized one-point collocation. Through normal forms, it will be shown that each such bifurcation gives rise to an exactly corresponding one in its discretization. However, spurious behavior can arise from period doubling cascades and from singular sets induced by the numerical methods.

Phil Heath (MUN)
Using algebra to count fixed and periodic points.

A fixed point of a continuous map f: X ® X is a point x Î X such that f(x) = x. A periodic point is a fixed point of fⁿ for some n where fⁿ is the nth iterate of f. Thus a fixed point is a periodic point of period 1.

The name of the game is to use a simple algebraic tool to estimate the number M(fⁿ) of periodic points of maps g that are "deformable" to f. This talk, given at an elementary level and therefore accessible to senior undergraduates, is designed to give an intuitive feeling for the connections between the algebra and the geometry that lies at the heart of that branch of Algebraic Topology called Nielsen theory.

Serpil Kocabiyik (MUN)
A finite difference calculation for an oscillating inclined elliptic cylinder

The unsteady flow past an inclined elliptic cylinder which starts translating and oscillating impulsively from rest in a viscous fluid is numerically investigated at a Reynolds number of R=10³. Flow is incompressible and two-dimensional, and the cylinder oscillations are harmonic. These oscillations are only allowed in the magnitude which is less than the constant translational velocity and in the transverse direction to that of the uniform translation. The investigation is based on an implicit finite difference scheme for integrating the unsteady Navier-Stokes equations together with the mass-conservation equation in their vorticity stream function formulation. A boundary-layer type transformation is adopted to scale out the singular nature in the vorticity at the start of the motion. A non-inertial coordinate transformation is used so that the grid mesh remains fixed relative to the accelerating cylinder. Present calculations are performed at a sufficiently large oscillation amplitude to induce separation. The effect of the maximum oscillatory-to-translational velocity ratio on the laminar asymmetric wake evolution are studied. The time variation of the in-line and transverse force coefficients are also presented.

Mashhoor Refai (Yarmouk)
On some properties of graded rings

Let G be a group and R be a G-graded ring. Methods of graded rings have proven to be successful tools in the structure theory for commutative and non commutative rings. Many studies in group graded rings assume R to be a strongly graded ring. These rings have been studied by E.C. Dade,where they were called Clifford Systems. In this paper we define three successively stronger properties that a grading may have. Then we investigate the relationship between these new strong gradings and the stronger nondegenerate and faithful properties.

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