Org: Dale Rolfsen (UBC)
- ALEJANDRO ADEM, Department of Mathematics, UBC
On spaces of homomorphisms
In this lecture we describe properties of Hom(Q,G), where Q is a
suitable discrete group and G is a Lie group.
- KRISTINE BAUER, University of Calgary, Calgary, AB T3l 2W9
The topological Lie operad
The classical Lie operad is an operad of modules whose algebras are
Lie algebras. Algebras which have the homotopy type of a Lie algebra
are called strongly homotoply Lie algebras (or sometimes
L¥-algebras), and there is another operad which detects these.
We construct a topological version of the L¥ operad, whose
algebras are topological spaces.
- STEVEN BOYER, UQAM
- IAN HAMBLETON, McMaster University, Hamilton, ON
Free actions on products of spheres
Which finite groups G can act freely and smoothly on a product of
spheres Sn ×Sn? In the talk we will present old and new
results on this problem-the answer is not yet known.
- GABRIEL INDURSKIS, UQAM, Dept. de mathématiques, Case postale 8888,
succursale Centre-Ville, Montréal, QC H3C 3P8
On the character varieties of manifolds obtained from the
Whitehead link exterior by Dehn filling
A well-known method due to Riley characterizes the p-reps of the
fundamental group of the exterior of a 2-bridge knot by the roots of a
one-variable polynomial. (A p-rep of such a group is a
representation with values in SL(2,C) which is parabolic on the
peripheral subgroup.) We describe how to generalize this method to
find the p-reps for all Dehn fillings on one boundary component of
the Whitehead link exterior in terms of the filling slope. This is
done by taking a "detour" through the eigenvalue variety of the
unfilled manifold and using elimination theory to find a polynomial
whose roots characterize the p-reps of the filled manifold. As an
application, we determine the minimal Culler-Shalen norm for all such
fillings and use this to make some statements about the structure of
their character varieties.
- RICK JARDINE, University of Western Ontario
This talk defines and gives applications of cocycle categories. The
path components of such categories define morphisms in homotopy
categories arising in a variety of algebraic and geometric settings.
Applications in non-abelian cohomology theory, including the homotopy
classification of gerbes, will be displayed.
- RICHARD KANE, University of Western Ontario
Invariant Theory and Lie Groups
The mod p cohomology of a Lie group is a Hopf algebra,
i.e., both an algebra and a coalgebra. It is well known,
going back to the work of Borel and Chevaley in the 1950's, that the
rational cohomology of a connected Lie group G and of its
classifying space BG can be determined from a knowledge of the
invariant theory of the Weyl group of G. This same result holds in
mod p cohomology provided p is not a torsion prime for G (p is
a torsion prime if p torsion appears in the integral cohomology of
G). Kac and Peterson introduced the concept of generalized
invariants of a Weyl group and demonstrated that generalized
invariants determine the mod p cohomology of G when p is a
torsion prime. We will consider the relation between the generalized
invariants of G and the coalgebra structure of the mod p
cohomology of G.
- ROBION KIRBY, Univ. of California, Berkeley, CA 94720-3840
Singular Lefschetz fibrations
Auroux, Donaldson and Katzarkov have shown that smooth 4-manifolds
with near symplectic forms are singular Lefschetz pencils. I will
discuss this work and attempts to extend it to other 4-manifolds.
This is joint work with David Gay.
- ELENA KUDRYAVTSEVA, University of Calgary, Dept. of Math. and Stat., Calgary,
AB T2N 1N4
On coincidence points of mappings of the torus into a surface
For an arbitrary pair of continuous maps (f,g) of the 2-torus T
into an arbitrary surface S, the Wecken property for the coincidence
problem is proved. This means that there exist homotopic maps f¢,g¢
such that each Nielsen class of coincidence points of (f¢,g¢)
consists of one point and has a non-vanishing index. Moreover, every
non-vanishing index is equal to ±1, and every non-vanishing
semi-index of Jezierski is equal to 1, if S is neither the sphere
nor the projective plane.
Joint work with S. Bogatyi and H. Zieschang.
- VICTOR NUNEZ, Cimat
Classical drawings of branched coverings
Given a branched covering j: S3 ® (S3,k), it
is an interesting and very difficult problem to determine the link
type of j-1(k) Ì S3. If k is drawn in an
n-bridge presentation, that is, if there is a 3-ball B Ì S3
such that k is the union of n properly embedded arcs in B and
n arcs on ¶B, it is tempting to try to recover
j-1(k) from a drawing of j-1(B)-an abstract
drawing, not an embedding of j-1(B) in S3. It is well
known that, if j-1(B) is also a 3-ball, this is possible.
If j-1(B) is a handlebody of positive genus, an arbitrary
drawing of j-1(B) is generally misleading.
We give a description of how to embed j-1(B) in S3 in
the general case, and, therefore, a complete criterion to recover the
link type of j-1(B) from an embedding of j-1(B)
in S3. We also give some applications.
- DORETTE PRONK, Dalhousie University
The Orbifold Construction
Orbifolds were originally defined as differentiable manifolds with
singularities that can be described as quotients of an open subset of
Euclidean space by the action of a finite group. Orbifolds have
proved their usefulness in various contexts and today we have
analytic, algebraic, topological, and differentiable orbifolds. This
leads us to ask the following questions:
- what kind of results are applicable to all orbifolds?
- in what kind of categories can one define orbifolds?
- is there an orbifold construction?
- is there a natural class of orbifold morphisms?
We will begin to answer these questions from an abstract categorical
view point, but we will also describe some of the concrete geometrical
This is joint work with Robin Cockett from the University of Calgary.
- ANTONIO RAMIREZ, University of British Columbia, 1984 Mathematics Rd.,
Vancouver, BC V6T 1Z2
Open-closed string topology
The area of string topology began with a construction by Chas and
Sullivan of previously undiscovered algebraic structure on the
homology H*(LM) of the free loop space of an oriented manifold M.
Among other results, Chas and Sullivan showed that H*(LM), suitably
regraded, carries the structure of a graded-commutative algebra. The
product pairing was subsequently extended by Cohen and Godin into a
form of topological quantum field theory (TQFT). Open-closed string
topology, first sketched by Sullivan, arises when considering spaces
of paths in M with endpoints constrained to lie on given
submanifolds (the so-called D-branes). In this talk, I describe a
way to extend the TQFT structure of string topology into an analogue
of TQFT which incorporates open strings. The method of construction
is homotopy theoretic, and it makes use of constrained mapping spaces
from fat B-graphs (which I define) into the ground manifold M.
- DALE ROLFSEN, UBC, Vancouver, BC V6T 1Z2
Ordering knot groups
Classical knot groups are known to be right-orderable, by a theorem of
Howie and Short. This means that the elements of the group may be
given a strict total ordering which is invariant under right
multiplication. Some knot groups can be given an ordering which is
invariant under multiplication on both sides; the figure-eight is an
example, as shown by B. Perron and the speaker. Others, such as torus
knot groups, do not enjoy a 2-sided ordering. For most knots, the
question of the existence of 2-sided orderings is still open. I will
discuss this problem, including some new techniques, the conjecture
that all knot groups are virtually orderable, and why we should care
about the question.
- LAURA SCULL, UBC
The Equivariant Fundamental Groupoid
I will discuss a Seifert-Van Kampen Theorem for the equivariant
- DONALD STANLEY, University of Regina, Department of Mathematics, College West
307.14, Regina, Saskatchewan
Refining Poincaré Duality
We refine Poincaré duality by showing that closed manifolds satisfy
Poincaré duality at the chain level. More precisely we prove that
every commutative differential graded algebra whose cohomology is a
simply-connected Poincaré duality algebra is quasi-isomorphic to one
whose underlying algebra is simply-connected and satisfies Poincaré
duality in the same dimension. We apply our result to the study of
CDGA models of configuration spaces on a closed manifold.
- JENS VON BERGMANN, University of Calgary
Compactness for Moduli Spaces of H-Holomorphic
We prove compactness of the moduli space of H-holomorphic
maps with varying complex structure on the domain into a certain
subclass of stable almost contact mansifolds. The compactness
statement differs from that for J-holomorphic maps as one needs a
compactification of the domains that is different from Deligne-Mumford
and one needs to fix the homotopy class of maps rather than just the
homology class. This result is needed for the compactness of the
moduli space of pseudoholomorphic maps into folded symplectic
manifolds and its possible generalization to all contact manifolds is
the missing link in Hofer's scheme to prove the Weinstein conjecture.
- GENEVIEVE WALSH, University of Texas at Austin, Dept. of Math., Austin, Texas
Which knots are great?
A great circle link is a link of geodesic circles in S3. We say
that a knot is great if its complement is commensurable with
the complement of a great circle link. All great circle link
complements are fibered. Therefore, if a knot is great, its
complement is virtually fibered. Provably great knots include:
two-bridge knots, spherical Montesinos knots, and torus knots. We
will discuss these knots and speculate on the greatness of other
- LIAM WATSON, UQAM mathématiques, Montréal, QC H3C 3P8
Tangle surgery and the Jones polynomial
Eliahou, Kauffman, and Thistlethwaite have given examples of
non-trivial links (with 2 or more components) that have trivial Jones
polynomial. However, it is still unknown if there is a non-trivial
knot with Jones polynomial 1. As this question remains open, I will
present a construction for producing pairs of prime knots with the
same Jones polynomial that uses machinery similar to that of Eliahou,
Kauffman, and Thistlethwaite.
- PETER ZVENGROWSKI, University of Calgary
Recent Progress in the Span of Smooth Manifolds
The span of a smooth manifold M is a classical invariant, defined as
the maximal number of (pointwise) linearly independent tangent vector
fields on M. We shall also consider the related concepts of stable
span and immersion codimension. Some recent progress (with
D. Crowley) showing that these invariants can depend on the smoothness
structure of M will be described, as well as some recent progress
(with J. Korbas and P. Sankaran) related to the span of a specific
family of manifolds, the projective Stiefel manifolds Xn,r.
In the former case we shall give examples of manifolds in
dimension 15 and higher where the span, stable span, and immersion
codimension can be different for different smoothness structures. In
the latter case we shall show how various techniques can be applied,
in particular the ring structure in (complex) K-theory, to obtain
refined estimates of the span of Xn,r.