Geometry and Topology

Speaker: Kathryn Hess (ETH, Lausanne)

"Power maps in algebra and topology"

Time: 15:30

Room: MC 108

(Joint work with J. Rognes, Oslo)

In this talk I will explain the construction and properties of a certain chain complex H(t) associated to a given fixed twisting cochain t. Since this construction generalizes that of both the Hochschild complex of an associative algebra and the coHochschild complex of a coassociative coalgebra, we call H(t) the Hochschild complex of t.

I'll give conditions under which H(t) admits power maps extending the usual power maps on a Hopf algebra. In particular, it turns out that both the Hochschild complex of any cocommutative Hopf algebra and the coHochschild complex of the normalized chain complex of a double suspension admit power maps, which are algebraic models for the topological power maps on free loop spaces. This algebraic model of the topological power map is a crucial element of the construction of our model for computing spectrum homology of topological cyclic homology of spaces.

Colloquium

Speaker: Kathryn Hess (Ecole Polytechnique Federale de Lausanne)

"Free loop spaces in topology and physics"

Time: 15:30

Room: MC 108

In this talk I will outline a few of the important roles   that free loop spaces play in topology and mathematical physics.  In  particular, I will talk about enumeration of geodesics on manifolds, as well as about the relationship between Hochschild homology and free loop spaces.  Moreover I will sketch how a free loop space gives rise to a homological conformal field theory, via string topology.

Algebra Seminar

Speaker: Priyavrat Deshpande (Western)

"Finiteness properties of groups and Morse theory"

Time: 14:30

Room: MC 106

Two fundamental finiteness conditions in group theory are the properties of being finitely generated and of finitely presented. More general finiteness conditions using geometric and homological properties of groups were introduced by C.T.C. Wall. Until recently, it was unknown whether the homological conditions implied the geometric conditions. M. Bestvina and N. Brady showed that this is not the case. They constructed counterexamples using right angled Artin groups and a piecewise linear version of the Morse theory. Purpose of this talk is to introduce these groups and see some applications.

Geometry and Topology

Speaker: David Barnes (Western)

"Splitting monoidal stable model categories"

Time: 15:30

Room: MC 107

If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit S forms a commutative ring. An idempotent e of this ring will split the homotopy category. We prove that provided the localised model structures exist, this splitting of the homotopy category comes from a splitting of the model category, that is, C is Quillen equivalent to the product of C localised at the object eS and C localised at the object (1-e)S. This Quillen equivalence is strong monoidal and is symmetric when the monoidal product of C is.

Stable Homotopy

Speaker: Dan Christensen (Western)

"An Introduction to Stable Homotopy Theory"

Time: 14:00

Room: MC 107

The stable homotopy category of spectra was created to capture those phenomena which are "stable". I will motivate the interest in this category by giving examples of stable phenomena, including generalized cohomology theories, Steenrod operations, the Freudenthal suspension theorem, Spanier-Whitehead duality and cobordism.

Geometry and Topology

Speaker: Paul Goerss (Northwestern)

"Serre duality for topological modular forms"

Time: 15:30

Room: MC 107

Serre duality is a twisted form of Poincare' duality satisfied by a very special class of projective schemes. For a variety of reasons the moduli stack of elliptic curves doesn't meet the hypotheses needed and doesn't quite exhibit Serre duality; however, when we consider a derived version of this stack -- replacing the structure sheaf by a sheaf of ring spectra -- suddenly and mysteriously the duality reappears. The purpose of this talk is to explain these ideas and calculations. This is a meditation on work of Hopkins, Miller, Lurie, Behrens, and many others.

Analysis Seminar

Speaker: Shengda Hu (Waterloo)

"Virtual integration in moduli problems"

Time: 15:30

Room: MC 108

We will discuss a method of carrying out integration on moduli spaces defined from Fredholm systems. The main point is that we do away with the regularity assumptions that is generally used in such a setup to obtain smooth manifolds (or orbifolds). We will also discuss some applications for such an integration. Most of the constructions will be elementary and will be motivated from finite dimensional examples. This is based on works of Chen-Li-Tian and works in progress with Chen and Hyvrier.

Pizza Seminar

Speaker: Gord Sinnamon (Western)

"Beyond the Do-It-Yourself Fractal"

Time: 16:30

Room: MC 107

Iterating a map on the plane can produce complicated self-similar patterns resulting in fractal pictures. The map that produces the do-it-yourself fractal operates just on points with integer coordinates and is simple enough to be worked out by hand. I will use it to explain how self-similarity arises naturally in fractal pictures.

When our simple map is extended to operate on the whole plane, things become much more complicated. A bit of Calculus, a bit of Number Theory, and plenty of Linear Algebra will be needed to look at the question, "What Happens Eventually?"

Noncommutative Geometry

Speaker: Sheldon Joyner (Western)

"An algebraic version of the Knizhnik - Zamolodchikov equation"

Time: 15:30

Room: MC 106

In this informal talk, I will outline new work using elementary ideas by means of which an algebraic analogue of the usual KZ equation is exhibited. This difference equation arises via the introduction of Hurwitz polyzeta functions and suitable regularizations. A difference equation analogue of the notion of connection will also be presented, built around the algebraic KZ equation, for which a generating series for Hurwitz polyzeta functions is a "flat section". Also, the algebraic independence of the Hurwitz polyzeta functions will be demonstrated, along with a construction of regularizations of polyzeta functions using limits.

Colloquium

Speaker: Lionel Nguyen van The (University of Calgary)

"When complete disorder is impossible"

Time: 15:30

Room: MC 108

Consider six points in the plane and connect (or not) each pair of those at random. Many graphs can be obtained that way but all of them share the following property: there are three points where the pairs are all connected or never connected. This simple combinatorial result is at the base of what is called Ramsey theory, a theory that describes the appearance of unavoidable patterns and nowadays touches areas of mathematics that are seemingly very far from combinatorics. The purpose of this talk will be to present some of those connections taken from metric geometry and topological group theory.

Algebra Seminar

Speaker: Martin Pinsonnault (Western)

"Maximal tori in symplectomorphism groups of 4-manifolds"

Time: 14:30

Room: MC 106

Let M be a closed symplectic manifold and denote by Ham its group of Hamiltonian diffeomorphisms. When equipped with the standard smooth topology, this is an infinite dimensional Fréchet Lie group. It is generally believed that Ham is "tamer" than the diffeomorphism group Diff(M) and constitutes an intermediate object between compact Lie groups and more general diffeomorphism groups. To develop a better understanding of this principle, one may look at maximal Hamiltonian actions by tori or, in other words, to classify symplectic conjugacy classes of maximal compact tori in Ham. In this talk, we will show that for 4-dimensional symplectic manifolds, there are at most finitely many of those conjugacy classes. As a by-product, we will also prove that the rational cohomology algebra of the symplectomorphism group of a generic blow-up is not finitely generated.

Geometry and Topology

Speaker: Graham Denham (Western)

"The Bernstein-Gelfand-Gelfand correspondence and combinatorics of free resolutions"

Time: 15:30

Room: MC 107

The Bernstein-Gelfand-Gelfand (BGG) correspondence is a an equivalence between the derived categories of graded modules over a polynomial ring and exterior algebra, respectively. Eisenbud, Fl\/oystad and Schreyer (2003) derive an explicit version of the BGG correspondence that, among other things, clarifies the relationship between free resolutions of graded modules over the exterior algebra and the cohomology of coherent sheaves on projective space.

On the other hand, work of Jan-Erik Roos and Maurice Auslander gives a filtration of a graded module over a polynomial ring with supports that decrease in dimension, via a suitable Grothendieck spectral sequence. I will describe some joint work with Hal Schenck in which we consider the interplay between these two constructions. We find applications to some topological spaces with cohomology rings generated in degree 1.

Stable Homotopy

Speaker: Enxin Wu (Western)

"Generalized Cohomology Theories and Brown Representability"

Time: 14:00

Room: MC 107

To motivate the stable homotopy category of CW-complexes, I will talk about generalized cohomology theories and Brown representability in more detail.

Colloquium

Speaker: Alvaro Rittatore (Universidad de Uruguay)

"The endomorphisms monoid of a homogeneous vector bundle"

Time: 15:30

Room: MC 108

Recently it has been proved that any (not necesssarily affine) normal algebraic monoid M is a closed submonoid of the endomophism monoid of a conveniently chosen homogeneous vector bundle over A(M), the Albanese variety of M. This result suggests that the category of homogeneous vector bundles over the abelian variety A(M) is the natural setting for a representation theory of M. Such a theory has not yet been developed, even in the case when M is an algebraic group.

The main goal of this talk is to present the basic facts about the structure of the endomorphism monoid of a homogeneous vector bundle over an abelian variety. These endomorphism monoids play a role similar to the endomorphism monoid of a vector space (the monoid of n by n matrices) in the affine case.

This is a joint work with Leticia Brambila-Paz.

Algebra Seminar

Speaker: David Jeffrey (Western)

"Inverse functions and matrix functions: connection with Lambert W"

Time: 15:30

Room: MC 106

This talk combines two ideas: inverse functions and matrix functions.

Given a function f(z), it is a standard procedure to define an inverse function through the solution of the equation f(z)=w. If we denote the inverse function by invf, then we can write z=invf(w). The definition is often complicated by the fact that there are multiple solutions of the equation.

Given a scalar function f(z), there are standard procedures for extending the definition to a matrix argument. For example, integer powers of a matrix, or the exponential of a matrix.

There are two ways in which one can arrive at a definition of the inverse of a matrix function. First, one can define the scalar inverse and then extend the definition to a matrix argument; second, one can extend the definition of the equation f(z)=w to matrices and then consider its solutions.

These definitions are discussed in the context of the Lambert W function, which is the inverse of the function f(z) = z*exp(z).