Geometry and Topology

Speaker: Nicole Lemire (Western)

"Equivariant Birational Properties of Algebraic Tori"

Time: 15:30

Room: MC 107

We examine the equivariant birational linearisation problem for algebraic tori equipped with a finite group action. We also study bounds on degree of linearisability, a measure of the obstruction for such an algebraic torus to be linearisable. We connect these problems to earlier work with Vladimir Popov and Zinovy Reichstein on the classification of the simple algebraic groups which are Cayley and on determining bounds on the Cayley degree of an algebraic group, a measure of the obstruction for an algebraic group to be Cayley.

Noncommutative Geometry

Speaker: Farzad Fathizadeh (York University)

"The Calculus of Pseudodifferential Operators 1"

Time: 12:30

Room: MC 107

This series of lectures provides an introduction to the basic calculus of pseudodifferential operators defined on Euclidean spaces. We will start by reviewing the space of Schwartz functions, the convolution, the Fourier transform, and their basic properties. Then we prove two important results for studying pseudodifferential operators: the Fourier inversion formula and the Plancherel theorem. We will proceed by finding an asymptotic expansion for the symbol of formal adjoint and composition of pseudodifferential operators. We will end the lectures by introducing a notion of ellipticity and constructing parametrices for elliptic pseudodifferential operators.

Analysis Seminar

Speaker: Jana Marikova (McMaster)

"O-minimal fields and convex valuations"

Time: 15:30

Room: MC 107

An o-minimal structure is a structure with a dense linear order in which there are as few definable subsets of the line as possible (namely just finite unions of points and intervals). This condition ensures rather nice topological properties of the definable sets in an o-minimal structure, the archetypical example here being the semialgebraic sets.

In order to understand the definable sets in an o-minimal field R, it is often helpful to understand the convex valuations on R in terms of the usually simpler residue field and value group. We shall discuss some related results, focusing mainly on the residue field.

Graduate Seminar

Speaker: Priyavrat Deshpande (Western)

"Better ways of cutting cheese, in all dimensions"

Time: 16:30

Room: MC 107

What is the maximum number of pieces of a cheese (or of a pizza) you can cut with $n$ cuts? A study of these kinds of problems goes back to the work of Jacob Steiner in the early 19th century. Over the years mathematicians have studied various aspects and generalizations of this problem. Questions of this type are collectively known as "topological dissection problems".

The aim of my talk is to introduce a unified way to solve a class of dissection problems. This new approach has helped in solving the topological dissection problems in a vast generality. The solution of this seemingly simple problem involves the use of a fundamental dimensionless invariant and some measure theory on posets.

I will try to explain this new approach with the help of simple diagrams and intuitive ideas avoiding technicalities. This is a part of an ongoing research project called "arrangements of submanifolds".

Noncommutative Geometry

Speaker: Arash Pourkia (Western)

"Cyclic Cohomology 1"

Time: 14:30

Room: MC 107

Cyclic (co)homology is the noncommutative analogue of de Rham (co)homology and as such plays an important role in noncommutative geometry and its applications (in operator algebras, index theory, ...) A variant of it, topological Hochschild and cyclic homology, plays an important role in algebraic K-theory as well. Arash will give a series of lectures on the subject (2 hours per week), starting from basic material and gradually building towards more advanced stuff. Outline: 1. Basic homological algebra in abelian categories 2. Hochschild (co)homology; computations (Hochschild-Kostant-Rosenberg, group algebras) 3. Cyclic (co)ohomology, Connes' spectral sequence; computations (relation with de Rham, group algebras); cyclic category. 4. K-theory and K-homology, 5. Connes-Chern character 6. An index formula 7. Applications to idempotent conjectures.

The basic texts to follow are: 1. Cyclic Homology, J. L. Loday 2. Noncommutative Geometry, A. Connes 3. Noncommutative Differential Geometry, Publication math. IHES, 1985, A. Connes. 4. Basic noncommutative geometry, Masoud Khalkhali

Algebra Seminar

Speaker: Lex Renner (Western)

"Conjugacy in $M_n(R)$ where $R$ is a DVR"

Time: 14:30

Room: MC 107

If $R$ is a field then the conjugacy class of $x\in M_n(R) = End(V)$ is determined by its rational canonical form using the theory of modules over the PID $R[T]$. If $R$ is a discrete valuation ring then the situation is more complicated, even if the characteristic polynomial of $x\in M_n(R)$ is irreducible over the quotient field $K$ of $R$. We discuss the following questions. (1) What further assumptions on $x$ and $R$ are useful? (e.g. $x$ semisimple, $R$ Henselian) (2) How do we sort out non-conjugate elements of $M_n(R)$ that become conjugate in $M_n(K)$? (3) Are some conjugacy classes of $M_n(R)$ better than others? (4) To what extent can $x$ be measured against a canonical form?

Geometry and Topology

Speaker: Craig Westerland (Univ. of Minnesota)

"TBA"

Time: 15:30

Room: MC 107

Noncommutative Geometry

Speaker: Ali Moatadelro (Western)

"Representation theory of compact quantum groups with examples 1"

Time: 09:30

Room: MC 106

In this series of lectures, we will discuss basic examples of compact quantum groups and their (finite dimensional) representations. We will start with reviewing the classical theory. We shall classify all finite dimensional irreducible representations of compact Lie groups SU(2) and SU(3). Then we will proceed to the general theory of representation of compact Lie groups and will discuss several important results including the highest weight theory, the Peter-Weyl decomposition theorem, and also the Borel-Weil-Bott construction of representations. Finally, we will see how much of the theory holds in the quantum case.

Geometry and Topology

Speaker: Paul Goerss (Northwestern)

"On the chromatic splitting conjecture"

Time: 15:30

Room: MC 107

In the chromatic take on stable homotopy theory, the homotopy type of a finite $p$-local spectrum $X$ is reassembled from its localizations with respect to the various Morava $K$-theories. In the early 1990s, Hopkins proprosed a brash conjecture for how the reassembly process works. I'll review the conjecture and the state of the art -- including a verfication of the conjecture at $p=3$ and chromatic level $2$, where the question is not simply algebraic and where there has been a proposed counterexample. This is joint work with Hans-Werner Henn.

Geometry and Topology

Speaker: Tom Baird (Memorial)

"GKM-sheaves and equivariant cohomology"

Time: 10:30

Room: MC 107

Let $T$ be a compact torus. Goresky, Kottwitz and Macpherson showed that for a large and interesting class of $T$-equivariant projective varieties $X$, the equivariant cohomology ring $H_T^*(X)$ may be encoded in a graph, now called the GKM-graph, with vertices corresponding to the fixed points of $X$ and edges labeled by the weights, $Hom(T, U(1))$.

In this lecture, we explain how the GKM construction can be generalized to any finite $T$-CW complex. This generalization gives rise to new mathematical objects: GKM-hypergraphs and GKM-sheaves. If time permits, we will show how these methods were used to resolve a conjecture concerning the moduli space of flat connections over a non-orientable surface.

Noncommutative Geometry

Speaker: Farzad Fathizadeh (York University)

"The Calculus of Pseudodifferential Operators 2"

Time: 12:30

Room: MC 107

This series of lectures provides an introduction to the basic calculus of pseudodifferential operators defined on Euclidean spaces. We will start by reviewing the space of Schwartz functions, the convolution, the Fourier transform, and their basic properties. Then we prove two important results for studying pseudodifferential operators: the Fourier inversion formula and the Plancherel theorem. We will proceed by finding an asymptotic expansion for the symbol of formal adjoint and composition of pseudodifferential operators. We will end the lectures by introducing a notion of ellipticity and constructing parametrices for elliptic pseudodifferential operators.

Analysis Seminar

Speaker: Franklin Vera Pacheco (Toronto)

"Strict desingularizations - the semi simple normal crossings case"

Time: 15:30

Room: MC 107

When resolving singularities of an algebraic variety one produces a smooth model and a birational map to the original variety. The desingularization is said to be strict when this map only changes singular points, i.e. it is an isomorphism over the smooth points. Sometimes it is needed to preserve other singularities besides the smooth points. One may want to get an isomorphism over the simple normal crossings points, or over the normal crossings points, or any other family of singularity types. These desingularizations may or may not exist. We will talk about a way to approach the construction of these desingularizations in the case of semi simple normal crossings singularities (the analogue of simple normal crossings on a non normal space).

Pizza Seminar

Speaker: Jason Haradyn (Western)

"Historical Perspectives on the Riemann Hypothesis"

Time: 16:30

Room: MC 107

In this talk, we will discuss some of the history behind the Riemann hypothesis, including its relation to the distribution of primes, attempts at a proof over the years and its appearance and importance in many areas of mathematics. This will lead to surprising real life examples where the Riemann hypothesis applies, such as quantum physics.

Noncommutative Geometry

Speaker: Arash Pourkia (Western)

"Cyclic Cohomology 2"

Time: 14:30

Room: MC 107

Cyclic (co)homology is the noncommutative analogue of de Rham (co)homology and as such plays an important role in noncommutative geometry and its applications (in operator algebras, index theory, ...) A variant of it, topological Hochschild and cyclic homology, plays an important role in algebraic K-theory as well. Arash will give a series of lectures on the subject (2 hours per week), starting from basic material and gradually building towards more advanced stuff. Outline: 1. Basic homological algebra in abelian categories 2. Hochschild (co)homology; computations (Hochschild-Kostant-Rosenberg, group algebras) 3. Cyclic (co)ohomology, Connes' spectral sequence; computations (relation with de Rham, group algebras); cyclic category. 4. K-theory and K-homology, 5. Connes-Chern character 6. An index formula 7. Applications to idempotent conjectures. The basic texts to follow are: 1. Cyclic Homology, J. L. Loday 2. Noncommutative Geometry, A. Connes 3. Noncommutative Differential Geometry, Publication math. IHES, 1985, A. Connes. 4. Basic noncommutative geometry, Masoud Khalkhali

Colloquium

Speaker: Sergey Arkhipov (University of Toronto)

"Determinants of infinite dimensional vector spaces and central extensions of formal loop groups."

Time: 15:30

Room: MC 107

We start by considering the notions of a torsor and a gerbe for a discrete Abelian group A (over a discrete set X). We describe central extensions of a group G by A as multiplicative A-torsors on X=G. A categorification of this construction is given by the notion of a gerbal central extension of a group G by an Abelian group A. We classify gerbal A-central extensions of the group G by A-valued 3-cocycles of G.

Then we recall the construction of determinantal gerbe Det(V) of an infinite dimensional Tate vector space V (e.g. V=k((s)) ) due to Kapranov. We explain that the obstruction of Det(V) to be GL(V)-equivariant provides the well known central extension of GL(V) by k^*. Given a 2-Tate vector space V, e.g. V=k((s))((t)), we consider the determinantal 2-gerbe 2-Det(V). The obstruction of 2-Det(V) to be GL(V)-equivariant provides a gerbal central extension of GL(V) by k^*. This construction leads to gerbal central extensions of double loop groups.

Colloquium

Speaker: Jean-Francois Lafont (Ohio State University)

"TBA"

Time: 15:30

Room: MC 108

TBA

Colloquium

Speaker: Jean-François Lafont (Ohio State University)

"TBA"

Time: 15:30

Room: MC 108

Colloquium

Speaker: Jean-François Lafont (Ohio State University)

"TBA"

Time: 15:30

Room: MC 107

Algebra Seminar

Speaker:

"No Seminar"

Time: 14:30

Room: MC 107

Colloquium

Speaker: Spiro Karigiannis (University of Waterloo)

"TBA"

Time: 15:30

Room: MC 107

TBA

Noncommutative Geometry

Speaker: Ali Moatadelro (Western)

"Representation theory of compact quantum groups with examples, lecture 2. Irreducible representations of SU(3)"

Time: 09:30

Room: MC 106

In this series of lectures, we will discuss basic examples of compact quantum groups and their (finite dimensional) representations. We will start with reviewing the classical theory. We shall classify all finite dimensional irreducible representations of compact Lie groups SU(2) and SU(3). Then we will proceed to the general theory of representation of compact Lie groups and will discuss several important results including the highest weight theory, the Peter-Weyl decomposition theorem, and also the Borel-Weil-Bott construction of representations. Finally, we will see how much of the theory holds in the quantum case.

Geometry and Topology

Speaker: Sanjeevi Krishnan (Penn)

"Cubical approximation for directed topology"

Time: 15:30

Room: MC 107

Topological spaces - such as classifying spaces of small categories and spacetimes - often admit extra temporal structure. Such "directed spaces" often arise as geometric realizations of simplicial sets and cubical sets; the temporal structure encodes orientations of simplices and 1-cubes. Directed spaces rarely decompose as homotopy colimits of simpler directed spaces. Nevertheless, we present simplicial and cubical approximation theorems for a homotopy theory of directed spaces. In our directed setting, ordinal subdivision plays the role of barycentric subdivision and cubical sets equipped with coherent compositions of higher cubes serve as analogues of Kan complexes. We consequently show that geometric realization induces an equivalence between certain weak homotopy diagram categories of cubical sets and directed spaces. As applications, we show that directed analogues of homotopy groups of spheres are uninteresting, sketch constructions of a (more interesting) cubical singular cohomology theory for directed spaces, and calculate such "directed cohomology" monoids for various directed spaces of interest.

Noncommutative Geometry

Speaker: Farzad Fathizadeh (York University)

"The Calculus of Pseudodifferential Operators 3"

Time: 12:30

Room: MC 107

This series of lectures provides an introduction to the basic calculus of pseudodifferential operators defined on Euclidean spaces. We will start by reviewing the space of Schwartz functions, the convolution, the Fourier transform, and their basic properties. Then we prove two important results for studying pseudodifferential operators: the Fourier inversion formula and the Plancherel theorem. We will proceed by finding an asymptotic expansion for the symbol of formal adjoint and composition of pseudodifferential operators. We will end the lectures by introducing a notion of ellipticity and constructing parametrices for elliptic pseudodifferential operators.

Analysis Seminar

Speaker: Tatyana Foth (Western)

"Higher order automorphic forms"

Time: 15:30

Room: MC 107

A higher order automorphic form is a generalization of the notion of a classical automorphic form. I will discuss the definition and I will review some recent results.

Pizza Seminar

Speaker: Masoud Khalkhali (Western)

"The Ubiquitous Regular Representation"

Time: 16:30

Room: MC 107

The idea of (left) regular representation has its origins in group theory and a well known theorem of Arthur Cayley from 19th century which is part of any introduction to group theory. The goal of this talk is to highlight a few of its ramifications, extensions, and applications in different areas of mathematics, including analysis and algebra. After a quick discussion of the general idea of representation, I shall try to point out - by way of examples drawn from different fields - how universal, as well as simple and useful, the idea of regular representation is. I shall make every effort to make this talk as self contained as possible for an undergraduate talk.

Noncommutative Geometry

Speaker: Arash Pourkia (Western)

"Cyclic Cohomology 3"

Time: 14:30

Room: MC 107

Cyclic (co)homology is the noncommutative analogue of de Rham (co)homology and as such plays an important role in noncommutative geometry and its applications (in operator algebras, index theory, ...) A variant of it, topological Hochschild and cyclic homology, plays an important role in algebraic K-theory as well. Arash will give a series of lectures on the subject (2 hours per week), starting from basic material and gradually building towards more advanced stuff. Outline: 1. Basic homological algebra in abelian categories 2. Hochschild (co)homology; computations (Hochschild-Kostant-Rosenberg, group algebras) 3. Cyclic (co)ohomology, Connes' spectral sequence; computations (relation with de Rham, group algebras); cyclic category. 4. K-theory and K-homology, 5. Connes-Chern character 6. An index formula 7. Applications to idempotent conjectures. The basic texts to follow are: 1. Cyclic Homology, J. L. Loday 2. Noncommutative Geometry, A. Connes 3. Noncommutative Differential Geometry, Publication math. IHES, 1985, A. Connes. 4. Basic noncommutative geometry, Masoud Khalkhali

Geometry and Topology

Speaker: Kirill Zaynullin (Ottawa)

"TBA"

Time: 15:30

Room: MC 107

Algebra Seminar

Speaker: Rick Jardine (Western)

"The Kunneth spectral sequence"

Time: 14:30

Room: MC 107

The Kunneth spectral sequence for abelian sheaf cohomology is displayed and discussed. Computational applications of this spectral sequence for the etale cohomology of classifying spaces of algebraic groups will also be displayed.

Noncommutative Geometry

Speaker: Ali Moatadelro (Western)

"Representation theory of compact quantum groups with examples, lecture 3. Irreducible representations of SU(3), continued."

Time: 09:30

Room: MC 106

In this series of lectures, we will discuss basic examples of compact quantum groups and their (finite dimensional) representations. We will start with reviewing the classical theory. We shall classify all finite dimensional irreducible representations of compact Lie groups SU(2) and SU(3). Then we will proceed to the general theory of representation of compact Lie groups and will discuss several important results including the highest weight theory, the Peter-Weyl decomposition theorem, and also the Borel-Weil-Bott construction of representations. Finally, we will see how much of the theory holds in the quantum case.

Analysis Seminar

Speaker: Rahim Moosa (Waterloo)

"Real-analytic versus complex-analytic families of complex-analytic sets"

Time: 15:30

Room: MC 107

Suppose M is a compact complex manifold. Model theory (a branch of mathematical logic) provides at least two approaches to the study of the complex-analytic subsets of Cartesian powers of M, roughly corresponding to whether one focuses on the real or complex structure on M. We can view M as definable in the structure R_an; that is, as a real globally subanalytic manifold. On the other hand, we can work in the Zariski-type structure CCM where M is the universe and there are predicates for all complex-analytic subvarieties of Cartesian powers of M. The two approaches lead to different notions of a "definable family" of complex-analytic subsets. I will give a geometric characterization, obtained in joint work with Sergei Starchenko in 2008, of when these two notions coincide, in terms of the Barlet or Douady spaces. As a consequence one has that for M Kaehler the two notions coincide.

Graduate Seminar

Speaker: Richard Gonzales (Western)

"The equivariant Chern character"

Time: 16:30

Room: MC 107

A classical result of Atiyah and Hirzebruch establishes a deep connection between K-theory and cohomology, via the Chern character. The purpose of my talk is to describe this relation in precise terms, and give an overview of its generalizations to the equivariant setting. Along the way we introduce a new class of objects, coming from algebraic geometry, on which many of these classical topological techniques could be successfully applied.

Noncommutative Geometry

Speaker: Arash Pourkia (Western)

"Cyclic Cohomology 4"

Time: 14:30

Room: MC 107

Cyclic (co)homology is the noncommutative analogue of de Rham (co)homology and as such plays an important role in noncommutative geometry and its applications (in operator algebras, index theory, ...) A variant of it, topological Hochschild and cyclic homology, plays an important role in algebraic K-theory as well.

We will give a series of lectures on the subject (2 hours per week), starting from basic material and gradually building towards more advanced stuff. Outline: 1. Basic homological algebra in abelian categories 2. Hochschild (co)homology; computations (Hochschild-Kostant-Rosenberg, group algebras) 3. Cyclic (co)ohomology, Connes' spectral sequence; computations (relation with de Rham, group algebras); cyclic category. 4. K-theory and K-homology, 5. Connes-Chern character 6. An index formula 7. Applications to idempotent conjectures.

The basic texts to follow are: 1. Cyclic Homology, J. L. Loday 2. Noncommutative Geometry, A. Connes 3. Noncommutative Differential Geometry, Publication math. IHES, 1985, A. Connes. 4. Basic noncommutative geometry, Masoud Khalkhali

Algebra Seminar

Speaker: Marc Moreno Maza (Western)

"Triangular decomposition of semi-algebraic systems"

Time: 14:30

Room: MC 107

Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. We propose adaptations of these tools focusing on solutions of the real analogue: semi-algebraic systems.

We show that any such system can be decomposed into finitely many so-called "regular semi-algebraic systems". We propose two specifications of such a decomposition and present corresponding algorithms. Under some assumptions, one type of decomposition can be computed in singly exponential time with respect to the number of variables. We have implemented our algorithms and the experimental results illustrate their effectiveness. A software demonstration will conclude this talk.

This is a joint work with Changbo Chen (UWO), James H. Davenport (Bath U.), John P. May (Maplesoft), Bican Xia (Peking U.) and Xiao Rong (UWO).

The corresponding article is published in the Proceedings of the 2010 International Symposium of Symbolic and Algebraic Computation (ISSAC'10) and available at ww.csd.uwo.ca/~moreno/Publications/118_paper.pdf