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?