Graduate Seminar

Speaker: Mitsuru Wilson (Western)

"EXISTENCE OF DEFORMATION QUANTIZATION ON POISSON MANIFOLDS"

Time: 11:20

Room: MC 106

The origin of deformation quantization goes back to as far as 1969 in its purely algebraic form. When applied this construction to the algebra $C^{\infty}(M)$ of smooth complex valued functions on a manifold $M$ , if exists, one obtains a quantization,making the space $C^{\infty}(M)$ noncommutative. Roughly speaking, the construction proceeds as follows: using the algebra $C^{\infty}(M)$ of complex valued smooth functions on $M$, one defines a new product $\star$ depending on some formal quantization parameter $\hbar$.This new product is viewed as formal power series in $\hbar$,thus defining a new algebra $C^{\infty}(M)[[\hbar ]]$ over the ring $\mathbb{C}[[\hbar]]$. An example of such a product called Weyl-Moyal product on $\mathbb{R}^{N}$ arises naturally from its Poisson structure. Under any new multiplication, $\frac{f\star g -g\star f}{\hbar}\vert_{\hbar\longrightarrow 0} = \{f,g\}$. In fact, M. Kontsevich proved that if $M$ has a Poisson bracket, then $M$ admits a nontrivial deformation quantization.I will sketch the proof of Kontsevich in the simplest case $M = \mathbb{R}^{n}$. As much as time is allotted, I will give as many applications of Kontsevich celebrated result as possible.

Dept Oral Exam

Speaker: Asghar Ghorbanpour (Western)

"Rationality of spectral action for Robertson-walker metrics and geometry of determinant line bundle for the nonocmmutative two torus"

Time: 13:00

Room: MC 107

In nonocmmutative geometry, the geometry of a space is given via a spectral

triple $(\mathcal{A,H},D)$.

In this approach the geometric information is encoded in the spectrum of $D$.

To extract this spectral information, one should study the spectral action $\Tr f(D/\Lambda)$.

This function is very closely related to classical spectral functions such as the heat trace $\Tr (e^{-tD^2})$ and the spectral zeta function $\Tr(|D|^{-s})$.

The main focus of this talk is on the methods and tools that can be used to extract the spectral information.

Applying the pseudodifferential calculus and the heat trace techniques, in addition to computing the newer terms of the spectral action, we prove the rationality of this spectral action, which was conjectured by Chamseddine and Connes.

In the second part of the talk, we define the canonical trace for Connes' pseudodifferential calculus on the noncommutative torus and use it to compute the curvature of the determinant line bundle for the noncommutative torus.

At the end, the Euler-Maclaurin summation formula will be used to compute the spectral action of a Dirac operator (with torsion) on the Berger spheres $\mathbb{S}^3(T)$.

Analysis Seminar

Speaker: Baris Ugurcan (Western)

"Dilation Theorems and Non-commutative Stochastic Processes"

Time: 14:30

Room: MC 107

We survey (including our results) the dilation theorems in operator algebras in various settings and in particular talk about their appearance in non-commutative stochastic processes. We talk about the well-known correspondence between semigroups and stochastic processes in the commutative case and survey how this correspondence can be generalized to non-commutative setting by using dilation theorems. We also mention how the correspondence (in the commutative case) arises in developing analysis on non-smooth spaces such as fractals.

Homotopy Theory

Speaker: Martin Frankland (Western)

"A univalent model in simplicial sets"

Time: 13:00

Room: MC 107

We will review the notion of model of dependent type theory, with prescribed constructors. Then we will describe work of Kapulkin, Lumsdaine, and Voevodsky producing a model in the category of simplicial sets, for which the Univalence Axiom holds.

Algebra Seminar

Speaker: Lila Kari (Western)

"Map of Life: A quantitative method for measuring and visualizing species' relatedness"

Time: 14:30

Room: MC 107

We introduce a novel method to computationally measure the distance between any two species based on unrelated short fragments of their genomic DNA. These pairwise species' distances are used to compute and output a two-dimensional "Map of Life", wherein each species is a point and the geometric distance between any two points reflects the degree of relatedness between the corresponding species. Such maps present compelling visual representations of relationships between species and could be used for species' classifications, new species identification, as well as for studies of evolutionary history.

Colloquium

Speaker: Abdellah Sebbar (University of Ottawa)

"*cancelled*"

Time: 15:30

Room: MC 108

TBA