Graduate Seminar

Speaker: Nicholas Meadows (Western)

"the Hilbert scheme of Cohen-Macauley curves"

Time: 11:20 - 12:30

Room: MC 106

After reviewing very quickly some algebraic geometry, I will define the Hilbert scheme which parameterizes closed subschemes of projective space $P_{k}^{n}$ and state its basic properties, for k an algebraically closed field of characteristic 0. I will then define various notions of deformations (deformations sheaves, deformations over the dual numbers etc). Finally, I will use obstruction theory for a local ring to prove a lower bound on the dimension of irreducible components of the Hilbert scheme of Cohen-Macauley curves of genus g and degree d in $P_{k}^{3}$

Geometry and Topology

Speaker: Chris Kapulkin (Western)

"Internal languages for higher categories"

Time: 15:30 - 16:30

Room: MC 107

Every category $C$ looks locally like a category of sets, and further structure on $C$ determines what logic one can use to reason about these "sets". For example, if $C$ is a topos, one can use full (higher order) intuitionistic logic. Similarly, one expects that every higher category looks locally like a higher category of spaces. A natural question then is: what sort of logic can we use to reason about these "spaces"? It has been conjectured that such logics are provided by variants of Homotopy Type Theory, a formal logical system, recently proposed as a foundation of mathematics by Vladimir Voevodsky. After explaining the necessary background, I will report on the progress towards proving this conjecture.

Analysis Seminar

Speaker: Thomas Ransford (U. Laval)

"Capacity and coverings"

Time: 14:30 - 15:30

Room: MC 107

I shall discuss two elementary inequalities relating capacity to coverings. They provide an approach to determining whether a set has positive capacity and, if so, to estimating the value of the capacity. (Joint work with Quentin Rajon, Jeremie Rostand and Alexis Selezneff).

Homotopy Theory

Speaker: Cihan Okay (Western)

"Homotopy groups of the circle"

Time: 12:00 - 13:30

Room: MC 106

(Note the unusual day, time, and room.) I will talk about homotopy type theoretic proofs of a well known topological fact that the fundamental group of the circle is the set of integers. There are two closely related proofs. The homotopy-theoretic proof follows a similar reasoning used in the classical proof in topology, whereas the encode-decode proof is more type theoretic.

Colloquium

Speaker: David Riley (Western)

"Hopf algebra actions, gradings, and identical relations"

Time: 15:30 - 16:30

Room: MC 107

I will begin by discussing how and when the action of a Hopf algebra $H$ on an algebra $A$ can be viewed as a grading of $A$. For example, if $G$ is a finite group and $H$ is the dual of the group algebra $K[G]$, then $A$ is an $H$-algebra precisely when $A$ is group-graded by $G$. I will then discuss the identical relations of an algebra with a Hopf algebra action. In particular, I will address the following question: when does the existence of an $H$-identity on $A$ imply the existence of an ordinary polynomial identity on $A$?

Algebra Seminar

Speaker: Martin Frankland (Western)

"Locally presentable categories and applications"

Time: 14:30 - 15:30

Room: MC 107

We will survey some characterization theorems for locally presentable categories and variants thereof. Then we will discuss some applications of locally presentable categories to homological and homotopical algebra.

Graduate Seminar

Speaker: Andrew Day (Western)

"Black Holes and the Schwarzschild metric"

Time: 11:20 - 12:30

Room: MC 106

I this talk we will derive the first black hole solution to the Einstein field equations as was done by Karl Schwarzschild in 1915. We will discuss the strange properties of this spacetime with the help of new coordinate systems, Killing vectors, and Penrose diagrams. We will then present the Birkhoff and uniqueness theorems which tell us there are only four different black hole solutions in 4 dimensions. If time permits we will quickly review the casual structure of these other spacetimes.

The only thing I will be assuming for the talk is that everyone is reasonably familiar with differential geometry.

Geometry and Topology

Speaker: Alexander Neshitov (Univ. of Ottawa)

"Framed Correspondences and the Milnor-Witt K-theory"

Time: 15:30 - 16:30

Room: MC 107

The theory of framed motives developed by Garkusha and Panin based on ideas by Voevodsky, gives a tool to construct fibrant replacements of spectra in A^1-homotopy category. In the talk we will discuss how this construction gives an identification of the motivic homotopy groups of the base field with its Milnor-Witt K-theory. In fact, this identification can be done in the same manner as the theorem of Suslin-Voevodsky which identifies motivic cohomology of the base field with Milnor K-theory.

Analysis Seminar

Speaker: Pinaki Mondal (Weizmann Institute of Science)

"Newton-type diagrams for singular flags and counting number of solutions of polynomials"

Time: 14:30 - 15:30

Room: MC 107

The Newton diagram of a polynomial or analytic function is a powerful tool for studying its behaviour near a point. We introduce a "global version" of Newton diagram of a polynomial (or analytic function) f at a subvariety in order to study behaviour of f near generic points of the subvariety. We apply this notion to the "affine Bezout-problem" of counting number of isolated solutions (in C^n) of a system of n polynomials and show that it is possible to arrive at the exact count by a recursive formula which involves at each step mixed volume of the faces of these Newton-type diagrams with respect to various (possibly singular) "flags of subvarieties". This in particular is a natural extension of the Bernstein-Kushnirenko-Khovanskii approach to the affine Bezout-problem.

Colloquium

Speaker: Ilya Shapiro (University of Windsor)

"Extensions, gerbes and duality"

Time: 14:30 - 15:30

Room: MC 107

We will discuss a correspondence between $G$-graded algebras and $S^1$-gerbes on action groupoids.  This was motivated by a question asked during my last visit to Western.

Homotopy Theory

Speaker: Gaohong Wang (Western)

"Fiber sequences and the Hopf fibration"

Time: 13:00 - 14:30

Room: MC 107

We introduce fiber sequences and the Hopf fibration $S^3 \to S^2$ in HoTT, and we show that the Hopf fibration induces equivalences between the homotopy groups $\pi_k$ of $S^3$ and $S^2$ for $k>2$. We also deduce that $\pi_2(S^2)$ is equivalent to $Z$. We will review the background on truncations and connectedness in this talk too.

Algebra Seminar

Speaker: Nicole Lemire (Western)

"Postponed until February 12, 2015, 3:30 p.m."

Time: 14:30 - 15:30

Room: MC 107

Graduate Seminar

Speaker: Baran Serajelahi (Western)

"n-plectic quantization"

Time: 11:20 - 12:30

Room: MC 106

In this talk I will discuss results obtained with Tatyana Barron. We suggest a way to quantize, using Berezin-Toeplitz quantization (also refered to as Kahler quantization), a compact hyperkahler manifold (equipped with a natural 3-plectic form), or a compact integral Kahler manifold of complex dimension n regarded as a (2n-1)-plectic manifold. In each of these cases the n-plectic structure is derived from the symplectic structure, so it is intuitively clear that we should be able to obtain a quantization by way of the usual Kahler quantization of the symplectic form. We show that the quantization has reasonable semiclassical properties. I will begin by giving an overview of the idea of quantization and in particular I will discuss the necessary background from Berezin-Toeplitz quantization. I will then review the main results of Berezin-Toeplitz quantization (due to Martin Bordemann, Eckhard Meinrenken, Martin Schlichenmaier, which can be found in their paper “Toeplitz Quantization of Kahler Manifolds and gl(N), N--> \infty Limits”) that we have generalized. Finally I will review our results and indicate the proofs.

Analysis Seminar

Speaker: Thomas Bloom (University of Toronto)

"Random Matrices and Potential Theory"

Time: 14:30 - 15:30

Room: MC 107

Ben Arous and A.Guionnet gave (1995) the first large deviation result for the Gaussian Unitary ensemble.This was subsequently extended to general Unitary matrix ensembles. I will discuss a method for obtaining such results using potential theory and, time permitting,its extension to other mathematical models. This is joint work with N.Levenberg and F.Wielonsky. No prior knowledge of random matrices will be assumed.

Homotopy Theory

Speaker: Karol Szumilo (Western)

"Freudenthal suspension theorem"

Time: 13:00 - 14:30

Room: MC 107

I will present a proof of the Freudenthal suspension theorem in HoTT. Unlike many other proofs in HoTT, this one is not merely an adaptation of a classical argument, but involves genuinely new techniques.

Algebra Seminar

Speaker: Tatyana Barron (Western)

"Vector-valued automorphic forms"

Time: 14:30 - 15:30

Room: MC 107

Modular forms are complex-valued functions on the upper-half plane $H$ that have certain properties related to the action of $SL(2,Z)$ on $H$ (or another discrete subgroup of $G=SL(2,R))$. Theory of modular forms has always included modular forms with values in a complex vector space, too. Dimension of spaces of vector-valued automorphic forms has been computed, in certain arithmetic situations, not only for $G=SL(2,R)$ but also for other linear groups (e.g. $G=SU(n,1)$, $Sp(n,1))$. This will be a survey-style talk, with the intention to to have a glimpse at a certain part of the theory of vector-valued automorphic forms, including some classical results and some very recent results.

Graduate Seminar

Speaker: Mitsuru Wilson (Western)

"EXISTENCE OF DEFORMATION QUANTIZATION ON POISSON MANIFOLDS"

Time: 11:20 - 12:30

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 - 15:00

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 - 15:30

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 - 14:30

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 - 15:30

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 - 16:30

Room: MC 108

TBA