Random Matrix Theory Seminar

Speaker: Nathan Pagliaroli (Western)

"Exploring Free Probability I"

Time: 14:40

Room: MC 106

The field of Free Probability was first started in the 1980’s by Dan-Virgil Voiculescu. He was investigating a property called freeness in the context of operator algebras. Eventually these ideas developed in a way to study non-commutative probability spaces. This later lead to connections with random matrices. Properties and concepts in Free Probability often have direct analogues with those in classical probability. We will begin with developing the notation of free independence which corresponds to independence in classical probability theory.

Geometry and Combinatorics

Speaker: Dan Bath (KU Leuven)

"Logarithmic Comparison Theorems for Hyperplane Arrangements, Twisted or Otherwise"

Time: 15:30

Room: Zoom

In the 1990s, Terao and Yuzvinsky conjectured that reduced hyperplane arrangements satisfy the Logarithmic Comparison Theorem, asserting that the logarithmic de Rham complex computes the cohomology of the arrangement's complement. Essentially, this replaces the Brieskorn algebra in Brieskorn's Theorem with the logarithmic de Rham complex. We prove this conjecture by, among other things, sharply bounding the Castelnuovo--Mumford regularity of logarithmic j-forms of a central, essential, reduced arrangement. Time permitting we will discuss how to extend this untwisted Logarithmic Comparison Theorem to a twisted version. Here the twisted logarithmic de Rham complex computes the cohomology of the arrangement's complement with coefficients the rank one local system corresponding to the twist. Unlike the twisted Orlik--Solomon algebra, which can only computes a subset of the rank one local systems on the complement, this generalization computes all such rank one local systems.

Algebra Seminar

Speaker: Juan Esteban Rodriguez Camargo (ENS de Lyon)

"Solid locally analytic representations of p-adic Lie groups"

Time: 14:30

Room: ZOOM

In this joint work with Joaquin Rodrigues Jacinto, we develop the theory of locally analytic representations from the perspective of Condensed mathematics of Clausen-Scholze. Taking as inspiration foundational works on the subject from Lazard/Schneider - Tetelbaum/Emerton, etc., we reprove and generalize some cohomological comparisons between continuous, locally analytic and Lie algebra cohomology.

Random Matrix Theory Seminar

Speaker: Nathan Pagliaroli (Western)

"Exploring Free Probability II"

Time: 14:40

Room: MC 106

We will explore examples of Freeness and Voiculescu's original motivation for its introduction.

Algebra Seminar

Speaker: Christian Maire (Université de Franche-Comté, Besançon, France)

"On Galois representations with open image unramified outside p"

Time: 14:30

Room: ZOOM

In this lecture we show the existence of odd Galois representations with open image and unramified outside p. We combine some properties of the extensions of number fields with restricted ramification, with embedding tools in the context of lifting mod p Galois representations.

Geometry and Combinatorics

Speaker: Chris Kapulkin (Western)

"Cubical setting for discrete homotopy theory"

Time: 15:30

Room: MC 107

Discrete homotopy theory, introduced by H. Barcelo and collaborators, is a homotopy theory of (simple) graphs. Homotopy invariants of graphs have found numerous applications, for instance, in the theory of matroids, hyperplane arrangements, topological data analysis, and time series analysis. Discrete homotopy theory is also a special instance of a homotopy theory of simplicial complexes, developed by R. Atkin, to study social and technological networks.

I will report on the joint work with D. Carranza (arXiv:2202.03516) on developing a new foundation for discrete homotopy theory, based on the homotopy theory of cubical sets. We use this foundation to prove the conjecture of Babson, Barcelo, de Longueville, and Laubenbacher from 2006 relating homotopy groups of a graph to the homotopy groups of a certain cubical complex associated to it, as well as a discrete homotopy theory analogue of the Hurewicz theorem.

Algebra Seminar

Speaker: Senate Meeting - No Seminar (Western)

"no seminar"

Time: 14:30

Room: MC 108

Transformation Groups Seminar

Speaker: Kumar Shukla (Western)

"The integration formula of Atiyah-Bott and Berline-Vergne"

Time: 10:30

Room: MC 204

Let $T$ be a torus. On a manifold with a $T$-action which has finitely many fixed points, the integration of an equivariantly closed form can be evaluated as a finite sum of its moment map on the fixed point set. We will give a proof of this localization formula for circle action (assuming certain facts about finite-dimensional representations of circle). We will then work out an explicit example of integrating volume form of the 2-sphere using this formula.

Analysis Learning Seminar

Speaker: Michael Francis (Western)

"Nef foliations of complex surfaces"

Time: 09:30

Room: MC 107

We will discuss what it means for a holomorphic foliation (possibly with singularities) of a complex surface into complex curves to be “nef”. Then we will look at some examples. These nef foliations are important for a classification program proposed by Brunella and others in the spirit of the Enriques classification of complex surfaces.

Algebra Seminar

Speaker: Vlad Serban (École Polytechnique Fédérale de Lausanne)

"New sphere packing lower bounds"

Time: 14:30

Room: ZOOM

We give a brief overview of the best lower bounds on the packing density that can be obtained from lattice sphere packings in n-dimensional Euclidean space, focusing on large n. We then show how, using division rings, various existing constructions can be extended and new effective lower bounds can be uncovered in many dimensions. This is joint work with Nihar Gargava.

Random Matrix Theory Seminar

Speaker: Luuk Verhoeven (Western)

"Gauge theory from classical to fuzzy"

Time: 14:40

Room: MC 106

We'll take a look at physical models coming from gauge theory. Starting from (parts of) the standard model as usual, followed by the NCG example of almost commutative manifolds and onto the goal of a gauge theory over a ``fuzzy'' manifold as in Perez-Sanchez.

Transformation Groups Seminar

Speaker: Kumar Shukla (Western)

"Applications of Localization Formula"

Time: 10:30

Room: MC 204

We will go over some applications of the Atiyah-Bott localization formula. We will evaluate the integral of the volume form of $S^2$ as a sum over the fixed point set of the rotation action of $S^1$ on $S^2$. A more non-trivial application will be in the enumerative problem of counting number of lines in a cubic surface in $\mathbb{CP}^3$. We will explain why this problem is well-posed and then compute the number using Atiyah-Bott formula.