webnotice

Mark Hamilton, Mount Allison University

Toric degenerations and independence of polarization

In the theory of geometric quantization, one essential ingredient is the choice of a "polarization"; a natural question is then whether the resulting quantization depends on this choice.  One recent approach to the question of "independence of polarization" is using a deformation of complex structure to "deform" one polarization into another.  Originally applied to smooth toric varieties, this has also been applied to a broader class of examples, such as flag varieties, by using a toric degeneration. 

In this talk I will present an overview of this program (including a short introduction to the key ideas of geometric quantization), and mention several examples of its application, including flag manifolds, more general varieties, and moduli spaces of flat connections (work in progress).

MC 5403

Kunjakanan Nath, IECL Nancy, France

Circle method and binary correlation problems

One of the key problems in number theory is to understand the correlation between two arithmetic functions. In general, it is an extremely difficult question and often leads to famous open problems like the Twin Prime Conjecture, the Goldbach Conjecture, and the Chowla Conjecture, to name a few. In this talk, we will discuss a few binary correlation problems involving primes, square-free integers, and integers with restricted digits. The objective is to demonstrate the application of Fourier analysis (aka the circle method) in conjunction with the arithmetic structure of the given sequence and the bilinear form method to solve these problems.

Zoom: https://uwaterloo.zoom.us/j/94276302733?pwd=stZaTKvufL02c5UlpyubhpXYkTSDoN.1

Meeting ID: 942 7630 2733 Passcode: 144512

Joey Lakerdas-Gayle, University of Waterloo

Fundamentals of Computability Theory 4

We will continue working through some examples of injury arguments, following Robert Soare's textbook.

MC 5403

Anand Pillay, University of Notre Dame du Lac

On theories of "nice" fields equipped with a generic derivation

There is a growing body of work on differential fields which are NOT differentially closed but nevertheless have a tractable model theory. I will discuss various results, including a description of definable groups and analogues of algebraic D-groups.

MC 5479

Faisal Romshoo, University of Waterloo

Special Lagrangian Geometry

I will talk about special Lagrangian submanifolds, which have garnered considerable interest in several areas in differential geometry and theoretical physics. In particular, I will describe some examples of special Lagrangian submanifolds explicitly.

MC 5479

Matthew Wiersma, University of Waterloo

Entropies and Poisson boundaries of random walks on groups with rapid decay

Let $G$ be a countable group and $\mu$ a probability measure on $G$. The Avez entropy of $\mu$ provides a way of quantifying the randomness of the random walk on $G$ associated with $\mu$. We build a new framework to compute asymptotic quantities associated with the $\mu$-random walk on $G$, using constructions that arise from harmonic analysis on groups. We introduce the notion of \emph{convolution entropy} and show that, under mild assumptions on $\mu$, it coincides with the Avez entropy of $\mu$ when $G$ has the rapid decay property. Subsequently, we apply our results to stationary dynamical systems consisting of an action of a group with the rapid decay property on a probability space, and give several characterizations for when the Avez entropy coincides with the Furstenberg entropy of the stationary space. This leads to a characterization of Zimmer amenability for stationary dynamical systems whenever the acting group has the property of rapid decay.

This talk is based on joint work with B. Anderson-Sackaney, T. de Laat and E. Samei.

MC 5417 or Zoom link below

https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09

Anand Pillay, University of Notre Dame

Quasirandomness of definable subsets of algebraic groups over finite fields

We give an arithmetic version of Tao’s algebraic regularity lemma (which was itself an improved Szemerédi regularity lemma for graphs uniformly definable in finite fields). In the arithmetic regime the objects of study are pairs (G,D) where G is a group and D an arbitrary subset. We obtain optimal results, namely that the algebraic regularity lemma holds for the associated bipartite graph (G,G,E) where E(x,y) is xy−1 ∈ D, witnessed by a the decomposition of G into cosets of a (uniformly definable) small index normal subgroup H of G. We compare to results of Green and Gowers. (This is joint work with Atticus Stonestrom.)

MC 5501

Riley Thornton, Carnegie Mellon University

Topological weak containment

Weak containment is a notion from ergodic theory with a wide variety of applications-- in dynamics, combinatorics, group theory, model theory, and beyond-- and a correspondingly wide variety of equivalent definitions. In this talk, I'll report on a project to adapt the theory to topological dynamics.

MC 5479

Jesse Huang, University of Waterloo

Birational coherent constructible correspondence

A major progress towards the Homological Mirror Symmetry (HMS) conjecture of Kontsevich is a version of HMS for toric varieties proved by Fang-Liu-Treumann-Zaslow and Kuwagaki using constructible sheaves, following an approach originally introduced by Bondal. These results suggest that Bondal's approach can be reinvested as a powerful tool to investigate fundamental algebraic questions pertaining to the birational geometry of toric varieties, and have inspired recent works of Hanlon-Hicks-Lazarev and my works with Favero, both used Bondal's map to obtain short resolutions of the diagonal by a specific collection of line bundles. In this talk, I will discuss these results and their connections to noncommutative resolutions of toric singularities and the broader goal to establish birational toric HMS.

MC 5417

Jacques Van Wyk, University of Waterloo

“The” Generalised Levi-Civita Connection

I will discuss the notions of generalised metrics and generalised connections in generalised geometry. A generalised connection has an associated torsion tensor, so one may ask, if given a generalised metric G, whether there is a torsion-free connection D compatible with G; this is the analogue of the Levi-Civita connection. We will see that there are infinitely many such connections D, that is, there is no unique “generalised Levi-Civita connection,” a striking difference from the situation for Riemannian geometry.

MC 5479