Events

Joey Lakerdas-Gayle, University of Waterloo

Symmetrically indivisible and elementarily indivisible structures

A first order structure M is indivisible if for every colouring of M into two colours, there is a monochromatic substructure N of M that is isomorphic to M. We will consider two stronger properties: M is symmetrically indivisible if N can be chosen so that every automorphism of N extends to an automorphism of M; and M is elementarily indivisible if N can be chosen to be an elementary substructure of M. We will discuss Model-Theoretic methods developed by Kojman and Geschke (2008), Hasson, Kojman, and Onshuus (2009), and Meir (2019) to study the relationships between these notions.

MC 5403

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

Adina Goldberg, University of Waterloo

Synchronous Quantum Games

We recast nonlocal games using string diagrams, allowing for a natural extension to quantum games (with bipartite question and answer states). We define strategies in this setting and show that synchronous quantum games require synchronized players to win. We give examples of some quantum games on quantum graphs and see that these require quantum homo/isomorphisms to win. (The talk is based on a preprint ``Quantum games and synchronicity'' (https://arxiv.org/abs/2408.15444). This work is inspired by Musto, Reutter, and Verdon's paper ``A compositional approach to quantum functions'', and relies heavily on the reference ``Categories for Quantum Theory'' by Heunen and Vicary for string diagrams in quantum information.)

MC 5417

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

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

Viktor Majewski, Humboldt University Berlin

Resolutions of Spin(7)-Orbifolds

In Joyce’s seminal work, he constructed the first examples of compact manifolds with exceptional holonomy by resolving flat orbifolds. Recently, Joyce and Karigiannis generalised these ideas in the G2 setting to orbifolds with Z2-singular strata. In this talk I will present a generalisation of these ideas to Spin(7) orbifolds and more general isotropy types. I will highlight the main aspects of the construction and the analytical difficulties.

MC 5479

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

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

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

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