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Jashan Bal, University of Waterloo

Veech's theorem

We present Veech's theorem which states that for every nontrivial locally compact group there exists a compact Hausdorff space on which it acts continuously and freely. As a consequence, we obtain that no nontrivial locally compact group is extremely amenable.

MC 5403

Kieran Mastel, University of Waterloo

The weighted algebra approach to constraint system games

Entanglement allows for correlations between spatially separated experiments that are not possible classically. One way to study the computational power of entanglement is via nonlocal games. I will discuss my recent works with Eric Culf and William Slofstra on constraint system games. Different types of perfect entangled strategies for these games can be understood as representations of the algebra of the underlying constraint system. The weighted algebra formalism, introduced by Slofstra and me, extends this to non-perfect strategies. Using this formalism we can show that classical reductions between constraint systems are sound against quantum provers, which allows us to prove the RE-completeness of some constraint system games and to show that MIP* admits two prover perfect zero knowledge proofs.

MC 5417

Kira Bruschke, University of Watelroo, Centre for Career Development

Career Decision

The Career Talks seminar series aims to provide valuable advice and guidance for current graduate students. In this final seminar, a career advisor from the Centre for Career Development will provide a framework for career-decision making and resources for identifying career options.

MC 5417

Aleksa Vujicic, University of Waterloo

The Spine of a Fourier Algebra

Given a locally compact group G, one can define the Fourier and Fourier-Stieltjes algebras A(G) and B(G), which in the abelian case, are isomorphic to L1(G^) and M(G^) respectively. The Fourier algebra A(G) is typically more tractable than B(G), and often easier to describe. A notable exception is when B(G) = A(G), which occurs precisely when G is compact.
The spine of a Fourier Algebra A*(G), introduced by M. Ilie and N. Spronk, is a subalgebra of B(G) which contains all A(H)∘η  where η : G → H is a continuous homomorphism.
It has been shown that for G = Qp ⋊ Op*, that B(G) = A*(G), despite not being compact.
We also explore G = Qp^2 ⋊ Op*, where we have shown that although B(G) is strictly larger than A*(G), they are close to being similar.

MC 5417

Joey Lakerdas-Gayle, University of Waterloo

Infinitary Continuous Logic II

We will prove a continuous analog of Scott's Isomorphism Theorem using the Scott analysis for metric structures developed by Ben Yaacov, Doucha, Nies, and Tsankov.

MC 5479

Rahim Moosa, University of Waterloo

Model theory and the birational geometry of algebraic vector fields

I will try to illustrate how and why model theory (itself a branch of mathematical logic) can sometimes have something to say about algebraic geometry. I will focus on some results of mine (along with Jim Freitag and Remi Jaoui), from the last few years, on the birational geometry of algebraic vector fields.

MC 5479

Adina Goldberg, University of Waterloo

Synchronous and quantum games: Graphical and algebraic methods

This is a mathematics thesis that contributes to an understanding of nonlocal games as formal objects. With that said, it does have connections to quantum information theory and physical operational interpretations.

Nonlocal games are interactive protocols modelling two players attempting to win a game, by answering a pair of questions posed by the referee, who then checks whether their answers are correct. The players may have access to a shared quantum resource state and may use a pre-arranged strategy. Upon receiving their questions, they can measure this state, subject to some separation constraints, in order to select their answers. A famous example is the CHSH game of [Cla+69], where making use of shared quantum entanglement gives the players an advantage over using classical strategies.

This thesis contributes to two separate questions arising in the study of synchronous nonlocal games: their algebraic properties, and their generalization to the quantum question-and-answer setting. Synchronous games are those in which players must respond with the same answer, given the same question.

First, we study a synchronous version of the linear constraint game, where the players must attempt to convince the referee that they share a solution to a system of linear equations over a finite field. We give a correspondence between two different algebraic objects modelling perfect strategies for this game, showing one is isomorphic to a quotient of the other. These objects are the game algebra of [OP16] and the solution group of [CLS17]. We also demonstrate an equivalence of these linear system games to graph isomorphism games on graphs parameterized by the linear system.

Second, we extend nonlocal games to quantum games, in the sense that we allow the questions and answers to be quantum states of a bipartite system. We do this by quantizing the rule function, games, strategies, and correlations using a graphical calculus for symmetric monoidal categories applied to the category of finite dimensional Hilbert spaces. This approach follows the overall program of categorical quantum mechanics. To this generalized setting of quantum games, we extend definitions and results around synchronicity. We also introduce quantum versions of the classical graph homomorphism [MR16] and isomorphism [Ats+16] games, where the question and answer spaces are the vertex algebras of quantum graphs, and we show that quantum strategies realizing perfect correlations for these games correspond to morphisms between the underlying quantum graphs.

MC 2009 or Zoom: https://uwaterloo.zoom.us/j/92051331429?pwd=fl6rjZHC4X7itlJpaJaxwpfzJINQvG.1

Yidi Wang, University of Waterloo

Local-global principles on stacky curves and its application in solving generalized Fermat equations. 

The primitive solutions of certain generalized Fermat equations, i.e., 
Diophantine equations of the form Ax^p+By^q = Cz^r, can be studied as 
integral points on certain stacky curves. In a recent paper by Bhargava and 
Poonen, an explicit example of such a curve of genus 1/2 violating 
local-global principle for integral points was given. However, a general 
description of stacky curves failing the local-global principle is 
unknown. In this talk, I will discuss our work on finding the primitive 
solutions to equation of the form when (p, q, r) = (2,2,n) by studying local-global principles for integral points on stacky curves constructed from such equations. 
The talk is based on a joint project with Juanita Duque-Rosero, 
Christopher Keyes, Andrew Kobin, Manami Roy and Soumya Sankar. 

MC 5417

Zahra Janbazi, University of Toronto

Extensions of Birch-Merriman and Related Finiteness Theorems

A classical theorem of Birch and Merriman states that, for fixed n, the set of integral binary n-ic forms with fixed nonzero discriminant breaks into finitely many GL(2, Z)-orbits. In this talk, I’ll present several extensions of this finiteness result.

In joint work with Arul Shankar, we study a representation-theoretic generalization to ternary n-ic forms and prove analogous finiteness theorems for GL(3,Z)-orbits with fixed nonzero discriminant. We also prove a similar result for a 27-dimensional representation associated with a family of K3 surfaces.

In joint work with Sajadi, we take a geometric perspective and prove a finiteness theorem for Galois-invariant point configurations on arbitrary smooth curves with controlled reduction. This result unifies classical finiteness theorems of Birch–Merriman, Siegel, and Faltings.

MC 5479

Aleksa Vujicic, University of Waterloo

Fourier Algebras of Semi-Direct Product Groups of Local Fields

We look at Fourier Algebras of Semi-Direct Product Groups of Local Fields.

MC 5403