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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

Nicolas Banks, University of Waterloo

Non-Trivial Theorems with Trivial Proofs

One of the most fruitful things we can do as mathematicians is to think deeply about simple things. As students and researchers, perhaps we come across results with simple proofs and believe that not much can be learned from them. In this talk, I will challenge this misconception by diving into three important, non-trivial theorems with seemingly trivial proofs - Desargue's Theorem of planar geometry, the finite intersection property of compact sets, and Lagrange's Theorem from group theory. These will demonstrate three reasons that a profound truth need not be complicated.

MC 5501

(snacks at 17:00)

Francisco Villacis, University of Waterloo

Computing the Quantum Cohomology

In this talk, I will compute the quantum cohomology ring of projective space and of the Grassmannian. If time permits, I will outline the computation of the quantum cohomology of generic quintic threefolds and their connections to the count of rational curves of a given degree on these.

MC 2017

Kaleb D Ruscitti, University of Waterloo

Real Analytic Varieties and Singularities

Analytic varieties have the flavour of algebraic geometry, but are also foreign in many ways. Of course, over the complex numbers, Serre showed that analytic and algebraic varieties are strongly related. Over the real numbers however, things are more interesting.

In this talk I will review the definition of analytic completion, analytic spaces, and their relationship to algebraic varieties. Then I will focus on the real case, and talk about singularities of real analytic spaces and real normal crossings divisors.

MC 5479

Kuntal Banerjee, University of Waterloo

Very stable and wobbly loci for elliptic curves

We explore very stable and wobbly bundles, twisted in a particular sense by a line bundle, over complex algebraic curves of genus 1. We verify that twisted stable bundles on an elliptic curve are not very stable for any positive twist. We utilize semistability of trivially twisted very stable bundles to prove that the wobbly locus is always a divisor in the moduli space of semistable bundles on a genus 1 curve. We prove, by extension, a conjecture regarding the closedness and dimension of the wobbly locus in this setting. This conjecture was originally formulated by Drinfeld in higher genus.

MC 5501

Kain Dineen, University of Waterloo

Symplectic Capacities and Rigidity

As an application of Gromov's non-squeezing theorem, we'll prove that the symplectomorphisms (and anti-symplectomorphisms) of (ℝ^2m, 𝜔_0) are exactly the diffeomorphisms that additionally preserve the capacity of every compact ellipsoid. If time permits, then we will use this to prove that if a sequence of symplectomorphisms of any symplectic manifold (M, 𝜔) converges in the C^0-sense to a diffeomorphism 𝜓, then 𝜓*𝜔 = ± 𝜔.

MC 5479

Erik Séguin, University of Waterloo

A Selected Topic on Fourier-Stieltjes Algebras of Locally Compact Hausdorff Groups

We discuss a particular selected topic on Fourier-Stieltjes algebras of locally compact Hausdorff groups. Time permitting, we may complete the proof a lemma.

MC 5403