Events

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

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

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

Elizabeth Cai, University of Waterloo

Mirror Symmetry Seminar: Isomorphism Between Small Analytical Neighborhoods of Points on (n − s − 1)-dim Stratum, Open Ball and Affine Toric Variety

In Batryrev's construction on dual polyhedra and mirror Symmetry for Calabi-Yau hypersurfaces in toric varieties, when he introduces regularity conditions for hypersurfaces, he proposes a theory implied by the definition of ∆-regular, in which sugguests that there exists an analytical isomorphism from small analytical neighbourhoods of points on a (n − s − 1)-dimensional stratum Zf,σ ⊂ Zf,Σ to products of a (s − 1)-dimensional open ball and a small analytical neighbourhood of the point pσ on the (n − s)-dimensional affine toric variety Aσ,N(σ). This theory and its corollaries help obtain a simultanious resolution of all members of the family F(∆). 

MC 2017

Annie Lafrance, University of Waterloo

Introduction to p-approximation property for locally compact groups

We will introduce the p-approximation property and show that if G has the p-approximation property, then the algebra of convoluters is the algebra of pseudomeasures.

MC 5403

Spiro Karigiannis, University of Waterloo

Organizational Meeting

We will plan out the DG working seminar for the May to August summer period. The plan is to have two talks per week, from 1:00pm to 2:15pm and from 2:30pm to 3:34pm.

MC 5403

Soham Chakraborty, École Normale Supérieure

Measured groupoids and the Choquet-Deny property

A countable discrete group is called Choquet-Deny if for every non-degenerate probability measure on the group, the corresponding space of bounded harmonic functions is trivial. Recently a complete characterization of Choquet-Deny groups was obtained by Frisch, Hartman, Tamuz and Ferdowsi. In this talk, we will look at the extension of the Choquet-Deny property to the framework of discrete measured groupoids. Our main result gives a complete characterisation of this property in terms of the associated measured equivalence relation and the isotropy groups of the groupoid. This talk is based on a joint work with Tey Berendschot, Milan Donvil, Mario Klisse and Se-Jin Kim.

MC 5417 or Join on Zoom

Joaco Prandi, University of Waterloo

Bounding the Local Dimension of the Convolution of Measures

Let mu be a finite measure on a metric space X. Then the local dimension of the measure mu at the point x in the support of mu is given by

dim_{loc}mu(x)=lim_r ln(B(x,r))}\ln(r)

In a sense, dim_{loc}mu(x) represents how much mass there is around the point x. The bigger the local dimension, the less mass there is. In this talk, we will explore how the local dimension of the convolution of two measures mu and nu can be bounded by the local dimension of one of the measures. This is based on joint work with Kevin Hare.

MC5417

Enric Solé-Farré, University College London

The Hitchin and Einstein indices of cohomogeneity one nearly Kahler manifolds

Nearly Kähler manifolds are Riemannian 6-manifolds admitting real Killing spinors. They are the cross-sections of Riemannian cones with holonomy G2. Like the Einstein equation, the nearly Kähler condition has a variational interpretation in terms of volume functionals, first introduced by Hitchin in 2001.

The existence problem for nearly Kähler manifolds is poorly understood, and the only currently known inhomogeneous examples were found in 2017 by Foscolo and Haskins using cohomogeneity one methods. For one of their examples, we establish non-trivial bounds on the coindex of the Hitchin-type and Einstein functionals. We do this by analysing the eigenvalue problem for the Laplacian on coclosed primitive (1,1)-forms under a cohomogeneity-one symmetry assumption.

MC 5417