webnotice

Laindon Burnett, University of Waterloo

A definable criterion for definability

In 2001, A.A. Muchnik proved the surprising result that for any n, there is a formula within Presburger arithmetic which takes in a predicate A and is true if and only if A is definable in Presburger arithmetic; that is, within this setting, the property of being definable is itself definable. We will go over the proof of this result and, time permitting, discuss its applications to automata theory.

MC 5403

Jennifer Zhu, University of Waterloo

Limits of Quantum Graphs

Quantum graphs were originally introduced as confusability graphs of quantum channels by Duan, Severini, and Winter. Weaver generalized a quantum graph to any weak-* closed operator system $\mathcal V \subseteq B(\mathcal H)$ that is bimodule over the commutant of some von Neumann algebra $\mathcal M \subseteq B(\mathcal H)$. To date, there seem to be two notions of quantum graph morphism. Weaver introduced and Daws extended a notion of CP morphism of quantum graphs. Musto, Reutter, and Verdon have also defined classical morphisms of quantum graphs in finite dimensions which agrees with CP morphisms in finite dimensions. Notably, however, these morphisms are not UCP maps between operator systems of the respective quantum graphs.

      Using a characterization of quantum relations as left ideals in the extended Haagerup tensor product, we will obtain a notion of quantum graph morphism (and hence limit) using the categories of von Neumann algebras and operator spaces. Time permitting, we will show that this limit recovers profinite classical graphs.

QNC 1507

Elan Roth, University of Waterloo

A Continuation of Random Binary Sequences

Filling some gaps left by last week's presenter, we'll show the K is a minimal information content measure and finally conclude that 1-Randomness and ML-Randomness are equivalent.

MC 5403

Facundo Camano, University of Waterloo

Dimensional Reduction of S^1-Invariant Instantons on the Multi-Taub-NUT

In this talk I will discuss the dimensional reduction of S^1-invariant instantons on the multi-Taub-NUT space to singular monopolos on R^3. I will first introduce the multi-Taub-NUT space, followed up by a discussion on S^1-equivariant principal bundles. Next, I will go over the natural decomposition of S^1-invariant connections into horizontal and vertical pieces, and then show how the self-duality equation reduces to the Bogomolny equation under said decomposition. I will then show how the smoothness of the instanton over the NUT points determines the asymptotic conditions for the singular monopole. Finally, I will go over the reverse construction: starting with a singular monopole on R^3 and building up to an S^1-invariant instanton on the multi-Taub-NUT space.

MC 5403

Shintaro Fushida-Hardy, University of Waterloo

Constructing Lagrangians in S2xS2

We investigate the existence of non-orientable Lagrangian surfaces in symplectic S^2xS^2s. On one hand, it is known that a Klein bottle embedded in S^2xS^2 cannot be Lagrangian in almost half of the possible symplectic structures on S^2xS^2. On the other hand, we use trisection-inspired methods to describe a general construction of Lagrangian surfaces, and ask "what is the minimal genus such that a non-orientable Lagrangian surface of said genus exists in every symplectic S^2xS^2?". This is joint work with Laura Wakelin.

MC 5417

Qirui Li, Pohang University of Science and Technology / Toronto

The higher linear Arithmetic Fundamental Lemma over function fields

The study of special values and derivatives of automorphic L-functions reveals deep connections between arithmetic geometry and harmonic analysis. A central theme is the Arithmetic Fundamental Lemma (AFL), which predicts precise identities between orbital integrals and intersection numbers of cycles. While methods based on perverse sheaves have achieved remarkable results in the function field case, they often obscure the underlying local geometry.

In this talk I will present recent progress on the higher linear AFL through the framework of the Relative Trace Formula (RTF). This approach provides explicit structural links between analytic orbital integrals and local intersection theory, enabling direct local proofs beyond global sheaf-theoretic methods. I will also outline several new directions: extending the AFL by adding conductors, and exploring their applications to global conjectures on derivatives of L-functions, including variants of Gross–Zagier type formulas.

MC 5417

Mateusz Wasilewski, Polish Academy of Sciences

Quantum graphs and their symmetries

I will present an overview of the theory of quantum graphs, which form a natural generalization of graphs from the point of view of operator algebras/quantum information. I will discuss several approaches to the theory, each coming with its own motivation. Just like in the classical case, studying symmetries is extremely important and this naturally leads to quantum groups. It turns out that to some extent one can go back: for a rich class of quantum groups one can construct quantum graphs, whose symmetries are given by the original quantum group. It is the quantum analogue of Frucht's theorem.

MC 5501

Noé de Rancourt , Université de Lille

Big Ramsey degrees of metric structures

Distortion problems, from Banach space geometry, ask about the possibility of distorting the norm of a Banach space in a significant way on all of its subspaces. Big Ramsey degree problems, from combinatorics, are about proving weak analogues of the infinite Ramsey theorem in structures such as hypergraphs, partially ordered sets, etc. Those two topics, coming back to the seventies, have quite disjoint motivations but share a surprisingly similar flavour. In a ongoing work with Tristan Bice, Jan Hubička and Matěj Konečný, as a step towards the unification of those two topics, we developped an analogue of big Ramsey degrees adapted to the study of metric structures (metric spaces, Banach spaces...). Our theory allows us to associate to certain metric structures a sequence of compact metric spaces quantifying their default of Ramseyness. In this talk, I'll present our theory and its motivations and illustrate it on the examples of the Banach space $\ell_\infty$ and the Urysohn sphere. If time permits, links with topological dynamics will also be discussed.

QNC 1507 or Join on Zoom

Thomas Sinclair, Purdue University

Model theory of metric lattices

We propose a general first-order framework for studying geometric lattices within the model theory of metric structures. As an application we develop a novel continuous limiting theory for finite partition lattices and discuss potential implications to their asymptotic combinatorics. This is joint work with Jose Contreras-Mantilla.

QNC 1507 or Join on Zoom

Paul Cusson, University of Waterloo

Moduli spaces of monopoles part 2

We continue discussing Euclidean SU(n)-monopoles, now in the case n >= 3, and we aim to describe their moduli spaces using spaces of rational maps from the projective line to flag varieties

MC 5403