Events

Jack Jia, University of Waterloo

Langlands Correspondence : Local, Global, and Possibly Geometric.

The classical global Langlands Correspondence is one of the deepest phenomenon in mathematics: In GL_1, it is equivalent to class field theory; in GL_2, it subsumes the modularity conjecture, which leads to the proof of Fermat’s last theorem. (Disclaimer: I will not talk about these). In this talk, I will explain what the local and global L.C. means, and talk about the geometrization if time permits.

Join us for the Grad colloquium to meet your fellow graduate students. Refreshments will start at 5:00, and the talk will start at 5:30.

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

Kaleb Ruscitti, University of Waterloo

Correspondence between logarithmic connections and framed parabolic bundles on the blow up of a nodal Riemann surface

In this seminar I will explain how a Mehta-Seshadri type correspondence between logarithmic connections and parabolic vector bundles works for a specific setting of interest. That is the blow up of the complex curve xy=t at the nodal point.

MC 5403

Guillaume Dumas, University of Maryland

Boundedness of weak quasi-cocycles for higher rank simple groups

If G is a second countable locally compact group, the Delorme-Guichardet theorem states that Kazhdan property (T) is equivalent to the fixed-point property for continuous affine isometric actions on Hilbert spaces—that is, every 1-cocycle with values in a Hilbert space is bounded. Many rigidity statements rely on property (T): for example, morphisms of G into R are trivial. However, it does not provide tools for studying quasi-homomorphisms, since these maps do not respect the group structure. In order to study this class of maps, Ozawa introduced wq-cocycles, which respect a cocycle identity up to a bounded error. A group is said to have property (TTT) if all wq-cocycles are bounded. In this talk, I will discuss the relationship between this property and other more analytical forms of “almost” property (T). I will also explain how to prove that a group possesses this property, with a focus on simple groups and their lattices.

QNC 1507 or Join on Zoom

Christine Eagles, University of Waterloo    

Algebraic Independence of solutions to multiple Lotka-Volterra systems          

A major problem in recent applications of the model theory of DCF_0 is determining when a given system of algebraic differential equations defines a strongly minimal set. A definable set S is strongly minimal if it is infinite and for any other definable set R (over any set of

parameters), either S\cap R or S\setminus R is finite. In joint work with Yutong Duan and Leo Jimenez, we classify exactly when the solution set to a Lotka-Volterra system is strongly minimal. In the strongly minimal case, we classify all algebraic relations between Lotka-Volterra systems and show that for any distinct solutions x_1,...,x_n (not in the algebraic closure of the base field F), trdeg(x_1, ..., x_m/F) = 2m.

MC 5403

Siyuan Lu, McMaster University 

Interior C^2 estimate for Hessian quotient equation

In this talk, we will first review the history of interior C^2 estimates for fully nonlinear equations. As a matter of fact, very few equations admit this property, not even the Monge-Ampère equation in dimension three or above. We will then present our recent work on interior C^2 estimate for Hessian quotient equation. We will discuss the main idea behind the proof. If time permits, we will also discuss the Pogorelov-type interior C^2 estimate for Hessian quotient equation and its applications.

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

Spiro Karigiannis, University of Waterloo

Schwarz Lemma for Smith maps

I will discuss a generalized Schwarz Lemma for Smith maps, proved recently by Broder-Iliashenko-Madnick, and explain how it generalizes the classical Schwarz Lemma from complex analysis.

MC 5403

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

Jon Cheah, University of Hong Kong

An advertisement of cluster algebras

Cluster algebras have had many surprising links with many areas of mathematics beyond their original purpose in studying total positivity. In this expository talk, we consider two discrete dynamical systems, namely Markov triples and Coxeter--Conway friezes. While the study of these examples predated that of cluster theory, we will see how the latter provides a conceptual explanation for the intergrality and positivity phenomena.

MC 5479

(snacks at 16:00)