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Dynamics, Geometry and Groups Seminar

Monday, September 25th, 2023

Time: 11:00 a.m.  Place: Jeffery Hall, Room 319

Speaker: Thomas Barthelmé (Queen's University)

Title: Partially hyperbolic diffeomorphisms in dimension 3.

Abstract: I'll talk about recent progress around the problem of classifying partially hyperbolic diffeomorphisms on 3-manifolds. In particular I'll discuss a special class of such diffeomorphisms called collapsed Anosov flows, how they are conjectured to give all partially hyperbolic diffeomorphisms, and the construction of a new type of such example (in progress, with Federico Salmoiraghi).

Math & Stats Department Colloquium

Friday, September 22nd, 2023

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Helge Gloeckner (Paderborn)

Title: Flows, diffeomorphism groups, and regularity

Abstract: Infinite-dimensional Lie groups provide a framework to study parameter-dependence of solutions to differential equations on a (say) compact manifold M, even in situations where a time-dependent vector field is only measurable in time. In the talk, an introduction to infinite-dimensional Lie groups is provided. Concepts of regularity of Lie groups are described. In the case of Lie groups of smooth or real-analytic diffeomorphisms, these are intimately related to the parameter-dependence of flows of differential equations on M.

Bio: Helge Glöckner studied Mathematics and Physics at Darmstadt and Imperial College London. Following research at Erlangen and Darmstadt, he received a Ph.D. in Mathematics in 1999. As a post-doc, Glöckner held researcher positions at Darmstadt and Göttingen, as well as a position of visiting assistant professor at LSU (Baton Rouge). In 2007, he became a Heisenberg Professor for Infinite-Dimensional Analysis and Geometry at Paderborn University, financed by the German Research Council (DFG). The latter was transformed into a permanent full professorship in 2012.

Curves Seminar

Thursday, September 21st, 2023

Time: 4:00 p.m.  Place: Jeffery Hall, Room 102

Speaker: Sonja Ruzic

Title: Total positivity: A motivation for cluster algebras

Abstract: In this talk, we cover the concept of total positivity and total non-negativity of n x n matrices. We look at general square matrices, Grassmannians, and basic affine spaces. We do not define the cluster algebra yet. However, these examples include notions which will appear in the formal definition of a cluster algebra. This talk will aim to cover Chapter 1 of the notes and serve as motivation for the study of cluster algebras.

Number Theory Seminar

Tuesday, September 19th, 2023

Time: 3:00 p.m.  Place: Jeffery Hall, Room 319

Speaker: Brad Rodgers (Queen's University)

Title: A survey on random matrix theory and the Riemann zeta-function

Abstract: In this talk we will survey some of the history of how random matrix theory has come into the study of the Riemann zeta-function, focusing in particular on questions surrounding moments and local spacings between zeros, and the arithmetical inputs for studying such questions. If there is sufficient time I will touch on recent work relating approximations of arithmetic functions to random matrix theory.

Algebra & Geometry Seminar

Monday, September 25th, 2023

Time: 4:30 p.m.  Place: Jeffery Hall, Room 422

Speaker: Tianyuan Xu (Haverford College)

Title: 2-roots for simply-laced Weyl groups.

Abstract: We introduce and study “2-roots”, which are symmetrized tensor products of orthogonal roots of Kac–Moody algebras. We concentrate on the case where W is the Weyl group of a simply laced Y-shaped Dynkin diagram with three branches of arbitrary finite lengths a, b and c; special cases of this include types D_n , E_n (for arbitrary n ≥ 6), and affine E_6 , E_7 and E_8.

We construct a natural codimension-1 submodule M of the symmetric square of the reflection representation of W, as well as a canonical basis B of M that consists of 2-roots. The 2-roots of W and the basis B share many similarities with the roots and the simple roots of W. In particular, we show that every 2-root is a linear combination of the basis B with coefficients of like sign. We also describe a natural partial order on the set of 2-roots and explicitly describe the "highest 2-roots", the maximal 2-roots with respect to this order. (This is joint work with Richard Green.)

Algebra & Geometry Seminar

Monday, September 18th, 2023

Time: 4:30 p.m.  Place: Jeffery Hall, Room 422

Speaker: Charles Paquette (RMC)

Title: The study of bricks in representation theory, and how they arose from cluster algebras.

Abstract: In this talk, I will discuss some of the implications that the theory of cluster algebras had in representation theory of finite dimensional algebras. In particular, it led to the study of some particular indecomposable modules. One such family is the set of bricks, where a brick is a module having a division ring as endomorphism ring. These modules also play a fundamental role in the study of geometric invariant theory for finite dimensional algebras, or in the study of torsion theory. Although important objects, there are still many open questions concerning bricks. My plan is to explore a few of those, from different perspectives, and provide some partial answers.

Number Theory Seminar

Tuesday, September 12th, 2023

Time: 3:00 p.m.  Place: Jeffery Hall, Room 319

Speaker: M. Ram Murty (Queen's University)

Title: ARITHMETICAL FUNCTIONS OF SEVERAL VARIABLES

Abstract: We will survey several results in the theory of arithmetical functions of several variables, a subject that is not often treated in number theory textbooks though it should be!

Algebra & Geometry Seminar

Monday, September 11th, 2023

Time: 4:30 p.m.  Place: Jeffery Hall, Room 422

Speaker: Sasha Zotine (Queen's University)

Title: Computing Higher Direct Images of Toric Morphisms

Abstract: Sheaf cohomology is a ubiquituous tool in algebraic geometry for understanding the structure of varieties---but how does one actually get one's hands on cohomology? In this talk, I will discuss computing sheaf cohomology (and higher direct images) of toric varieties, which translate geometry into combinatorics. This translation is far more accessible and amenable to computation, allowing us to get a more tangible grasp of the abstract constructions. In particular, I implemented an algorithm for computing the higher direct images of toric morphisms for line bundles in Macaulay2, which I will demonstrate.

Math & Stats Department Colloquium

Friday, September 8th, 2023

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Yevgeny Liokumovich (U of T)

Title: Minimal surfaces, metric geometry and scalar curvature

Abstract: Which manifolds admit metrics of positive scalar curvature? What are the geometric properties of such metrics? Minimal surfaces play a key role in the recent progress made in answering these questions. These developments also inspired new results about a macroscopic analog of scalar curvature in the geometry of metric spaces. I will describe the interplay between scalar curvature and macroscopic scalar curvature, and explain how this relationship yields new findings in both areas.