Curves Seminar

Thursday, September 28th, 2023

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

Speaker: Calvin Fletcher

Title: Total positivity of flag minors

Abstract: Last week we saw our first piece of motivation for studying cluster algebras. In this talk we explore another motivational example, namely, total positivity of flag minors. These are an important subset of matrix minors and are related to the basic affine space by invariant theory. In much the same modus as last week we will construct a ring corresponding to these flag minors, study its generators and see how these generators form extended clusters with important combinatorial data. We will also compare and contrast the examples given to help motivate further the topic of cluster algebras.

Math & Stats Department Colloquium

Friday, September 29th, 2023

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

Speaker: Federico Salmoiraghi (Queen’s University)

Title: Foliation, contact structures and Anosov flows in dimension 3.

Abstract: An example of the beautiful intertwine between hyperbolic dynamics, foliation theory, and contact geometry is given by an Anosov flow. Geometrically an Anosov flow is defined by two transverse invariant foliations with expanding and contracting behaviours. Much of our understanding of the structure of an Anosov flow relies on the study of the leaves space of the invariant foliations. Mitsumatsu first noticed that an Anosov vector field also belongs to the intersection of two transverse contact structures rotating towards each other. After giving the necessary background, I will show how to use this point of view to address questions in the theory of surgery on Anosov flows.

Bio: Federico Salmoiraghi joined Queen’s as a Coleman postdoctoral fellow in 2022, before that he was postdoc at the Technion from 2019 to 2022 after receiving his PhD from Liousiana State University in 2019. Prof. Salmoiraghi’s work is at the intersection of contact geometry and dynamical systems.

Number Theory Seminar

Tuesday, September 26th, 2023

Time: 4:00 p.m.  Place: Jeffery Hall, Room 319

Speaker: Mike Roth (Queen's University)

Title: The Bugeaud-Corvaja-Zannier theorem and extensions.

Abstract: Two integers a and b are called multiplicatively independent if the only solution (m,n) to a^m = b^n is (m,n)=(0,0). In 2003 Bugeaud, Corvaja, and Zannier proved that, if a and b are multiplicatively independent, then for every epsilon > 0 the inequality log gcd(a^n -1, b^n -1) < (epsilon) n holds for all but finitely many n > 0. In this talk we will discuss the BCZ result and give a new proof of their theorem. This proof makes it easier to see the key ingredients which make the argument work. The method of proof applies equally well to prove an extension of the theorem due to Corvaja and Zannier, and a more general extension due to Aaron Levin.

This is joint work with David McKinnon at Waterloo.

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!