Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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CYMS Seminar - Richard Gottesman (Queen's University)

Thursday, September 20th, 2018

Time: 11:30 a.m - 12:20 p.m Place: Jeffery Hall 422

Speaker: Richard Gottesman (Queen's University)

Title: Vector-Valued Modular Forms on $\Gamma_{0}(2)$C

Abstract: The collection of vector-valued modular forms form a graded module over the graded ring of modular forms. I will explain how understanding the structure of this module allows one to show that the component functions of vector-valued modular forms satisfy an ordinary differential equation whose coefficients are modular forms. In certain cases, we can use a Hauptmodul to transform such a differential equation into a Fuchsian differential equation on the projective line minus three points. We then are able to use the Gaussian hypergeometric series to explicitly solve this differential equation.
Finally, we make use of these ideas together with some algebraic number theory to study the prime numbers that divide the denominators of the Fourier coefficients of the component functions of vector-valued modular forms.

Number Theory - Zhen Zhao (Queen's University)

Tuesday, September 18th, 2018

Time: 10:00 a.m.  Place: Jeffery Hall 422

Speaker: Zhen Zhao (Queen's University) 

Title: Asymptotics of the dimensions of irreducible representations of the symmetric group

Abstract: We will discuss the results of the 1985 paper of S. V. Kerov and A. M. Vershik giving the asymptotic rates growth for the dimensions of the largest and 'typical' dimensions of the irreducible representations of the symmetric group of order n.

Geometry & Representation - Tianyuan Xu (Queen's University)

Monday, September 17th, 2018

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Tianyuan Xu (Queen's University)

Title: Broken lines and the topological ordering of the alternating quiver of type A

Abstract:  The Positivity Conjecture in cluster algebra theory states that the coefficients of the Laurent expansion of any cluster variable in a cluster algebra are always positive integers. In 2014, Gross, Hacking, Keel and Kontsevich constructed a so-called Theta function basis to prove the conjecture for all cluster algebras of geometric type. A key ingredient in the construction of the Theta functions is the broken line model. In this talk, we will discuss the broken lines associated to the alternating quiver of type A, with an emphasis on relating its combinatorial properties to the topological ordering of the quiver, the partial order obtained by taking the transitive and reflexive losure of the relation “v<w if v->w is an edge” on the vertices of the quiver.

The talk is based work in progress with Ba Nguyen, David Wehlau and Imed Zaguia.

Department Colloquium - Anthony Bloch (University of Michigan)

Anthony Bloch, University of Michigan

Friday, September 14th, 2018

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

Speaker: Anthony Bloch (University of Michigan)

Title: Control and Geometry of Quantum systems with Dissipation

Abstract: In this talk we discuss aspects of the mathematics, control and geometry of quantum control systems interacting with their environment. In particular we discuss the control of a finite-dimensional dissipative Lindblad system by considering its orbit and interorbit dynamics. This entails considering the geometry of the system and its unitary orbits, the structure of the Lindblad operator, and the convexity associated with the density equation. Applications are given to constructing pure states. We discuss controllability and also discuss optimality and optimal control in this setting.

Anthony Bloch (University of Michigan): Anthony M. Bloch is the Alexander Ziwet Collegiate Professor of Mathematics and current department chair at the University of Michigan. He received his B.Sc.~in Applied Mathematics and Physics from the University of the Witwatersrand, Johannesburg, an M.~S.~in Physics from CalTech, an M.~Phil in Control Theory and Operations Research from Cambridge and a Ph.~D in Applied Mathematics from Harvard. He has received many awards including a Presidential Young Investigator Award, a Guggenheim Fellowship and a Simons Fellowship and is Fellow of the IEEE, SIAM and the AMS.

Dynamics, Geometry, & Groups - Jacob Russell (CUNY)

Friday, September 14th, 2018

Time: 10:30 a.m Place: Jeffery Hall 422

Speaker: Jacob Russell (CUNY)

Title: The geometry of groups via their boundaries

Abstract: Gromov revolutionized the study of finitely generated groups by purposing the study of groups as geometric objects. The success of this geometric viewpoint has inspired a whole program of classifying groups geometrically. From the geometry of the group, one can construct various boundaries; topological spaces which record the geometry of the group "at infinity". One of these boundaries, the Morse boundary, is particularly nice as the group has a natural action on it by homeomorphisms. The Morse boundary can also be equipped with a natural cross-ratio and we will discuss how the topology of the boundary coupled with this cross-ratio is actually sufficient to encode the entire geometry of the group.

CYMS Seminar - Yasuhiro Goto (Hokkaido University)

Thursday, September 13th, 2018

Time: 11:30 a.m - 12:20 p.m Place: Jeffery Hall 422

Speaker: Professor Yasuhiro Goto (Hokkaido University of Education, Hakodate)

Title: Formal groups of low dimensional Calabi-Yau varieties

Abstract: Calabi-Yau varieties are associated with formal groups of dimension one. When they are defined over an algebraically closed field of positive characteristic, the formal groups are classified by the height. Using weighted Delsarte varieties of low dimensions (say, $2$, $3$ or $4$), we describe how to compute the height of their formal groups and show various numerical data for them.

Number Theory - M. Ram Murty (Queen's University)

Tuesday, September 11th, 2018

Time: 10:00 a.m.  Place: Jeffery Hall 422

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

Title: Some reflections on Hasse's inequality (Part 1)

Abstract: In 1935, Hasse proved the Riemann hypothesis for zeta functions attached to elliptic curves (mod p) which was originally conjectured in the 1922 doctoral thesis of Emil Artin. There are two papers, one by Davenport in 1931 and another by Mordell in 1933 that discuss elementary approaches to this conjecture that give suprisingly non-trivial estimates. Both papers make use of a clever averaging argument that later appears in Bombieri's proof of Weil's theorem on the Riemann hypothesis for curves. We will give a motivated (and somewhat leisurely) discourse on these developments.

Geometry & Representation - Charles Paquette (Queen's/RMC)

Monday, September 10th, 2018

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Charles Paquette (Queen's/RMC)

Title: A quiver construction of some subalgebras of asymptotic Hecke algebras

Abstract:  Lusztig defines an asymptotic Hecke algebra J from a Coxeter system (W,S). This is an algebra that is defined using the Kazhdan-Lusztig (KL) basis of the corresponding Hecke algebra of (W,S). Even though these KL bases are generally hard to understand, there is a two-sided cell C of W that gives rise to a nice subalgebra J_C of J having rich combinatorics and whose algebraic description does not use KL bases. We will see that J_C has a presentation using a quiver with relations, and this allows one to study the representation theory of J_C (and of J) from another perspective. Using quiver representations, we will see that the classification of simple modules, which falls into three categories (finite type, bounded type and unbounded type), can be characterized completely using the shape of the weighted graph G of (W,S).

This is joint work with I. Dimitrov, D. Wehlau and T. Xu.

Department Colloquium - Oleg Bogoyavlenskij (Queen's University)

Oleg Bogoyavlenskij, Queen's University

Friday, September 7th, 2018

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

Speaker: Oleg Bogoyavlenskij (Queen's University)

Title: Counterexamples to Moffatt’s statements on vortex knots

Abstract: One of the well-known problems of hydrodynamics is studied - the problem of classification of vortex knots for ideal fluid flows. In the literature there are known Moffatt's statements that all torus knots $K_{m,n}$ for all rational numbers $m/n$ $(0 < m/n < \infty)$ are realized as vortex knots for each one of the considered axisymmetric fluid flows. We prove that actually such a uniformity does not exist because it does not correspond to the facts. Namely, we derive a complete classification of all vortex knots realized for the fluid flows studied by Moffatt and demonstrate that the real structure of vortex knots is much more rich because the sets of mutualy non-isotopic vortex knots realized for different axisymmetric fluid flows are all different.

Dynamics, Geometry, & Groups - Camille Horbez

Friday, September 7th, 2018

Time: 10:30 a.m Place: Jeffery Hall 422

Speaker: Camille Horbez (Laboratoire de Mathématiques d’Orsay)

Title: Growth under automorphisms of hyperbolic groups

Abstract: Let G be a finitely generated group, let S be a finite generating set of G, and let f be an automorphism of G. A natural question is the following: what are the possible asymptotic behaviors for the length of f^n(g), written as a word in the generating set S, as n goes to infinity, and as g varies in the group G?

We investigate this question in the case where G is a torsion-free Gromov hyperbolic group. Growth was completely described by Thurston when G is the fundamental group of a hyperbolic surface, and can be understood from Bestvina-Handel’s work on train-tracks when G is a free group. We address the case of a general torsion-free hyperbolic group. We show in particular that every element g has a well-defined exponential growth rate under iteration of f, and that only finitely many exponential growth rates arise as g varies in G.

This is a joint work with Rémi Coulon, Arnaud Hilion and Gilbert Levitt.