Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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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.

Curves Seminar - Eric Han (Queen's University)

Thursday, August 30th, 2018

Time: 1:30-2:30 p.m Place: Jeffery Hall 422

Speaker: Eric Han (Queen's University)

Title: The Hilbert Scheme of points on a surface and a related combinatorial problem

Abstract: We will introduce the idea of a Hilbert scheme, and in particular the Hilbert scheme of points on a surface. We will also briefly discuss a problem about the ‘limits of multiple points’, which is most properly expressed as the closure of a certain locus in the Hilbert scheme of points.

Number Theory - Richard Gottesman (Queen's University)

Wednesday, August 29th, 2018

Time: 2:30-3:20p.m.  Place: Jeffery Hall 422

Speaker: Richard Gottesman (Queen's University)


Abstract: I will give an introduction to vector-valued modular forms and describe my research on the arithmetic of vector-valued modular forms with respect to a representation of Gamma_0(2). 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 project line minus three points. We then are able to use 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 - Anup Dixit (Queen's University)

Wednesday, August 22nd, 2018

Time: 2:30-3:20p.m.  Place: Jeffery Hall 422

Speaker: Anup Dixit (Queen's University)


Abstract: In 1975, S. Voronin proved a fascinating result on Riemann zeta-function, which states that every non-vanishing holomorphic function on a compact set in the critical strip $1/2  \Re(s)  1$ is well approximated by vertical shifts of the zeta function, infinitely often. This is called the universality property of the Riemann zeta-function. This property can be shown for many familiar L-functions, for instance all L-functions in the Selberg class are known to be universal. Moreover, functions such as the Hurwitz zeta-function or Lerch zeta-function, which are not elements in the Selberg class also satisfy the universality property. This motivated Y. Linnik and I. Ibragimov to conjecture that every Dirichlet series, with has an analytic continuation and satisfies some "growth condition" must be universal. In this talk, we will formulate this conjecture more precisely and prove some partial results towards the conjecture.

Number Theory - Arpita Kar (Queen's University)

Tuesday, August 14th, 2018

Time: 2:30-3:20p.m.  Place: Jeffery Hall 422

Speaker: Arpita Kar (Queen's University)


Abstract: In 1917, G.H Hardy and S. Ramanujan coined the phrase ``normal order" and proved that $\omega(n)$ has normal order $\log \log n$. (Here, $\omega(n)$ denotes the number of distinct prime factors of $n$.) In other words, they showed that $\omega(n) \approx \log \log n$ for all but $o(x)$ many integers $n \leq x$, as $x \to \infty$. In this talk, we will show that the size of this exceptional set is, in fact $O(\frac{x}{(\log x)^A)})$ for any $A>0$, improving upon the work of Hardy, Ramanujan and Tur\'an.

Dynamics, Geometry, & Groups - Derrick Wigglesworth (Fields Inst.)

Friday, August 10th, 2018

Time: 10:30 am Place: Jeffery Hall 422

Speaker: Derrick Wigglesworth (Fields Institute)

Title: Groups acting on trees

Abstract: I'll discuss several of the ways one can learn about groups via their actions on trees.  There will be many examples and pictures.  Then, we'll briefly discuss folding paths; a tool for understanding complicated actions. Finally, I'll mention some applications of folding paths.

Number Theory - Siddhi Pathak (Queen's University)

Wednesday, August 1st, 2018

Time: 2:30-3:20p.m.  Place: Jeffery Hall 422

Speaker: Siddhi Pathak (Queen's University)

Title: Adjoint of the Serre derivative, Poincaré series and special values of shifted Dirichlet series.

Abstract: The Serre derivative is a linear differential operator from the space of modular forms of weight k to the space of modular forms of weight k+2. In this talk, we compute the adjoint of the Serre derivative with respect to the Petersson inner product, as done by A. Kumar. Using this as a pretext, we will highlight the connection between the Serre derivative and special values of shifted Dirichlet series via the tool of Poincaré series.