Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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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.

Number Theory - Anup Dixit (Queen's University)

Tuesday, July 24th, 2018

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

Speaker: Anup Dixit (Queen's University)


Abstract: For a number field $K/ \mathbb{Q}$, the class number $h_K$ captures how far the ring of integers of $K$ is from being a PID. The study of class numbers is an important theme in number theory. In order to understand how the class number varies upon varying the number field, Siegel showed that the class number times the regulator tends to infinity in any family of quadratic number fields. Brauer extended this result to family of Galois extensions over $\mathbb{Q}$. This is the Brauer-Siegel theorem. Recently, Tsfasman and Vladut conjectured a Brauer-Siegel statement for an asymptotically exact family of number fields. In this talk, we prove the classical Brauer-Siegel and the generalized version in several unknown cases. We also discuss some effective versions of Brauer-Siegel in the classical setting.

Number Theory - Kalyan Chakraborty (Harish-Chandra Research Institute)

Wednesday, July 18th, 2018

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

Speaker: Kalyan Chakraborty (Harish-Chandra Research Institute, Allahabad, India)


Abstract: Class number of a number field and in particular that of a quadratic number field, is one of the fundamental and mysterious objects in algebraic number theory and related topics. I will discuss some results concerning the divisibility of the class numbers of certain families of real (respectively, imaginary) quadratic fields in both qualitative and quantitative aspects. I will also look at the 3-rank of the ideal class groups of certain imaginary quadratic fields. The talk will be based on some recent works done along with my collaborators.