Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Curves Seminar—Gregory G. Smith (Queen's University)

Wednesday, September 11th, 2019

Time: 4:00-5:30 p.m.   Place: Jeffery Hall 319

Speaker: Gregory G. Smith (Queen's University)

Title: Minimal free resolutions

Abstract: With the year-long objective of understanding the syzygies of generic canonical curves, we begin with a gentle introduction to minimal free resolutions of homogeneous ideals in a standard graded polynomial ring.

Number Theory Seminar - Seoyoung Kim (Queen's University)

Tuesday, September 10th, 2019

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

Speaker: Seoyoung Kim (Queen's University)

Title: The Sato-Tate conjecture and Nagao’s conjecture.

Abstract: Nagao’s conjecture relates the rank of an elliptic surface to a limit formula arising from a weighted average of fibral Frobenius traces, and it is further generalized for smooth irreducible projective surfaces by Hindry and Pacheco. We show that the Sato-Tate conjecture for abelian surfaces studied by Fit\'{e}, Kedlaya, Rotger, Sutherland implies Nagao’s conjecture for certain twist families hyperelliptic curves of genus 2. Moreover, one can relate analogous discussions for higher genus g to the non-vanishing result on the symmetric power $L$-functions, from which analogous proof will hold for certain cases.

Department Colloquium - Bahman Gharesifard (Queen's University)

Bahman Gharesifard (Queen's University)

Friday, September 6th, 2019

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

Speaker: Bahman Gharesifard (Queen's University)

Title: Fundamental Limits in Control and Optimization of Networked Systems.

Abstract: The emergence of network sciences within the disciplines of engineering, biological, and social systems has revealed numerous opportunities for sensing and feedback. This development, together with the availability of an abundance of useful data, has provided capabilities that allow for the execution of remarkably complex tasks, which cannot be handled by individual systems. I will provide an overview of some of the recent advancements on control and optimization of large-scale networked systems, mathematically modelled as dynamical systems with external inputs over graphs. The talk will focus on fundamental limits to decentralization; I will show graph-theoretic conditions that decentralization imposes on the controllability and stabilization of sparse systems, as well as a class of submodoular optimization problems. One key objective throughout the talk is to showcase the versatile set of mathematical tools that naturally enter the study of networked systems.

CYMS Seminar - Richard Gottesman (Queen's University)

Thursday, September 5th, 2019

Time: 2:30 p.m Place: Jeffery Hall 319

Speaker: Richard Gottesman (Queen's University)

Title: Vector-Valued Modular Forms and Modular Linear Differential Equations

Abstract: The sequence of denominators of the Fourier coefficients of a modular form on a congruence subgroup is always bounded. It has been conjectured that the converse is also true. We will consider this problem in the context of vector-valued modular forms and explain a strategy for proving such an unbounded denominator result. A key point is the importance of understanding the solutions of the modular linear differential equation at all of the cusps).

Dynamics, Geometry, & Groups - Catherine Pfaff (Queen's University)

Tuesday, August 27th, 2019

Time: 1:30 p.m Place: Jeffery Hall 319

Speaker: Catherine Pfaff (Queen's University)

Title: Typical Trees: An Out(F_r) Excursion.

Abstract: Random walks are not new to geometric group theory (see, for example, work of Furstenberg, Kaimonovich, Masur). However, following independent proofs by Maher and Rivin that pseudo-Anosovs are generic within mapping class groups, and then new techniques developed by Maher-Tiozzo, Sisto, and others, the field has seen in the past decade a veritable explosion of results. In a 2 paper series, we answer with fine detail a question posed by Handel-Mosher asking about invariants of generic outer automorphisms of free groups and then a question posed by Bestvina as to properties of R-trees of full measure in the boundary of Culler-Vogtmann outer space. This is joint work with Ilya Kapovich, Joseph Maher, and Samuel J. Taylor.

Dynamics, Geometry, & Groups - Marco Lenci (Universita di Bologna)

Friday, August 23rd, 2019

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Marco Lenci (Universita di Bologna)

Title: Infinite-volume mixing and the case of one-dimensional maps with an indifferent fixed point.

Abstract: I will first discuss the question of mixing in infinite ergodic theory, which will serve as a motivation for the introduction of the notions of "infinite-volume mixing". Then I will focus on a prototypical class of infinite-measure-preserving dynamical systems: non-uniformly expanding maps of the unit interval with an indifferent fixed point. I will show how the definitions of infinite-volume mixing play out in this case. As it turns out, the most significant property, and the hardest to verify, is the so-called global-local mixing, corresponding to the decorrelation in time between global and local observables. I will present sufficient conditions for global-local mixing, which will cover the most popular examples of maps with an indifferent fixed point (Pomeau-Manneville and Liverani-Saussol-Vaienti). If time permits, I will also present some peculiar limit theorems that can be derived for these systems out of the property of global-local mixing.

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

Wednesday, August 7th, 2019

Time: 3:30-4:30 p.m.  Place: Jeffery Hall 319

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

Title: AN INTRODUCTION TO THE CIRCLE METHOD.

Abstract: In 1916, in their ground-breaking paper deriving an asymptotic formula for the partition function, Hardy and Ramanujan introduced a powerful method in analytic number theory called the circle method. In a series of subsequent papers, Hardy and Littlewood developed the method to study Waring's problem, Goldbach's conjecture and other additive questions in number theory. I will present a gentle introduction to this method and indicate how it can be used to solve Waring's problem following an arrangement due to Linnik.

Number Theory Seminar - Akshaa Vatwani (IIT Gandhinagar)

Wednesday, July 31st, 2019

Time: 3:30-4:30 p.m.  Place: Jeffery Hall 319

Speaker: Akshaa Vatwani (IIT Gandhinagar)

Title: The Voronoi Summation Formula for a generalized divisor function.

Abstract: In 1904, Vorono\" {\i} formulated a conjecture for arithmetical functions, which is a generalization of the Poisson summation formula. He was able to establish this conjecture for the divisor function $d(n)$, giving a relation between Bessel functions and weighted sums of the divisor function. Such formulas have since been generalized to various arithmetical functions. We report on joint work with Atul Dixit, Bibekananda Maji and A. Sankaranarayanan, giving such a formula for the generalized divisor function:
$$
\sigma_{k,z} (n) = \sum_{d^k |n }d^z, \qquad \qquad k\in \mathbb N, \, z\in \mathbb{C}.
$$

Number Theory Seminar - Francesco Cellarosi (Queen's University)

Wednesday, July 24th, 2019

Time: 3:30-4:30 p.m.  Place: Jeffery Hall 319

Speaker: Francesco Cellarosi (Queen's University)

Title: COMPUTING KRONECKER SYMBOLS IN THE UNIVERSAL COVER OF SL_2(R).

Abstract: The Kronecker symbol generalizes Jacobi and Legendre symbols, and is intimately connected to the theory of Dirichlet characters. For co-prime integers $c$ and $d$ such that $c \equiv 0 \bmod 4$ and $d \equiv 1 \bmod 4$, I will explain an algorithm to compute the Kronecker symbol $(c/d)$ using a free group inside the universal cover of SL_2(R). Joint work with J. Griffin.

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