Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Number Theory - Steven Spallone (IISER Pune)

Wednesday, May 22nd, 2019

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

Speaker: Steven Spallone (IISER Pune)

Title: Divisibility of Character Values of Symmetric Groups

Abstract: Fix a permutation $\sigma$ in some symmetric group $S_k$, and consider it as sitting in $S_n$ for $n \geq k$. Also fix a positive integer $d$. The probability that an irreducible character $\chi$ of $S_n$ evaluated at $\sigma$ will be a multiple of $d$ approaches $100\%$ as $n$ approaches infinity. We will sketch a proof, which is joint work with Jyotirmoy Ganguly and Amritanshu Prasad.

Nonlocal problems in PDEs and geometry

May 20-24, 2019

Prof. Eleonora Cinti (Università di Bologna, Italy) will teach a 5-day mini-course aimed at graduate students and junior researchers at the intersection of Analysis and Geometry.

Dr. Cinti earned her Ph.D in 2010 and since then has worked at the Max Planck Institute in Leipzig, the Weierstrass Institute in Berlin, and various Italian universities (Pavia, Bologna, Torino). Eleonora Cinti's research focuses on nonlocal partial differential equations, geometric measure theory, and calculus of variations.

Schedule

The mini-course will be structured as follows:

  • Lecture 1 (Monday, May 20): Preliminaries: basic facts about the Laplacian and harmonic functions.
  • Lecture 2 (Tuesday, May 21): The fractional Laplacians: motivations and properties.
  • Problem Session (Wednesday, May 22).
  • Lecture 3 (Thursday, May 23): $s$-Harmonic functions and the Caffarelli-Silvestre extension theorem.
  • Lecture 4 (Friday, May 24): Geometry meets PDEs, a nonlocal phase transition model and nonlocal minimal surfaces.

Registration

If you want to attend the mini-course, please email nonlocal19@queensu.ca.

Control Theory Seminar - Prof. Melkior Ornik (UIUC)

Friday, May 17th, 2019

Time: 10:30 a.m Place: Jeffery 110

Speaker: Prof. Melkior Ornik (University of Illinois at Urbana-Champaign)

Title: Deception and Unpredictability in Stochastic Control

Abstract: In a number of adversarial scenarios, the success of an agent at achieving its objective rests on its use of a deceptive strategy: a strategy that enables the agent to progress towards its objective while manipulating the beliefs of the agent’s adversary about the nature of the agent. For instance, the agent may wish to instill incorrect beliefs about its location, identity, or objective, or it may simply wish to act seemingly unpredictably while still progressing towards its objective. In this talk, I will outline recent work on formalizing the notions of deception and unpredictability within the setting of Markov decision processes. I will begin by describing a basic approach that encodes deception through introducing a belief space for an adversary and a belief-induced reward objective, thus expressing deceptive strategies as control policies on a product state space. I will then discuss notions of unpredictability, deception, and counter-deception in scenarios with a temporal logic objective. I will relate unpredictability of an agent to the total Shannon entropy of its paths, and show that maximal unpredictability is achieved by following a policy that results in maximal total entropy of the induced Markov chain. Finally, I will express the notion of deception for temporal logic objectives using Kullback-Leibler divergence and show that optimal deceptive (for the agent) and counter-deceptive (for the adversary) policies can be synthesized as solutions of a convex optimization problem and a non-convex min-max problem, respectively. I will conclude with a brief discussion of open problems in the area of deceptive planning.

Number Theory - Anup Dixit (Queen's University)

Wednesday, May 15th, 2019

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

Speaker: Anup Dixit (Queen's University)

Title: On the distribution of the number of local prime factors of n

Abstract: Let $\omega(n)$ denote the number of distinct prime factors of $n$ and $\omega_y(n)$ denote the number of distinct prime factors of $n$ less than $y$. It was shown by Hardy and Ramanujan that typically $\omega(n)$ is $\log \log n$ up to an error term of $\sqrt{\log \log n}$. This was further generalized in the famous Erd\"{o}s-Kac theorem, which asserts that the probability distribution of $(\omega(n) - \log \log n)/(\sqrt{\log\log n})$ is the standard normal distribution.In this talk, we will prove analogous results for $\omega_y(n)$, which can be thought of as a local Erd\"{o}s-Kac theorem and describe its further implications. This is joint work with Prof. Ram Murty.

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

Wednesday, May 15th, 2019

Time: 2:00 p.m Place: Jeffery Hall 319

Speaker: Catherine Pfaff (Queen's University)

Title: Counting Conjugacy Classes: Groups Rebelling Against Dynamics.

Abstract: This talk will start with a gentle introduction to ways of viewing outer automorphisms of free groups and then will discuss joint work with Ilya Kapovich regarding counting conjugacy classes of these outer automorphisms. That is, inspired by results of Eskin and Mirzakhani counting closed geodesics of bounded length in the moduli space of a fixed closed surface, we consider a similar question in the Out(F_r) setting and discover bounds revealing behavior not present in the surface setting or in classical hyperbolic dynamical systems.

Number Theory - Siddhi Pathak (Queen's University)

Wednesday, May 8th, 2019

Time: 3:30-4:30 p.m.  Place: Jeffery Hall 110

Speaker: Siddhi Pathak (Queen's University)

Title: Convolution sums of values of the Lerch zeta-function

Abstract: In 1887, Lerch introduced the function, $\Phi(z, \alpha, s) := \sum_{n=0}^{\infty} \frac{ z^n } { {(n + \alpha)}^s },$ for $|z| = 1$, $0 < \alpha \leq 1$ and $\Re(s)>1$. This function is a generalization of the Riemann zeta-function, the Hurwitz zeta-functions as well as the polylogarithms. In this talk, we discuss convolution sum identities of values of the Lerch zeta-function at positive integers. This study is inspired by similar identities for values of the Riemann zeta-function, which were known to Euler, and leads one naturally into the realm of multiple Hurwitz zeta-functions. This is joint work with Prof. M. Ram Murty.

Probability Seminar - Sang-Gyun Youn (Queen's University)

Wednesday, May 1st, 2019

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Sang-Gyun Youn (Queen's University)

Title: Minimum output entropy and capacities of quantum channels.

Abstract:  In this talk, minimum output entropy (MOE) and Holevo/quantum capacity of quantum channels will be discussed. And I will sketch how the additivity conjecture of MOE was disproved by random unitary channels.

Free Probability and Random Matrices Seminar Webpage:

Control Theory Seminar - Prof. Jon Sensinger (UNB)

Monday, April 29th, 2019

Time: 10:30 a.m Place: Jeffery 110

Speaker: Prof. Jon Sensinger (UNB)

Title: Bottlenecks in rehabilitation human-machine interfaces: from mechanisms to control to human-machine interaction

Abstract: Humans can do amazing things compared with many robots. When humans interact with machines, it often leads to high expectations. These expectations are particularly high of human machine interfaces that try to assist (such as exoskeletons and prostheses) or rehabilitate (e.g., for stroke). Humans are complex, and the tasks they often wish to do require unique mechanisms and insightful control strategies. My personal bias is that solutions to these problems are often best solved using a control-theoretic framework.

This talk will highlight some of the mechanical and control bottlenecks that have limited the field, along with our contributions to help solve those problems. It will then turn to the field of computational motor control - a promising field that has used optimal stochastic feedback control theory to offer a compelling explanation for why humans move the way they do. The talk will briefly discuss the idea and some of the recent contributions by our group and others. From an engineering perspective, I will propose a holistic approach of including the person’s own capabilities, control strategies, and even level of interest, in the closed-loop design process. I will survey initial success of applying this approach to augmented sensory feedback, and lay out a vision for applying it to feedforward control as well. The talk will end by diving deeper into our most recent work developing a simple model of human adaptation. We've developed an inductive outcome measure that probes can infer from trial-by-trial data how confident people are in their feed-forward control. Challenges, limitations, and next steps will be discussed.

Dr. Jon Sensinger is the acting director of the Institute of Biomedical Engineering (IBME) at the University of New Brunswick and an associate professor in Electrical and Computer Engineering. Trained as a biomedical engineer and a clinical prosthetist, he directed the prosthesis design and control lab at the Rehabilitation Institute of Chicago and Northwestern University prior to coming to UNB. He has licenced several patents and is a cofounder of Coapt LLC, the first company to commercialize pattern recognition in the field of prostheses. He has a strong interest in seeing clinical problems through the lens of math - fusing theoretical paradigm shifts that result in meaningful clinical applications. As the acting director of IBME he directs a broad team comprising clinicians, scientists, engineers, professors, and graduate students who all share a passion to improve the lives of persons with disability. IBME has a 50+ year legacy of innovation in the field of prostheses and rehabilitation engineering, and Dr. Sensinger strives to maintain that focus as the field pushes the boundaries of rehabilitation engineering.

Number Theory - Chantal David (Concordia University)

Tuesday, April 23rd, 2019

Time: 1:30-2:30 p.m.  Place: Jeffery Hall 319

Speaker: Chantal David (Concordia University)

Title: Moments of cubic Dirichlet twists over function fields (Joint work with A. Florea and M. Lalin.)

Abstract: We obtain an asymptotic formula for the mean value of $L$--functions associated to cubic characters over $\F_q[T]$. We solve this problem in the non-Kummer setting when $q \equiv 2 \pmod 3$ and in the Kummer case when $q \equiv 1 \pmod 3$. The proofs rely on evaluating averages of cubic Gauss sums over function fields, which can be done using the theory of metaplectic Eisenstein series. In the non-Kummer setting, we display some explicit cancellation between the main term and the dual term coming from the approximate functional equation of the $L$--functions.

Dynamics, Geometry, & Groups - Reda Chhaibi (Toulouse)

Friday, April 12th, 2019

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

Speaker: Reda Chhaibi (Toulouse)

Title: Quantum SL_2, infinite curvature and Pitman's 2M-X Theorem.

Abstract: Pitman's theorem (1975) is an aesthetic theorem from probability theory, with geometry and representation theory related to SL_2, in disguise. Many proofs do exist, and the goal of this talk is to present a unified point of view regarding two proofs. - A proof by Bougerol and Jeulin - which generalizes to all semi-simple groups. They consider a Brownian motion on the symmetric space $H^3 = SL_2(\mathbb{C})/SU_2$, with varying curvature r and then take the limit $r \rightarrow \infty$. - Biane defined and studied quantum random walks on the enveloping algebra of SL_2, in the 90s. Then in the years 2000, he made the connection to the representation theory of the Jimbo-Drinfeld quantum group $\mathcal{U}_q(sl_2)$, in the crystal regime, i.e $q \rightarrow 0$.

Why should the crystal regime $q=0$ for quantum groups be related infinite curvature in symmetric spaces? The goal of this talk is to convince the audience that the parameter q, from the point of view of quantization and Kirillov's orbit method, is not a quantum parameter but indeed a curvature parameter. The simple relationship is q=e^{-r}, after a modification of the classical definition of quantum groups. I shall only mention the rank 1 group SL_2, and assume no knowledge of quantum groups, since they will have to be (re)defined anyway.

Joint work with F. Chapon.

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