Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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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. Jon Sensinger (UNB)

Monday, April 29th, 2019

Time: 13:00 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 23nrd, 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.

Department Colloquium - Jorge Cortes (UC San Diego)

Jorge Cortes (University of California, San Diego)

Friday, April 5th, 2019

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

Speaker: Jorge Cortes (University of California, San Diego)

Title: The Role of Network Structure in Controlling Complex Networks.

Abstract: Controllability of complex network systems is an active area of research at the intersection of network science, control theory, and multi-agent coordination, with multiple applications ranging from brain dynamics to the smart grid and cyber-physical systems. The basic question is to understand to what extent the dynamic behavior of the entire network can be shaped by changing the states of some of its subsystems, and decipher the role that network structure plays in achieving this. This talk examines this question in two specific instances: characterizing network controllability when control nodes can be scheduled over a time horizon and hierarchical selective recruitment in brain networks. Regarding controllability, we show how time-varying control schedules can significantly enhance network controllability over fixed ones, especially when applied to large networks. Through the analysis of a novel scale-dependent notion of nodal centrality, we show that optimal time-varying scheduling involves the actuation of the most central nodes at appropriate spatial scales. Regarding hierarchical selective recruitment, we examine network mechanisms for selective inhibition and top-down recruitment of subnetworks under linear-threshold dynamics. Motivated by the study of goal-driven selective attention in neuroscience, we build on the characterization of key network dynamical properties to enable, through either feedforward or feedback control, the targeted inhibition of task-irrelevant subnetworks and the top-down recruitment of task-relevant ones.

Jorge Cortes is a Professor with the Department of Mechanical and Aerospace Engineering at the University of California, San Diego. He received his Ph.D. degree in engineering mathematics from the Universidad Carlos III de Madrid, Spain, in 2001 and held postdoctoral positions at the University of Twente, The Netherlands, and at the University of Illinois at Urbana-Champaign, USA. He was an Assistant Professor with the Department of Applied Mathematics and Statistics at the University of California, Santa Cruz from 2004 to 2007. He is an IEEE Fellow and is currently its Director of Operations and an elected member (2018-2020) of its Board of Governors. He has received many prestigious award including the NSF CAREER award in 2006, the 2006 Spanish Society of Applied Mathematics Young Researcher Prize, the 2008 IEEE Control Systems Outstanding Paper Award, the 2009 SIAM Review SIGEST selection from the SIAM Journal on Control and Optimization, and the 2012 O. Hugo Schuck Best Paper Award in the Theory category. His current research interests include distributed control and optimization, network neuroscience, reasoning and decision making under uncertainty, resource-aware control, and multi-agent coordination in robotic, power, and transportation networks.

Math Club - Francesco Cellarosi (Queen's University)

Thursday, April 4th, 2019

Time: 5:30 - 6:30 p.m Place: Jeffery Hall 118

Speaker: Francesco Cellarosi (Queen's University)

Title: The irrationality of $\zeta(3)$.

Abstract:  This talk will discuss the famous theorem by R. Apéry who showed in 1978 that $\zeta(3)$ is irrational.
We will see a proof of this fact, due to F. Beuker in 1979, which only uses calculus.

Geometry & Representation - Ba Nguyen (Queen's University)

Monday, April 1st, 2019

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

Speaker: Ba Nguyen (Queen's University)

Title: A Combinatorial Model for Detecting Cluster Variables of Type D Cluster Algebras

Abstract:  Perfect Matchings of a Snake Diagram, which was developed by G. Musiker, R. Schiffler and L. Williams, can be employed to compute cluster variables of a type D cluster algebra. Based on that model we have developed a new model called Globally Compatible Sequence. In this talk, we will discuss how to find cluster variables using this model. To have a better understanding of those cluster variables, description of their denominator vectors will also be introduced.

Department Colloquium - Carolyn Gordon (Dartmouth College)

Carolyn Gordon (Dartmouth College)

Friday, March 29th, 2019

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

Speaker: Carolyn Gordon (Dartmouth College)

Title: Decoding geometry and topology from the Steklov spectrum of orbisurfaces.

Abstract: The Dirichlet-to-Neumann or "voltage-to-current" operator of, say, a surface $M$ with boundary is a linear map $C^\infty(\partial M)\to C^\infty(\partial M)$ that maps the Dirichlet boundary values of each harmonic function $f$ on M to the Neumann boundary values of $f$. The spectrum of this operator is discrete and is called the Steklov spectrum. The Dirichlet-to-Neumann operator also generalizes to the setting of orbifolds, e.g., cones. We will address the extent to which the Steklov spectrum encodes the geometry and topology of the surface or orbifold and, in particular, whether it recognizes the presence of orbifold singularities such as cone points.

This is joint work with Teresa Arias-Marco, Emily Dryden, Asma Hassannezhad, Elizabeth Stanhope and Allie Ray.

Prof. Gordon is an expert in spectral geometry. She obtained her PhD from Washington University in 1979, then went to the Technion institue and held positions at Lehigh University and Washington University before moving to Dartmouth where she is currently the Benjamin Cheney Professor of Mathematics.
Prof. Gordon was awarded an AMS Centennial Fellowship in 1990, the MAA Chauvenet prize in 2001 and was the 2010 Noether Lecturer. In 2012, she became a fellow of both the AMS and the American Association for the Advancement of Science. In 2017, she was selected to be a fellow of the AWM in the inaugural class.

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