Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Department Colloquium

Department Colloquium - Eugene A. Feinberg (Stony Brook University)

Eugene A. Feinberg (Stony Brook University)

Friday, October 11th, 2019

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

Speaker: Eugene A. Feinberg (Stony Brook University)

Title: Fatou's Lemmas for Varying Probabilities and their Applications to Sequential Decision Making.

Abstract: The classic Fatou lemma states that the lower limit of expectations is greater or equal than the expectation of the lower limit for a sequence of nonnegative random variables. This talk describes several generalizations of this fact including generalizations to converging sequences of probability measures. The three types of convergence of probability measures are considered in this talk: weak convergence, setwise convergence, and convergence in total variation. The talk also describes the Uniform Fatou Lemma (UFL) for sequences of probabilities converging in total variation. The UFL states the necessary and sufficient conditions for the validity of the stronger inequality than the inequality in Fatou's lemma. We shall also discuss applications of these results to sequential optimization problems with completely and partially observable state spaces. In particular, the UFL is useful for proving weak continuity of transition probabilities for posterior state distributions of stochastic sequences with incomplete state observations known under the name of Partially Observable Markov Decision Processes. These transition probabilities are implicitly defined by Bayes' formula, and general method for proving their continuity properties have not been available for long time. This talk is based on joint papers with Pavlo Kasyanov, Yan Liang, Michael Zgurovsky, and Nina Zadoianchuk.

Prof. Eugene Feinberg is currently a Distinguished Professor in the Department of Applied Mathematics and Statistics at Stony Brook University. Before coming to Stony Brook, he help positions at Moscow State University of Railway Transportation and Yale. He obtained his Ph.D. from Vilnius University, Lithuania.

Prof. Feinberg is a Fellow of INFORMS and has received several awards including the 2012 IEEE Charles Hirsh Award, the 2012 IBM Faculty Award, and the 2000 Industrial Associates Award from Northrop Grumman.

Department Colloquium - Diane Maclagan (Warwick)

Diane Maclagan (Warwick)

Friday, October 4th, 2019

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

Speaker: Diane Maclagan (Warwick)

Title: Geometry of the moduli space of genus zero curves.

Abstract: The moduli space $\overline{M}_{0,n}$ of stable genus zero curves with $n$ marked points is a beautiful space that has been intensively studied by algebraic geometers and topologists for over half a century. It arises from a simple geometric question ("How can we arrange $n$ points on a sphere?"), but is the first nontrivial case of several interesting families of varieties (higher genus curves, stable maps, ...) and phenomena. Despite the long history there are still many mysteries about this variety. I will introduce this moduli space, and discuss some combinatorial approaches to understanding it.

Diane Maclagan (Warwick) is a Professor of Mathematics at the University of Warwick. She received her PhD from UC Berkeley, and moved to Warwick from Rutgers, following postdocs at IAS and Stanford. Her research is in Combinatorial Algebraic Geometry, with a particular focus on Tropical Geometry.

Department Colloquium - Kathryn Mann (Cornell University)

Kathryn Mann (Cornell University)

Friday, September 27th, 2019

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

Speaker: Kathryn Mann (Cornell University)

Title: Structure theorems for actions of homeomorphism groups.

Abstract: The groups $\mathrm{Homeo}(M)$ and $\mathrm{Diff}(M)$ of homeomorphisms or diffeomorphisms of a manifold $M$ have many striking parallels with finite dimensional Lie groups. In this talk, I'll describe some of these, and explain new work, joint with Lei Chen, that gives an orbit classification theorem and a structure theorem for actions of homeomorphism and diffeomorphism groups on other spaces, analogous to some classical results for actions of locally compact Lie groups. As applications, we answer many concrete questions towards classifying all actions of $\mathrm{Diff}(M)$ on other manifolds (many of which are nontrivial, for instance $\mathrm{Diff}(M)$ acts naturally on the unit tangent bundle of $M$...) and resolve several threads in a research program initiated by Ghys. I'll aim to give both a broad overview and several toy applications in the talk.

Professor Kathryn Mann received her PhD from the University of Chicago in 2014, she then held post-doctoral positions at MSRI, UC Berkeley and the Institut de mathématiques de Jussieu before becoming a Manning Assistant Professor of Mathematics at Brown University. In 2019, she joined Cornell University. Among her accolades is an Alfred P. Sloan Foundation Fellowship (2019), an NSF Career Award (2019), the AWM-Birman Research Prize in Topology and Geometry (2019), the Kamil Duszenko Award (2019) and the Mary Ellen Rudin young researcher award (2017).

Department Colloquium - Xudong Chen (CU Boulder)

Xudong Chen (CU Boulder)

Friday, September 20th, 2019

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

Speaker: Xudong Chen (CU Boulder)

Title: Structure Theory for Ensemble Control and Estimation of Nonholonomic Systems.

Abstract: Ensemble control deals with the problem of using a finite number of control inputs to simultaneously steer a large population (in the limit, a continuum) of individual control systems. As a dual, ensemble estimation deals with the problem of using a finite number of measurement outputs to estimate the initial state of every individual system in the (continuum) ensemble. We introduce in the talk a novel class of ensembles of nonlinear control systems, termed distinguished ensemble systems. Every such system has two key components, namely a set of finely structured control vector fields and a set of co-structured observation functions. In the first half of the talk, we demonstrate that the structure of a distinguished ensemble system can significantly simplify the analysis of ensemble controllability and observability. Moreover, such a structure can be used as a principle for ensemble system design. In the second half of the talk, we address the issue about existence of a distinguished ensemble system for a given manifold. We will focus on the case where the underlying space of every individual system is an arbitrary semi-simple Lie group or its homogeneous space.

Professor Chen is an Assistant Professor at the University of Colorado, Boulder. Before that, he was a postdoctoral fellow in the Coordinated Science Lab at UIUC. He obtained his Ph.D. degree in Electrical Engineering from Harvard University in 2014. His research interests are in the area of control theory, stochastic processes, optimization, game theory and their applications in modeling and control of large-scale networked systems.

Department Colloquium - Sarah Mayes-Tang (University of Toronto)

Sarah Mayes-Tang (University of Toronto)

Friday, September 13th, 2019

Time: 2:30 p.m.  Place: Jeffery Hall 127

Speaker: Sarah Mayes-Tang (University of Toronto)

Title: Why We Share Our Stories: Identity, Participation, and Celebration of Women in Math.

Abstract: Despite remarkable contributions by women mathematicians, the participation and recognition of women in mathematics remains unacceptably low. Women are usually excluded from the popular images of mathematicians, and the number of women in our academic departments lags behind most other STEM disciplines. How can we transform mathematics into a field where women are accepted, valued, and visible? In this talk, I will argue that mathematical stories shape participation in mathematics and I will advocate for the value of celebrating stories of women mathematicians, amplifying stories of girls and women doing mathematics, and sharing our own stories.

Professor Sarah Mayes-Tang is a Queen's alumni, she then got her Ph.D. at the University of Michigan and worked at Quest University before moving to the University of Toronto in 2017. Her research interests are in commutative algebra and in Mathematics education.

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.

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.

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.

Department Colloquium - Maksym Radziwill (Caltech)

Maksym Radziwill (California Institute of Technology)

Friday, March 22nd, 2019

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

Speaker: Maksym Radziwill (California Institute of Technology)

Title: Recent progress in multiplicative number theory.

Abstract: Multiplicative number theory aims to understand the ways in which integers factorize, and the distribution of integers with special multiplicative properties (such as primes). It is a central area of analytic number theory with various connections to $L$-functions, harmonic analysis, combinatorics, probability etc. At the core of the subject lie difficult questions such as the Riemann Hypothesis, and they set a benchmark for its accomplishments. An outstanding challenge in this field is to understand the multiplicative properties of integers linked by additive conditions, for instance $n$ and $n+ 1$. A central conjecture making this precise is the Chowla-Elliott conjecture on correlations of multiplicative functions evaluated at consecutive integers. Until recently this conjecture appeared completely out of reach and was thought to be at least as difficult as showing the existence of infinitely many twin primes. These are also the kind of questions that lie beyond the capability of the Riemann Hypothesis. However recently the landscape of multiplicative number theory has been changing and we are no longer so certain about the limitations of our (new) tools. I will discuss the recent progress on these questions.

Maksym Radziwill graduated from McGill University in Montreal in 2009, and in 2013 took a PhD under Kannan Soundararajan at Stanford University in California. In 2013-2014, he was at the Institute for Advanced Study in Princeton, New Jersey as a visiting member, and in 2014 became a Hill assistant professor at Rutgers University. In 2016, he became an assistant professor at McGill. In 2018, he became Professor of Mathematics at Caltech.

Department Colloquium - Henri Darmon (McGill)

Henri Darmon (McGill)

Friday, March 15th, 2019

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

Speaker: Henri Darmon (McGill)

Title: Kronecker's Jugendtraum: a non-archimedean approach.

Abstract: Kronecker's Jugendtraum ("dream of youth") seeks to construct abelian extensions of a given field through the special values of explicit analytic functions, in much the same way that the values of the exponential function $e(x) = e^{2\pi i x}$ at rational arguments generate all the abelian extensions of the field of rational numbers. Modular functions like the celebrated $j$-function play the same role when the field of rational numbers is replaced by an imaginary quadratic field. An extensive literature, both classical and modern, is devoted to the special values of modular functions at imaginary quadratic arguments, known as singular moduli; and modular functions have also been central to such modern developments as Wiles' proof of Fermat's Last Theorem and significant progress on the Birch and Swinnerton--Dyer conjecture arising through the work of Gross--Zagier and Kolyvagin. I will describe some recent attempts, in colaboration with Jan Vonk, to extend Kronecker's Jugendtraum to real quadratic fields by replacing modular functions by mathematical structures called "$p$-adic modular cocycles". While they are still poorly understood, these objects exhibit many of the same rich arithmetic properties as modular functions.

Prof. Darmon obtained his PhD from Harvard in 1991, then went on to Princeton before coming to McGill University in 94 where he is now a James McGill Professor. Prof. Darmon received many awards and distinctions, including a Sloan Research Award in 96, the Prix Andre Aisenstadt in 97, the Killam Research Fellowship in 2008, and both the Cole prize and the CRM-Fields-PIMS prize in 2017. Prof. Darmon was elected fellow of the Royal Society of Canada in 2003.