Department of Mathematics and Statistics

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

Lorne Campbell Lectureship - Olgica Milenkovic (UIUC)

Olgica Milenkovic (UIUC)

This is the second in a lecture series named in honour of Lorne Campbell, emeritus professor in the department, made possible by a generous donation from alumnus Vijay K. Bhargava, Professor of Electrical and Computer Engineering at the University of British Columbia.

Wednesday, November 27th, 2019

Time: 3:30 p.m.  Place: Jeffery Hall 127

Speaker: Olgica Milenkovic (UIUC)

Title: String reconstruction problems in molecular storage.

Abstract: String reconstruction problems frequently arise in many areas of genomic data processing, molecular storage, and synthetic biology. In the most general setting, they may be described as follows: one is given a single or multiple copies of a coded or uncodedstring, and the string copies are subsequently subjected to some form of (random) processing such as fragmentation or repeated transmission through a noise-inducing channel. The goal of the reconstruction method is to obtain an exact or approximate version of the string based on the processed outputs. Examples of string reconstruction questions include reconstruction from noisy traces, reconstruction from substrings and k-decks and reconstruction from compositional substring information. We review the above and some related problems and then proceed to describe coding methods that lead to strings that can be accurately reconstructed from their noisy traces, substrings and compositions. (This is a joint work with Ryan Gabrys, Han Mao Kiah, Srilakshmi Pattabiraman and Gregory Puleo.

Olgica Milenkovic is a professor of Electrical and Computer Engineering at the University of Illinois, Urbana-Champaign (UIUC), and Research Professor at the Coordinated Science Laboratory. She obtained her PhD from the University of Michigan, Ann Arbor. Her research interests include coding theory, bioinformatics, machine learning and signal processing.

Among her accolades, she received an NSF CAREER grant, the DARPA Young Faculty Award, the Dean’s Excellence in Research Award, and several best paper awards. She was elected a UIUC Center for Advanced Study Associate and Willett Scholar (2013) and became a Distinguished Lecturer of the Information Theory Society (2015). She is an IEEE Fellow and has served as Associate Editor and Guest-Editor-in-Chief of several leading IEEE journals.

Department Colloquium - Alexei Novikov (Penn State)

Alexei Novikov (Penn State)

Friday, November 15th, 2019

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

Speaker: Alexei Novikov (Penn State)

Title: The Noise Collector for sparse recovery in high dimensions.

Abstract: The ability to detect sparse signals from noisy high-dimensional data is a top priority in modern science and engineering. A sparse solution of the linear system $Ax=b$ can be found efficiently with an $l_1$-norm minimization approach if the data is noiseless. Detection of the signal's support from data corrupted by noise is still a challenging problem, especially if the level of noise must be estimated. We propose a new efficient approach that does not require any parameter estimation. We introduce the Noise Collector (NC) matrix $C$ and solve an augmented system $Ax+Cy=b+e$, where $e$ is the noise. We show that the $l_1$-norm minimal solution of the augmented system has zero false discovery rate for any level of noise and with probability that tends to one as the dimension of $b$ increases to infinity. We also obtain exact support recovery if the noise is not too large, and develop a Fast Noise Collector Algorithm which makes the computational cost of solving the augmented system comparable to that of the original one. I'll introduce this new method and give its geometric interpretation.

Prof. Alexei Novikov obtained his Ph.D.~from Stanford in 1999 and then held postdoctoral positions at the IMA and at CalTech before joining the Pennsylvania State University where he is now a Professor in the Department of Mathematics. Prof. Novikov specializes in applied analysis and probability. His research has been supported by the NSF since 2006, as well as by the US--Israel Binational Science Foundation from 2005-2009.

Department Colloquium - Ari Arapostathis (UT Austin)

Ari Arapostathis (UT Austin)

Friday, November 8th, 2019

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

Speaker: Ari Arapostathis (UT Austin)

Title: Lower bounds on the rate of convergence for heavy-tailed driven SDEs motivated by large scale stochastic networks.

Abstract: We show that heavy-tailed Levy noise can have a dramatic effect on the rate of convergence to the invariant distribution in total variation. This rate deteriorates from the usual exponential to strictly polynomial under the presence of heavy-tailed noise. To establish this, we present a method to compute a lower bound on the rate of convergence. We should keep in mind that standard Foster-Lyapunov theory furnishes only an upper bound on this rate. To motivate the study of such systems, we describe how L\'evy driven stochastic differential equations arise in the study of stochastic queueing networks. This happens when the arrival process is heavy-tailed, or the system suffers asymptotically negligible service interruptions. We identify conditions on the parameters in the drift, the Levy measure and/or covariance function which result in subexponential and/or exponential ergodicity, and we show that these conditions are sharp. In addition, we show that for the queueing models described above with no abandonment, the rate of convergence to the stationary distribution in total variation is polynomial, and we provide a sharp quantitative characterization of this rate via matching upper and lower bounds. We conclude by presenting analogous results on convergence in the Wasserstein distance.

This talk is based on joint work with Hassan Hmedi, Guodong Pang and Nikola Sandric.

Ari Arapostathis is a professor in the Department of Electrical and Computer Engineering at The University of Texas at Austin, and holds the Texas Atomic Energy Research Foundation Centennial Fellowship in Electrical Engineering. He received his BS from MIT and his PhD from U.C. Berkeley, in 1982. He is a Fellow of the IEEE, and was a past Associate Editor of the IEEE Transactions on Automatic Control and the Journal of Mathematical Systems and Control. His research has been supported by several grants from the National Science Foundation, the Air-Force Office of Scientific Research, the Army Research Office, the Office of Naval Research, DARPA, the Texas Advanced Research/Technology Program, Samsung, and the Lockheed-Martin Corporation.

Department Colloquium - Atabey Kaygun (Istanbul Tech University)

Atabey Kaygun (Istanbul Technical University)

Friday, November 1st, 2019

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

Speaker: Atabey Kaygun (Istanbul Technical University)

Title: Noncommutative Geometry for Fun and Profit.

Abstract: There are whole fields of mathematics devoted to transferring problems of geometry and topology to commutative algebra (and vice versa) and solving them. This practice produced different "dictionaries'" of terms that tell us which type of objects in the realm of geometry or topology correspond to which other types of objects in the realm of algebra. In this talk, I am going to describe such a dictionary from the perspective of a homological algebraist who forgoes commutativity "for fun and profit" going through K-theory, cyclic and Hochschild homology, Hopf algebras, and quantum groups.

Prof. Atabey Kaygun works on homological and homotopical algebra in the context of noncommutative geometry. He obtained his Ph.D. from The Ohio State University in 2005. He was a postdoctoral fellow at the University of Western Ontario, KMMF-Warsaw University, Max-Plank-Institut fur Mathematik and University of Buenos Aires before joining the faculty of Bahcesehir University in 2009. He has been an associate professor at the Istanbul Technical University since 2016. Prof. Kaygun is currently on sabbatical and visiting Queen's University.

Department Colloquium - Jeremy Quastel (University of Toronto)

Jeremy Quastel (University of Toronto)

Friday, October 18th, 2019

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

Speaker: Jeremy Quastel (University of Toronto)

Title: The KPZ fixed point.

Abstract: The one dimensional KPZ universality class contains random growth models, directed random polymers, stochastic Hamilton-Jacobi equations (e.g.~the eponymous Kardar--Parisi--Zhang equation). It is characterized by unusual scale of fluctuations, some of which appeared earlier in random matrix theory, and which depend on the initial data, the explanation being that on large scales everything approaches a special scaling invariant Markov process, the KPZ fixed point, which turns out to be a new type of integrable system, leading to unexpected connections between probability and dispersive partial differential equations.

Prof. Jeremy Quastel specializes in probability theory, stochastic processes and partial differential equations. He obtained is Ph.D.~from the Courant Institute at NYU. He was a postdoctoral fellow at the MSRI in Berkeley, then was a faculty at UC-Davis until he returned to Canada in 1998, where he is now a professor at the University of Toronto and the current chair of the Mathematics department.

Among his accolades, Prof. Quastel received a Sloan Fellowship in 1996, was an invited speaker at the ICM in 2010, gave the Current Developments in Mathematics 2011 and St. Flour 2012 lectures, and was a plenary speaker at the International Congress of Mathematical Physics in Aalborg 2012. He is a fellow of the Royal Society of Canada.

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.