Department of Mathematics and Statistics

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

Department Colloquium - Laura DeMarco (Northwestern University)

Laura DeMarco, Northwestern University

Friday, March 2nd, 2018

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

Speaker: Laura DeMarco (Northwestern University)

Title: Complex dynamics and arithmetic equidistribution

Abstract: About 5 years ago, Matt Baker and I formulated a conjecture about the dynamics of rational maps on P1, connecting geometry and arithmetic in the moduli space of such maps. My goal is to present recent progress on the conjecture, illustrating some of the main ideas appearing in proofs of special cases. One important special case includes a result about torsion points on elliptic curves, and I hope to discuss how this case can be related to dynamical stability and the Mandelbrot set.

Laura DeMarco (Northwestern University): Laura DeMarco received her Ph.D in Mathematics from Harvard University in 2002 under the supervision of Curtis McMullen. From 2002 to 2007, she was at he University of Chicago (as L.E. Dickinson Instructor from 2002 to 2005, and Assistant Professor from 2005 to 2007). From 2007 to 2014 she was at the University of Illinois at Chicago (as Assistant Professor from 2007 to 2009, Associate Professor from 2009 to 2012, and Professor from 2012 to 2014). In 2014, Prof DeMarco joined Northwestern University. Her awards include the NSF Postdoctoral Fellowship at the University of Chicago (2003-2006), the Sloan Foundation Research Fellowship (2008-2010), the NSF Career Award (2008-2013), the Simons Foundation Fellowship (2015-2016), and the Ruth LyttleSatter Prize (2017). In 2012, she became Fellow of the American Mathematical Society. Laura DeMarco is an Invited Speaker at the International Congress of Mathematicians in Rio de Janeiro in 2018. Her research interest include dynamical systems, complex analysis, and arithmetic geometry. She mainly focuses on the dynamics of rational maps on P1 and their moduli spaces.

Department Colloquium - Catherine Pfaff (UC-Santa Barbara)

Catherine Pfaff, UC-Santa Barbara

Friday, February 16th, 2018

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

Speaker: Catherine Pfaff (University of California-Santa Barbara)

Title: A Nielsen-Thurston Inspired Story of Iterating Free Group Automorphisms and Efficiently Deforming Graphs

Abstract: While many fundamental contributions to the study of outer automorphisms of free groups date back to the early 20th century, the real explosion of activity in the eld came with two much more recent developments: the denition by Culler and Vogtmann of the deformation space of metric graphs on a surface, namely Outer Space, and the development by Bestvina, Feighn, and Handel of a train track theory for outer automorphisms of free groups. The explosion was a result of a new ability to study free group outer automorphisms using generalizations of techniques developed to study surface homeomorphisms (mapping classes) via their action on the deformation space of metrics on the surface (Teichmuller space). In our talk, we focus specically on a Nielsen-Thurston inspired story jointly studying: 1) outer automorphism conjugacy class invariants obtained by iteratively applying the automorphisms and 2) geodesics in Culler-Vogtmann Outer Space.

Catherine Pfaff, (University of California-Santa Barbara): Catherine Pfa obtained her Ph.D. in Mathematics from Rutgers University in 2012 under the supervision of Lee Mosher. Dr. Pfa was Postdoctoral Research Fellow at the Universite d'Aix-Marseille (2013-2014) and at the Universitat Bielefeld (2014-2015). Since 2015, she is Ky Fan Visiting Assistant Professor at the University of California, Santa Barbara. Catherine Pfa's research focuses on geometric group theory and geometric topology. In particular, she studies the outer automorphism group of the free group and Outer Space from a mapping class group perspective.

Department Colloquium - Qiang Zeng (Northwestern University)

Qiang Zeng, Northwestern University

Wednesday, February 14th, 2018

Time: 3:30 p.m.  Place: Jeffery Hall 234

Speaker: Qiang Zeng, Northwestern University

Title: Replica Symmetry Breaking for Mean Field Spin Glass Models

Abstract: In statistical physics, the study of spin glasses was initialized to describe the low temperature state of a class of magnetic alloys in the 1960s. Since then spin glasses have become a paradigm for highly complex disordered systems. Mean eld spin glass models were introduced as an approximation of the physical short range models in the 1970s. The typical mean eld models include the Sherrington- Kirkpatrick (SK) model, the (Ising) mix p-spin model and the spherical mixed p-spin model. Starting in 1979, the physicist Giorgio Parisi wrote a series of ground breaking papers introducing the idea of replica symmetry breaking (RSB), which allowed him to predict a solution for the SK model by breaking the symmetry of replicas innitely many times at low temperature. This is known as full-step replica symmetry breaking (FRSB). In this talk, we will show that Parisi's FRSB prediction holds at zero temperature for the more general mixed p-spin model. As a consequence, at positive temperature the level of RSB will diverge as the temperature goes to zero. On the other hand, we will show that there exist two-step RSB spherical mixed spin glass models at zero temperature, which are the rst examples beyond the replica symmetric, one-step RSB and FRSB phases. This talk is based on joint works with Antonio Aunger (Northwestern University) and Wei-Kuo Chen (University of Minnesota).

Qiang Zeng (Northwestern University): Qiang Zeng obtained his Ph.D. in Mathematics from the University of Illinois at Urbana-Champaign in 2014 under the supervision of Marius Junge and Renming Song. From 2014 to 2015 he was a Postdoctoral Fellow at Harvard University. In 2015, Dr. Song was a Postdoctoral Fellow at the Mathematical Sciences Research Institute in Berkeley, California. Since 2016, he is Boas Assistant Professor at Northwestern University in Evanston, Illinois. Qiang Zeng works at the interfaces of probability, functional analysis and mathematical physics. His main topic of study is noncommutative probability and spin glasses.

Department Colloquium - Brad Rodgers (University of Michigan)

Brad Rodgers, University of Michigan

Monday, February 12th, 2018

Time: 4:30 p.m.  Place: Jeffery Hall 234

Speaker: Brad Rodgers, University of Michigan

Title: Some Applications of Random Matrix Theory to Analytic Number Theory

Abstract: In this talk I'll survey some of the ways that ideas originating from the study of random matrices have had an impact on analytic number theory. I hope to discuss in particular: 1) the statistical spacing of zeros of the Riemann zeta function, and what this spacing has to say about arithmetic, 2) a resolution of conjectures of Saari and Montgomery about the distribution of Rudin-Shapiro polynomials, using a connection to random walks on compact groups, and 3) recent work on the de Bruijn-Newman constant; de Bruijn showed that the Riemann hypothesis is equivalent to the claim that this constant is less than or equal to 0, and I will describe recent work showing the constant is greater than or equal to 0, conrming a conjecture of Newman. This includes joint work with J. Keating, E. Roditty-Gershon, and Z. Rudnick; and with T. Tao.

Brad Rodgers (University of Michigan): Brad Rodgers obtained his Ph.D. in Mathematics from the University of California, Los Angeles in 2013 under the supervision of Terence Tao. From 2013 to 2015 he held a postdoctoral position at the Institut fur Mathematik at the Universitat Zurich. Since 2015, he is a Postdoc Assistant Professor at the University of Michigan. Dr. Rodgers's awards include the AMS-Simons Travel Grant (2013-2016) and a NSF research grant (2017-2020). His research interests include random matrix theory, analytic number theory. In particular, he focuses on the interaction of these disciplines with analysis, probability, and combinatorics.

Department Colloquium - Daniel Le (University of Toronto)

Daniel Le, University of Toronto

Friday, February 9th, 2018

Time: 3:30 p.m.  Place: Jeffery Hall 234

Speaker: Daniel Le, University of Toronto

Title: The geometry of Galois representations

Abstract: The arithmetic of number fields can be profitably studied through the representation theory of their absolute Galois groups. These representations exhibit a number of elegant and surprising phenomena, most famously the quadratic reciprocity law. Many of these phenomena are explained by the modularity conjecture of Langlands that all Galois representations come from modular forms. Startling progress towards this conjecture began with Taylor and Wiles's study of Galois deformation spaces. We give a construction of local models for some Galois deformation spaces coming from geometric representation theory, and describe some applications to modularity conjectures and congruences between modular forms. Much of what we discuss is joint work with Bao Le Hung, Brandon Levin, and Stefano Morra.

Department Colloquium - Jory Griffin (Queen’s University)

Jory Griffin

Friday, February 2nd, 2018

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

Speaker: Jory Griffin, Queen’s University

Title: The Lorentz Gas – Macroscopic Transport from Microscopic Dynamics

Abstract: The Lorentz Gas is microscopic model for conductivity in which a point particle representing an electron moves through an infinite array of scatterers representing the background medium. On the macroscopic scale the dynamics can instead be modelled by the linear Boltzmann transport equation, an irreversible equation where motion of particles appears to be stochastic. How can these two pictures be reconciled? Can we 'derive' the macroscopic picture from the microscopic one? I will talk about the solution to this problem as well as its quantum mechanical analogue where much less is currently known.

Jory Griffin (Queen's University): Jory Griffin received his Ph.D. in Mathematics from the University of Bristol in 2017 under the supervision of Jens Marklof. He recently joined the Department of Mathematics and Statistics at Queen's University as a Coleman Postdoctoral Fellow. Dr. Grin's research focuses on Mathematical Physics, specifically in the quantum propagation of wave packets in the presence of scatterers.

Department Colloquium - Milian Derpich (USM, Valparaiso, Chile)

Milan Derpich

Friday, January 26th, 2018

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

Speaker: Milian Derpich, Universidad Técnica Federico Santa Maria, Valparaiso, Chile

Title: The Differential Entropy Gain Created by Linear Time-Invariant Systems

Abstract: The differential entropy of a continuous-valued random variable quantifies the uncertainty associated with the latter, and plays a crucial role in many fundamental result of Information Theory. This talk will discuss how the differential entropy rate of a random process exciting a discrete-time linear time invariant (LTI) system relates to that of the random process coming out of it. First, an apparent contradiction between existing results characterizing the difference between these two differential entropy rates, referred to a 'differential entropy gain', will be exposed. It will then be shown how and when these results can be reconciled, presenting a geometric interpretation as well as novel results which quantify the differential entropy gain introduced by LTI systems. Finally, some of the implications of these results will be illustrated for three different problems, namely: the rate-distortion function for non stationary processes, an inequality in networked control systems, and the capacity of stationary Gaussian channels.

Milan S. Derpich (Universidad Tecnica Federico Santa Maria, Valparaiso, Chile): Milan S. Derpich received the 'Ingeniero Civil Electronico' degree from Federico Santa Maria Technical University, in Valparaso, Chile in 1999. Dr. He then worked by the electronic circuit design and manufacturing company Protonic Chile S.A. between 2000 and 2004. In 2009 he received the PhD degree in electrical engineering from the University of Newcastle, Australia. He received the Guan Zhao-Zhi Award at the Chinese Control Conference 2006, and the Research Higher Degrees Award from the Faculty of Engineering and Built Environment, University of Newcastle, Australia, for his PhD thesis. Since 2009 he has been with the Department of Electronic Engineering at UTFSM, currently as associate professor. His main research interests include rate-distortion theory, networked control systems, and signal processing. He has just started a sabbatical one-year visit to the Department of Mathematics and Statistics in Queen's University, Canada, as a visiting professor.

Department Colloquium - Svetlana Jitomirskaya (UC Irvine)

Svetlana Jitomirskaya

Friday, January 19th, 2018

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

Speaker: Svetlana Jitomirskaya, UC Irvine

Title: Lyapunov exponents, small denominators, arithmetic spectral transitions, and universal hierarchical structure of quasiperiodic eigenfunctions

Abstract: A very captivating question in solid state physics is to determine/understand the hierarchical structure of spectral features of operators describing 2D Bloch electrons in perpendicular magnetic fields, as related to the continued fraction expansion of the magnetic flux. In particular, the hierarchical behavior of the eigenfunctions of the almost Mathieu operators, despite significant numerical studies and even a discovery of Bethe Ansatz solutions has remained an important open challenge even at the physics level. I will present a complete solution of this problem in the exponential sense throughout the entire localization regime. Namely, I will describe, with very high precision, the continued fraction driven hierarchy of local maxima, and a universal (also continued fraction expansion dependent) function that determines local behavior of all eigenfunctions around each maximum, thus giving a complete and precise description of the hierarchical structure. In the regime of Diophantine frequencies and phase resonances there is another universal function that governs the behavior around the local maxima, and a reflective-hierarchical structure of those, a phenomena not even described in the physics literature. These results lead also to the proof of sharp arithmetic transitions between pure point and singular continuous spectrum, in both frequency and phase, as conjectured since 1994. The talk is based on papers joint with W. Liu.

Svetlana Jitomirskaya, (UC Irvine): Svetlana Jitomirskaya earned her Ph.D. in Mathematics from Moscow State University in 1991 under the
supervision of Ya.G. Sinai with a thesis on Spectral and Statistical Properties of Lattice Hamiltonians.  Her awards include the A.P. Sloan Research Fellowship (1996-2000), the AMS Satter Prize (2005), the EPSRC Fellowship at Cambridge University (2008), the Simons Fellowship (2014-2015), and the Aisenstadt Chair at the CRM in Montreal (2018). Prof. Jitomirskaya was also an invited speaker at the 2002 International Congress of Mathematicians in Beijing. She solved (with Artur Avila) the famous Ten Martini Problem in 2009. Her research focuses on Mathematical Physics and Dynamical Systems.

Department Colloquium - Yifan Cui (UNC Chapel Hill)

Yifan Cui

Friday, January 12th, 2018

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

Speaker: Yifan Cui, University of North Carolina at Chapel Hill

Title: Tree-based Survival Models and Precision Medicine

Abstract: In the first part, we develop a theoretical framework for survival tree and forest models. We first investigate the method from the aspect of splitting rules. We show that existing approaches lead to a potentially biased estimation of the within-node survival and cause non-optimal selection of the splitting rules. Based on this observation, we develop an adaptive concentration bound result which quantifies the variance component for survival forest models. Furthermore, we show with three specific examples how these concentration bounds, combined with properly designed splitting rules, yield consistency results. In the second part, we focus on one application of survival trees in precision medicine which estimates individualized treatment rules nonparametrically under right censoring. We extend the outcome weighted learning to right censored data without requiring either inverse probability of censoring weighting or semi-parametric modeling of the censoring and failure times. To accomplish this, we take advantage of the tree-based approach to nonparametrically impute the survival time in two different ways. In simulation studies, our estimators demonstrate improved performance compared to existing methods. We also illustrate the proposed method on a phase III clinical trial of non-small cell lung cancer.

Yifan Cui (University of North Carolina at Chapel Hill): Yifan Cui is a PhD candidate in the Department of Statistics and Operations Research at the University of North Carolina at Chapel Hill. He works under the co-supervision of Professors Michael Kosorok and Jian Hannig. His research interest include machine learning, tree-based methods, high-dimensional data, personalized medicine, fiducial inference, bayesian inference, causal inference, and survival analysis.

Special Colloquium - Zhenhua Lin (U. California Davis)

Zhenhua Lin

Wednesday, January 10th, 2018

Time: 3:30 p.m.  Place: Jeffery Hall 234

Speaker: Zhenhua Lin, University of California Davis

Title: Intrinsic Riemannian Functional Data Analysis

Abstract: Data of random paths on a Riemannian manifold is often encountered in real-world applications. Examples include trajectories of bird migration, the dynamics of brain functional connectivity, etc. To analyze such data, a framework of intrinsic Riemannian functional data analysis is developed, which provides a rigorous theoretical foundation for statistical analysis of random paths on a Riemannian manifold. The cornerstone of the framework is the Hilbert space of vector fields along a curve on the manifold, based on which principal component analysis and Karhunen-Loève expansion for Riemannian random paths are then established. The framework also features a proposal for proper comparison of vector fields along different curves, which paves the way for intrinsic asymptotic analysis of estimation procedures for Riemannian functional data analysis. Built on intrinsic geometric concepts such as vector field, Levi-Civita connection and parallel transport on Riemannian manifolds, the proposed framework embraces full generality of applications and proper handle of intrinsic geometric concepts. Based on the framework, functional linear regression models for Riemannian random paths are investigated, including estimation methods, asymptotic properties and an application to the study of brain functional connectivity.

Zhenhua Lin (University of California Davis): Zhenhua Lin obtained his Ph.D. in Statistics in 2017 from the University of Toronto. He recently joined the University of California, Davis, as a Postdoctoral Fellow. Dr. Lin research focuses on functional data analysis. Specifically, he works on locally sparse modelling for functional objects, adaptive representation of functional data, adaptive nonparametric functional regression, and application of manifold learning in functional data analysis. His interests also include the analysis of high-dimensional data and the analysis of data with complex structures (such as graphs, networks, matrices).

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