Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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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.

Number Theory - François Séguin (Queen's University)

Wednesday, January 24th, 2018

Time: 2:15 p.m.  Place: Jeffery Hall 319

Speaker: François Séguin (Queen's University)

Title: Heights of elliptic curves and the elliptic analogue of the two-variable Artin conjecture

Abstract: Similar to the way Lang and Trotter adapted Artin's primitive root conjecture in the case of elliptic curves, we consider this natural adaptation for the two-variable Artin Conjecture. In light of our recent results for the two-variable setting, we present similar, unconditional lower bounds for this elliptic analogue.

Free Probability Seminar - Rob Martin (University of Cape Town)

Tuesday, January 23rd, 2018

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

Speaker: Rob Martin (University of Cape Town)

Title: Non-commutative Clark measures for Free and Abelian multi-variable Hardy space

Abstract:  In classical Hardy space theory, there is a natural bijection between the Schur class of contractive analytic functions in the complex unit disk and Aleksandrov- Clark measures on the unit circle. A canonical several-variable analogue of Hardy space is the Drury-Arveson space of analytic functions in the unit ball of d-dimensional complex space, and the canonical non-commuting or free multi- variable analogue of Hardy space is the full Fock space over d-dimensional complex space. Here, the full Fock space is naturally identified with a non- commutative reproducing kernel Hilbert space of free or non-commutative ana- lytic functions acting on a several-variable non-commutative open unit ball. We will extend the concept of Aleksandrov-Clark measure, the bijection between the Schur class and AC measures, Clark’s unitary perturbations of the shift, Lebesgue decomposition formulas and additional related results from one to several commuting and non-commuting variables.

Free Probability and Random Matrices Seminar Webpage:

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.

Number Theory - Jung-Jo Lee (Queen's University)

Wednesday, January 17th, 2017

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

Speaker: Jung-Jo Lee

Title: The p-adic zeta function and Iwasawa’s main conjecture

Abstract: I would explain the role of the p-adic zeta function in describing the structure of certain Iwasawa module.

Note: I consider a series of talks, each "hopefully" self-contained.

Talk 2 : Euler system of cyclotomic units and Iwasawa's main conjecture

Talk 3 : Euler system of Heegner points and Birch and Swinnerton-Dyer conjecture

 

Free Probability Seminar - Neha Prabu (Queen's University)

Tuesday, January 16th, 2018

Time: 4:00-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Neha Prabu

Title: Semicircle distribution in number theory

Abstract:  In free probability theory, the role of the semicircle distribution is analogous to that of the normal distribution in classical probability theory. However, the semicircle distribution also shows up in number theory: it governs the distribution of eigenvalues of Hecke operators acting on spaces of modular cusp forms. In this talk, I will give a brief introduction to this theory of Hecke operators and sketch the proof of a result which is a central limit type theorem from classical probability theory, that involves the semicircle measure.

Free Probability and Random Matrices Seminar Webpage:

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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