Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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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).

Department Colloquium - Bob Ross Barmish (U. of Wisconsin-Madison)

Bob Ross Barmish

Friday, December 1st, 2017

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

Speaker: Bob Ross Barmish, University of Wisconsin-Madison

Title: When the expected value is not expected

Abstract: Our motivation for the research to be described is derived from the following fact: The expected value of a random variable~$X$, denoted~$\mathbb{E}(X)$, is often inconsistent with what human beings may actually expect based on psychological considerations. This is particularly important when predictions involving life-threatening situations arise. To bring this issue into sharp focus, this seminar begins with a set of questions related to the recent rampage of Hurricane Irma in the State of Florida. When the use of empirical data leads to an expected value forecast of storm surge wave height which is unduly pessimistic, will the "boy who cried wolf" effect be in play the next time a hurricane approaches the mainland? On the other hand, if the formally calculated expected wave height is too optimistic, might it be the case that many will take inadequate protective measures? Based on such considerations, we define a new alternative to~$\mathbb{E}(X)$ which we believe is quite useful for "mission critical" situations with downside risk being of paramount concern. We call this new metric the {\it Conservative Expected Value} and denote it by~$\mathbb{CEV}(X)$. In this talk, we provide the technical definition of the $\mathbb{CEV}$, compare it with the classical expected value and describe some aspects of the rich mathematical theory which accompanies it. We also include a description of some of the studies we have conducted using the~$\mathbb{CEV}$ to evaluate historical data.

Bob Ross Barmish (University of Wisconsin): B. Ross Barmish received his Ph.D. degree in electrical engineering from Cornell University in 1975. From 1975 to 1978, he was an Assistant Professor of Engineering and Applied Science at Yale University. From 1978 to 1984, he was as an Associate Professor of Electrical Engineering at the University of Rochester. He joined the University of Wisconsin{Madison in 1984, where he is currently a Professor ofelectrical and computer engineering. His research interests include robustness of dynamical systems, building a bridge between feedback control theory, and trading in complex financial markets. Prof. Barmish was a recipient of the Best Paper Award (1986-1989 and 1990-1992) from the International Federation of Automatic Control, and the 2013 Bode Prize by the IEEE Control Systems Society.

Free Probability Seminar - Pei-Lun Tseng (Queen's University)

Thursday, November 30th, 2017

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

Speaker: Pei-Lun Tseng

Title: Infinitesimal Laws of non-commutative random variables, II

Abstract:  In this talk, we will focus on a single type B variable, and introduce the corresponding infinitesimal law. In addition, we will also define the free additive convolution for infinitesimal laws, and to see the relation between the type B laws and the infinitesimal laws.

Free Probability and Random Matrices Seminar Webpage:

Special Colloquium - Jason Klusowski (Yale University)

Jason Klusowski

Wednesday, November 29th, 2017

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

Speaker: Jason Klusowski

Title: Counting motifs and connected components of large graphs via subgraph sampling

Abstract: Learning properties of large graphs from samples is an important problem in statistical network analysis. We revisit the problem of estimating the numbers of connected components in a graph of $N$ vertices based on the subgraph sampling model, where we observe the subgraph induced by $n$ vertices drawn uniformly at random. The key question is whether it is possible to achieve accurate estimation, i.e., vanishing normalized mean-square error, by sampling a vanishing fraction of the vertices. We show that it is possible by accessing only sublinear number of samples if the graph does not contains high-degree vertices or long induced cycles; otherwise it is impossible. Optimal sample complexity bounds are obtained for several classes of graphs including forests, cliques, and chordal graphs. The methodology relies on topological identities of graph homomorphism numbers, which, in turn, also play a key role proving minimax lower bounds based on constructing random instances of graphs with matching structures of small subgraphs. We will also discuss results for the neighborhood sampling model, where we observe the edges between the sampled vertices and their neighbors.

Jason Klusowski (Yale University): Jason M. Klusowski obtained his B.Sc. in Mathematics and Statistics in 2013 from the University of Manitoba, receiving the Robert Ross McLaughlin Scholarship in Mathematics, the Faculty of Science Medal in B.Sc., and the Governor General's Silver Medal. In 2013 he was received the NSERC Alexander Graham Bell Canada Graduate Scholarship and joined Yale University where he is currently a Ph.D. candidate in Statistics and Data Science, under the supervision of Prof. Andrew Barron. Jason Klusowski's research interests include the theoretical and computational aspects of neural networks, approximation algorithms for networks, high-dimensional function estimation, mixture models, and shape constrained estimation.

Number Theory - Multiple Speakers

Wednesday, November 29th, 2017

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

1st Speaker:  M. Ram Murty

Title:  Hilbert’s tenth problem over number fields

Abstract:   Hilbert's tenth problem for rings of integers of number fields remains open in general,

although a conditional negative solution was obtained by Mazur and Rubin assuming some unproved conjectures about the Shafarevich-Tate groups of elliptic curves. In this talk, we highlight how the non-vanishing of certain L-functions is related to this problem. In particular, we show that Hilbert's tenth problem for rings of integers of number fields is unsolvable assuming the automorphy of L-functions attached to elliptic curves and the rank part of the Birch and Swinnerton-Dyer conjecture. This is joint work with Hector Pasten.

2nd Speaker:  François Séguin

Title:  A lower bound for the two-variable Artin conjecture

Abstract:  In 1927, Artin conjectured that any integer other than -1 or a perfect square generates the multiplicative group Z/pZ× for infinitely many p. In a 2000 article, Moree and Stevenhagen considered a two-variable version of this problem, and proved a positive density result conditionally to the generalized Riemann Hypothesis by adapting a 1967 proof by Hooley for the original conjecture. During this talk, we will prove an unconditional lower bound for this two-variable problem. This is joint work with Ram Murty and Cameron Stewart.

3rd Speaker:  Arpita Kar

Title:  On a Conjecture of Bateman about $r_5(n)$

Abstract:  Let $r_5(n)$ be the number of ways of writing $n$ as a sum of five integer squares. In his study of this function, Bateman was led to formulate a conjecture regarding the sum $$\sum_{|j| \leq \sqrt{n}}\sigma(n-j^2)$$ where $\sigma(n)$ is the sum of positive divisors of $n$. We give a proof of Bateman's conjecture in the case $n$ is square-free and congruent to $1$ (mod $4$). This is joint work with Prof. Ram Murty.

4th Speaker:  Siddhi Pathak

Title:  Derivatives of L-series and generalized Stieltjes constants

Abstract:  Generalized Stieltjes constants occur as coefficients of (s-1)^k in the Laurent series expansion of certain Dirichlet series around s=1. The connection between these generalized Stieltjes constants and derivatives of L(s,f) for periodic arithmetical functions f, at s=1 is known. We utilize this link to throw light on the arithmetic nature of L'(1,f) and certain Stieltjes constants. In particular, if p is an odd prime greater than 7, then we deduce the transcendence of at least (p-7)/2 of the generalized Stieltjes constants, {gamma_1(a,p) : 1 \leq a < p }, conditional on a conjecture of S. Gun, M. Ram Murty and P. Rath.