Date |
Speaker |
Title & Abstract |
Host |
Oct. 29 |
Greg Blekherman
(Virginia Tech) |
Title.
Nonnegative Polynomials and Sums of Squares: Real Algebra meets Convex Geometry
Abstract. A multivariate real polynomial is non-negative if its value is at least zero for all points in $\mathbb{R}^n$. Obvious examples of non-negative polynomials are squares and sums of squares. What is the relationship between non-negative polynomials and sums of squares? I will review the history of this question, beginning with Hilbert's groundbreaking paper and Hilbert's 17th problem. I will discuss why this question is still relevant today, for computational reasons, among others. I will then discuss my own research which looks at this problem from the point of view of convex geometry. I will show how to prove that there exist non-negative polynomials that are not sums of squares via "naive" dimension counting. I will discuss the quantitative relationship between non-negative polynomials and sums of squares and also show that there exist convex polynomials that are not sums of squares. |
Bill Helton & Jiawang Nie |
Nov. 19 |
Prof. Herbert Heyer
(Univ. Tuebingen, Germany) |
Title.
Hypergroup stationarity of random fields
Abstract. Traditionally weak stationarity of a random field $\{X(t) : t\in \TTT\}$ over an index space $\TTT$ is defined with respect to a translation operation in $\TTT$. But this classical notion of stationarity does not extend to related random fields, as for example to the field of averages of $\{X(t): t\in \TTT\}$. In order to equip this latter field with a stationarity property one introduces a generalized translation in $\TTT$ which arises from a generalized convolution structure in the space $M^b(\TTT)$ of bounded measures on $\TTT$. There are two fundamental constructions providing such (hypergroup) convolution structures on the index spaces $\ZZZ_+$ and $\RRR_+$, in terms of polynomial sequences and families of special functions, respectively. In the present talk emphasis will be put on polynomially stationary random fields $\{X(n): n\in\ZZZ_+\}$ which were studied for the first time by R.~Lasser and M.~Leitner about 20 years ago. In the meantime the theory has developed interesting applications such as regularization, moving averages and prediction. For square-integrable radial random fields over graphs, J.P.~Arnaud has coined a notion of stationarity which yields spectral and Karhunen type representations. These fields are related to polynomially stationary random fields over $\ZZZ_+$, where the underlying polynomial sequence generates the Cartier-Dunau convolution structure in $M^b(\ZZZ_+)$. An analogous approach related to special function stationarity of random fields over $\RRR_+$ seems promising, but requires further progress. |
Patrick Fitzsimmons |
Nov. 26 | No colloquium. Thanksgiving Holiday. |
Date |
Speaker |
Title & Abstract |
Host |
Jan. 7 |
Prof. Rahul Pandharipande
(Princeton) |
Title.
Counting boxes
Abstract. I will give an introduction to 3-dimensional partitions and their role in modern counting problems: ideal sheaves, stable pairs, and holomorphic curves. Emphasis will be placed on the elementary examples that led to the discovery of the full structure. |
Mark Gross |
Jan. 28 |
Prof. Zuowei Shen
(National Univ. of Singapore) |
Title.
Frame based image restoration
Abstract. Efficient algorithms of image restoration and data recovery are derived by exploring sparse approximations of the underlying solutions by redundant systems such as wavelet frames and Gabor frames. Several algorithms and numerical simulation results for image restoration, compressed sensing, and matrix completion will be presented in this talk. |
Li-Tien Cheng & Bo Li |
Feb. 4 |
Prof. Michael Overton
(Courant, NYU) |
Title.
Nonsmooth, Nonconvex Optimization
Abstract. There are many algorithms for minimization when the objective function is differentiable, convex, or has some other known structure, but few options when none of the above hold, particularly when the objective function is nonsmooth at minimizers, as is often the case in applications. We describe two simple algorithms for minimization of nonsmooth, nonconvex functions. Gradient Sampling is a relatively new method that, although computationally intensive, has a nice convergence theory. The method is robust and the convergence theory has recently been extended to constrained problems. BFGS is an old method, developed for smooth problems, for which we have very limited theoretical results, but some remarkable empirical observations, extensive success in applications, and a rather bold conjecture. Limited Memory BFGS is a popular extension for large problems, and it too is applicable to the nonsmooth case, although our experience with it is more mixed. |
Philip Gill, Bill Helton, & Jiawang Nie |
Feb. 11 |
Prof. Richard Palais
(UC Irvine) |
Title.
Some Applications of Geometry to Computer Graphics---Payback
Abstract. For more than a dozen years I have been interested in Mathematical Visualization. This is the process of creating computer realizations of mathematical objects and then displaying them on a computer screen. For the most part, creating the computer "avatar" of the mathematical object is the "hard" and interesting part, and then displaying it comes relatively easily. But this is only because the field of Computer Graphics has developed many powerful and efficient algorithms that we can borrow and adapt for the display process. For the most part I have been a "consumer", but along the way I have noticed several places where quite sophisticated concepts from geometry can markedly improve the algorithms currently used in computer graphics, and in this talk I will discuss two of these. The first is how most efficiently to use a mouse to rotate a three-dimensional object on a computer screen. The second is how to sprinkle a large number of points on a surface embedded in three-space. Here, "sprinkle" means that the number of points in any region of the surface should be proportional to its area. |
Lei Ni & Nolan Wallach |
2:00 - 3:00, Feb. 18
(Special time) |
Prof. Greg Forest
(Math, UNC Chapel Hill) |
Title.
The Virtual Lung Project at UNC
Abstract. This lecture will provide an overview of the Virtual Lung Project at UNC, which has been active for several years. The goal is to build predictive tools for medical applications, with a major focus on mucus transport. The Cystic Fibrosis Center at UNC is the main driver of the project, which grounds our basic science projects in chemistry, mathematics and physics to clinical practice. I will give an overview of basic facts about lung function and dysfunction, and the approaches we have undertaken, including experimental, theoretical and numerical. I will give some detail in the second half of the talk about projects in my research group related to characterization of mucus viscoelastic properties at biologically relevant length and force scales, and to diffusive transport of foreign particles in mucus layers. |
Bo Li |
Feb. 18 |
Prof. Eyal Lubetsky
(Microsoft) |
Title. Random regular graphs: from random walks to geometry and back.
Abstract. The class of random regular graphs has been the focus of extensive study highlighting its excellent expansion properties. For instance, it is well known that almost every regular graph of fixed degree is essentially Ramanujan, and understanding this class of graphs sheds light on the general behavior of expanders. In this talk we will present several recent results on random regular graphs, focusing on the interplay between the spectrum, geometry and behavior of the simple random walk in these graphs. We will first discuss the relation between spectral properties and the abrupt convergence of the random walk to equilibrium, derived from precise asymptotics of the number of paths between vertices. Following the study of the graph geometry we proceed to its random perturbation via exponential weights on the edges (first-passage-percolation). We then show how this allows the derivation of various key features of the classical Erd\H{o}s-R\'enyi random graph near criticality, such as the asymptotics of the diameter of the largest component and the mixing time of the random walk on it. Based on joint works with Jian Ding, Jeong Han Kim, Yuval Peres and Allan Sly. |
Jacques Verstraete |
Feb. 25 |
Prof. Xiaofeng Shao
(UIUC) |
Title.
A selfnormalized approach to statistical inference for time series
Abstract. In the inference of time series (e.g. hypothesis testing and con^Ldence interval construction)m one often needs to obtain a consistent estimate for the asymptotic covariance matrix of a statistic. Or the inf erence can be conducted by using resampling (e.g. moving block bootstrap) and subsampling techniques. What is common for almost all the existing methods is that they involve the selection of a smoothing parameter . Some rules have been proposed to choose the smoothing parameter, but they may involve another user-chosen number, or assume a parametric model. In this talk, we introduce the so-called self-normalized (SN) app roach in the context of con^Ldence interval construction and change point detection. The self-normalized sta tistic does not involve any smoothing parameter and its limiting distribution is nuisance parameter free. T he ^Lnite sample performance of the SN approach is evaluated in simulated and real data examples. |
Dimitris Politis |
March 11 |
Prof. Ezra Getzler
(Northwestern Univ.) |
Title.
n-groups
Abstract. In this talk, we give a brief introduction to a natural generalization of groups, called n-groups. Just as discrete groups represent the homotopy types of acyclic spaces, n-groups realize homotopy types of connected topological spaces X such that pi_i(X)=0 for i>n. In this talk, we adopt the formalism of simplicial sets, and define n-groups as simplicial sets satisfying certain a filling condition (introduced by Duskin). In the first part of the talk, we explain what a 2-group look like: this material is contained in any textbook on simplicial sets. We indicate how 2-groups arise in topological quantum field theory. In the second part of the talk, we explain a generalization of Lie theory to n-groups, in which the role of Lie algebras is taken by differential graded Lie algebras, and the role of the ordinary differential equations underlying Lie theory is taken by the Maurer-Cartan equation for flat superconnections on simplices. |
Ben Weinkove |
Date |
Speaker |
Title & Abstract |
Host |
April 1
Room: AP&M 7421 |
Prof. Van Vu
(Rutgers Univ.) |
Title.
Inverse Littlewood-Offord theory, Smooth Analysis and the Circular Law
Abstract. A corner stone of the theory of random matrices is Wigner's semi-circle law, obtained in the 1950s, which asserts that (after a proper normalization) the limiting distribution of the spectra of a random hermitian matrix with iid (upper diagonal) entries follows the semi-circle law. The non-hermitian case is the famous Circular Law Conjecture, which asserts that (after a proper normalization) the limiting distribution of the spectra of arandom matrix with iid entries is uniform in the unit circle. Despite several partial results (Ginibre-Mehta, Girko, Bai, Edelman, Gotze-Tykhomirov, Pan-Zhu etc) the conjecture remained open for more than 50 years. In 2008, T. Tao and I confirmed the conjecture in full generality. I am going to give an overview of this proof, which relies on rather surprising connections between various fields: combinatorics, probability and theoretical computer science. |
Jozsef Balogh |
April 15 |
Prof.
Steve Zelditch
(Northwestern Univ.) |
Title.
Solution of Kac's problem for analytic plane domains with a symmetry
Abstract. Kac's `hear the shape of a drum" problem is the extent to which a plane domain is determined by its Dirichlet eigenvalues. I.e. is the map from domains to their spectra 1-1. We show that if the spectrum map is restricted to analytic plane domains with one up down symmetry (and an axis length fixed), then it is one-one. I.e. you can determine such a domain from its eigenvalues among other such domains. In joint work with Hamid Hezari, we also give a generalization to higher dimensions. |
Lei Ni |
April 22 |
Prof. Rudy Beran
(UC Davis) |
Title.
From Inadmissibility to Effective Regularization
Abstract. Charles Stein (1956) discovered that, under quadratic loss, the usual unbiased estimator for the mean vector of a multivariate normal distribution is inadmissible if the dimension $n$ of the mean vector exceeds two. Contemporaries claimed that Stein's results and the subsequent James-Stein estimator are counter-intuitive, even paradoxical, and not very useful. This talk reexamines such assertions in the light of arguments presented, sketched, or foreshadowed in Stein's beautifully written 1956 paper. Among often overlooked aspects of the paper are the asymptotic geometry of quadratic loss in high dimensions that makes Stein estimation transparent; asymptotic optimality results associated with Stein estimators; the explicit mention of practical multiple shrinkage estimators; and the foreshadowing of confidence balls centered at Stein estimators. Implications of these ideas underlie effective modern regularization estimators, among them, penalized least squares estimators with multiple quadratic penalties, running weighted means, nested submodel fits, and more. |
Dimitris Politis |
April 29 |
Prof. Benny Sudakov
(UCLA) |
Title.
Extremal Graph Theory and its applications.
Abstract. In typical extremal problem one wants to determine maximum cardinality of discrete structure with certain prescribed properties. Probably the earliest such result was obtain 100 years ago by Mantel who computed the maximum number of edges in a triangle free graph on n vertices. This was generalized by Turan for all complete graphs and became a starting point of Extremal Graph Theory. In this talk we survey several classical problems and results in this area and present some interesting applications of Extremal Graph Theory to other areas of mathematics. We also describe a recent surprising generalization of Turan's theorem which was motivated by question in Computational Complexity. |
Jacques Verstraete |
May 13 |
Prof. Pengfei Guan
(McGill Univ.) |
Title.
Fully nonlinear equations, elementary symmetric functions and convexity property of solutions
Abstract. We discuss convexity properties of solutions of fully nonlinear partial differential equations. The classical examples indicate that the level-sets of equilibrium potential in a convex domain is convex and the first eigenfunction of the Laplace equation in a convex domain is log-concave. Recently, there emerge two differente types of methods in the study of convexity of solutions of nonlinear equations. The macroscopic convexity principle is based on the convex hull of the solution. The microscopic convexity principle is based on constant rank type theorem for the Hessian matrix of the solution. The microscopic convexity principle is effective to treat nonlinear geometric differential equations on general manifolds. In the talk, we will explain how elementary symmetric functions can be used in a crucial way in this direction and what kind of "convexity" structural conditions are involved. The microscopic convexity principle shares close relationship with the Hamilton's maximum principle for general evolution equations. We will also discuss some related open problems. |
Ben Weinkove |
June 3
(Joint Math and MSED Colloquium) |
Prof. David Bressoud
(Macalester College) |
Title.
Issues of the Transition to College Mathematics
Abstract. Over the past quarter century, 2- and 4-year college enrollment in rst semester calculus has remained con- stant while high school enrollment in calculus has grown tenfold, from 60,000 to 600,000, and continues to grow at 6% per year. We have passed the cross-over point where each year more students study rst semester calculus in US high schools than in all 2- and 4-year colleges and universities in the United States. In theory, this should be an engine for directing more students toward careers in science, engineering, and mathematics. In fact, it is having the opposite eect. This talk will present what is known about the eects of this growth and what needs to happen in response within our high schools and universities. David Bressoud is the current President of the Mathematical Association of America. |
Jeff Rabin |