2010 - 2011
Department of Mathematics Colloquium
UC San Diego

Thursday, 4 - 5pm, AP&M 6402
(Unless Otherwise Stated)

Coordinator: Li-Tien Cheng

Fall Quarter, 2010

Date Speaker Title & Abstract Host
Oct. 21 Masakazu Kojima
(Tokyo Institute of Technology)
Title: Exploiting Structured Sparsity in Linear and Nonlinear Semidefinite Programs
Abstract: This talk summarizes conversion of large scale linear and nonlinear SDPs, which satisfies the sparsity characterized by a chordal graph structure , into smaller scale SDPs. The sparsity is classified in two types, the domain-space sparsity (d-space sparsity) for the symmetric matrix variable in the objective and/or constraint functions of the SDP, which is required to be positive semidefinite, and the range-space sparsity (r-space sparsity) for a linear or nonlinear matrix-inequality constraint of the SDP. Some numerical results on the conversion methods indicate their potential for improving the efficiency of solving various problems.
Jiawang Nie & Bill Helton
& Philip Gill
Oct. 28 Chun Liu
(Penn State University)
Title: Energetic Variational Approaches in the Modeling of Ionic Solutions and Ion Channels
Abstract: Ion channels are key components in a wide variety of biological processes. The selectivity of ion channels is the key to many biological process. Selectivities in both calcium and sodium channels can be described by the reduced models, taking into consideration of dielectric coefficient and ion particle sizes, as well as their very different primary structure and properties. These self-organized systems will be modeled and analyzed with energetic variational approaches (EnVarA) that were motivated by classical works of Rayleigh and Onsager. The resulting/derived multiphysics-multiscale systems automatically satisfy the Second Laws of Thermodynamics and the basic physics that are involved in the system, such as the microscopic diffusion, the electrostatics and the macroscopic conservation of momentum, as well as the physical boundary conditions. In this talk, I will discuss the some of the related biological, physics, chemistry and mathematical issues arising in this area.
Bo Li
Nov. 4 Noam Elkies
(Harvard University)
Title: How many points can a curve have?
Abstract: Diophantine equations, one of the oldest topics of mathematical research, remains the object of intense and fruitful study. A rational solution to a system of algebraic equations is tantamount to a point with rational coordinates (briefly, a "rational point") on the corresponding algebraic variety V. Already for V of dimension 1 (an "algebraic curve"), many natural theoretical and computational questions remain open, especially when the genus g of V exceeds 1. (The genus is a natural measure of the complexity of V; for example, if P is a nonconstant polynomial without repeated roots then the equation y^2 = P(x) gives a curve of genus g iff P has degree 2g+1 or 2g+2.) Faltings famously proved that if g>1 then the set of rational points is finite (Mordell's conjecture), but left open the question of how its size can vary with V, even for fixed g. Even for g=2 there are curves with literally hundreds of points; is the number unbounded? We will briefly review the structure of rational points on curves of genus 0 and 1, and then report on relevant work since Faltings on points on curves of given genus g>1.
Center for
Communications Research
& Cristian Popescu
Nov. 18 Herbert Heyer
(University of Tübingen)
Title:Radial Random Walks on Matrix Cones
Abstract: The present lecture is devoted to recent developments on random walks with spherical symmetry, a topic which was opened to research by J.F.C. Kingman in 1963, and which has developed wide-ranging applications through the work of W. Hazod, M. Rösler, and M. Voit. The analytic method to be described in the talk concerns generalized convolutions of measures on hypergroups, in particular on the self-dual commutative hypergroup of positive semidefinite (Hermitian) matrices. These hypergroups are defined via Bessel functions of higher rank. A typical application of the hypergroup setting is the study of Bessel random walks on matrix cones and their convergence to Wishart distributions.
Patrick Fitzsimmons
Dec. 2 at 3pm in AP&M 6402 Ioan Bejenaru
(University of Chicago)
Title:Schrödinger Maps
Abstract:I will introduce and motivate the Schrödinger Map problem. We'll then discuss several interesting questions about this equation, and also describe some of the recent progress in the field.
Jacob Sterbenz & Hiring Committee
Dec. 2 Tamas Szamuely
(University of Pennsylvania & The Renyi Institute of the Hungarian Academy of Science)
Title:On the arithmetic of 1-motives
Abstract:1-motives have been introduced by Deligne in the 1970s as the simplest kind of mixed motives; basically, they are the only ones having a simple and concrete geometric description. In recent years there has been a resurgence of interest in them, and they have been successfully applied to arithmetic questions as well, such as generalizations of the classical duality theorems of Poitou and Tate, or finding rational points on algebraic varieties. I shall explain some of these results.
Cristian Popescu

Winter Quarter, 2011

Date Speaker Title & Abstract Host
Jan. 13 Scott Morrison
(UC Berkeley)
Title:Classifying Fusion Categories
Abstract:Fusion categories are quantum analogues of finite groups. They describe certain topologically invariant 2-dimensional quantum systems, and may be relevant for building a quantum computer. Families of fusion categories can be constructed from quantum groups, subfactors or conformal field theory. Attempting to classify small examples requires techniques from analysis, combinatorics, representation theory and number theory. The classification results available so far reveal intriguing exotic examples and leave many questions.
Justin Roberts & Hiring Committee
Jan. 20 at 4pm in AP&M 5402 Bob Eisenberg
(Rush University)
Title:Ions in Channels
Abstract:Ion channels are irresistible objects for biological study because they are the 'nanovalves of life' controlling most biological functions, much as transistors control computers. Channels contain an enormous density of crowded charged spheres, fixed and mobile, and induced polarization charge as well. Ions in such conditions do not resemble the point particles of ideal solutions and cannot be described adequately by ideal gas, Debye-Hückel, or PNP (Poisson Nernst Planck) equations. Direct simulation of channel behavior in atomic detail is difficult if not impossible. Gaps in scales of time, volume, and concentration between atoms and biological systems are each ~10^12. All the gaps must be dealt with at once, because biology occurs on all the scales at once. Simple models are surprisingly successful in dealing with ion binding in three very different (and important) channels the voltage activated sodium channel that produces the signals of nerve and muscle and two cardiac calcium channels that controls contraction over a large range of conditions, suggesting that mathematical analysis is both possible and useful. Amazingly, the same model with the same three parameters accounts quantitatively for qualitatively different binding in a wide range conditions for two very different calcium and sodium channels. The binding free energy is an output of the calculation, produced by the crowding of charged spheres in a very small space. The model does not involve any traditional chemical 'quantum' binding energies at all. How can such a simple model give such specific results when crystallographic wisdom and chemical intuition says that selectivity depends on the precise structural relation of ions and side chains? The answer is that structure is a computed consequence of forces in this model and is very important, but as an output of the model, not as an input. Binding is a consequence of the 'induced fit' of side chains to ions and ions to side chains. Binding sites are self-organized and at their free energy minimum, forming different structures in different conditions. Channels function away from equilibrium. A variational approach is obviously needed to replace our equilibrium analysis of binding and one is well under way, treating ionic solutions as complex fluids with simple components, applying the energy variational methods of Chun Liu, used in electrorheology to deal with liquid crystals. Correlations are generated automatically by this variational method from the energy and dissipation of the components of the physical model without arbitrary coefficients.
Bo Li
Feb. 3 Han Xiao
(University of Chicago)
Title:Covariance Matrix Estimation For Time Series
Abstract:Covariance matrix is of fundamental importance in many aspects of statistics. Recently, there is a surge of interest on regularized covariance matrix estimation using banding, tapering and thresholding methods, in high dimensional statistical inference, where multiple iid copies of the random vector from the underlying multivariate distribution are required. In the context of time series analysis, however, it is typical that only one realization is available, so the current results are not applicable. In this talk, we shall exploit the connection between covariance matrices and spectral density functions using the idea in Toeplitz (1911) and Grenander and Szegö (1958), together with Wu (2005)'s recent theory on stationary processes, and establish sharp convergence rates for banded and tapered autocovariance matrix estimates. We also consider thresholded estimates that can better characterize sparsity if the true covariance matrix is sparse. Our results shed new lights on many classical problems, including the celebrated Wiener-Kolmogorov prediction theory for finite samples, interpolation and smoothing, as well as efficient estimation for linear models, among others.
Lily Xu & Hiring Committee
Feb. 10 Dhruv Mubayi
(University of Illinois at Chicago)
Title:Coloring Simple Hypergraphs
Abstract:Improvements of the obvious lower bounds on the independence number of (hyper)graphs have had impact on problems in discrete geometry, coding theory, number theory and combinatorics. One of the most famous examples is the result of Komlos-Pintz-Szemeredi (1982) on the independence number of 3-uniform hypergraphs which made important progress on the decades old Heilbronn problem. We give a sharp upper bound on the chromatic number of simple k-uniform hypergraphs that implies the above result as well as more general theorems due to Ajtai-Komlos-Pintz-Spencer-Szemeredi, and Duke-Lefmann-Rodl. Our proof technique is inspired by work of Johansson on graph coloring and uses the semi-random or nibble method.? This is joint work with Alan Frieze.
Jacques Verstraete
Feb. 18 at 4pm in AP&M 6402 Adrian Iovita
(Concordia University)
Title:A p-adic criterion for good reduction of curves over a p-adic field
Abstract:If A is an abelian variety over a p-adic field K then A has good reduction if and only if the p-adic Tate module of A is a crystalline representation of the absolute Galois group of K. As there are examples of curves over K with bad reduction whose Jacobian has good reduction, the Galois action on the p-adic etale cohomology of the curve does not determine its reduction. We will discuss these issues and point to a p-adic criterion of good reduction for curves.
Cristian Popescu
Feb. 18 at 2pm in AP&M 6402 Jeff Lagarias
(University of Michigan)
Title:Packing Space with Regular Tetrahedra
Abstract:The problem of the densest packing of space by congruent regular tetrahedra has a long history, starting with Aristotle's assertion that regular tetrahedra fill space, and continuing through its appearance in Hilbert's 18th Problem. This talk describes its history and many recent results obtained on this problem, including contributions from physicists, chemists and materials scientists. The current record for packing density is held by my former graduate student Elizabeth Chen, jointly with Michael Engel and Sharon Glotzer.
Fan Chung Graham
Feb. 24 at 2pm in AP&M 6402 Zhihua Su
(University of Minnesota)
Title:Envelope models: efficient estimation in multivariate linear regression
Abstract:This talk will introduce a new class of models which can lead to efficient estimation in multivariate analysis. Some members in the class include the basic envelope model, partial envelope model, inner envelope model, scaled envelope model, and heteroscedastic envelope model. They have the common word "envelope" in their names because they are all constructed by enveloping: use reducing subspaces to connect the mean function and the covariance function, so that the number of parameters can be reduced. The application of enveloping is very broad and can be used in many contexts to control parameterization. Theoretical results, simulations and a large number of data examples show that the efficiency gains obtained by enveloping can be substantial.
Lily Xu and Hiring Committee
Feb. 24 Ben Andrews
(Australia National University)
Title:On the fundamental gap of a convex domain
Abstract:The eigenvalues of the Laplacian (or Laplacian with potential) on a smoothly bounded domain domain are very natural quantities, arising as the fundamental tones of a drum, the rates of decay of diffusions, and the energy levels of quantum systems. I will discuss some of the history relating to inequalities for low eigenvalues, leading up to the proof of a conjecture of Yau and van den Berg for the `fundamental gap' or excitation energy of a convex domain.
Lei Ni
Mar. 10 James McKernan
Title:Symmetries of algebraic varieties
Abstract:We give a survey of what is known about how many symmetries an algebraic variety can possess. We start with some classical results, including those of Hurwitz, Noether and Riemann, to do with the automorphism group of the plane and the automorphism group of curves (or equivalently Riemann surfaces), and we end with some more recent results to do with the automorphism group of threefolds of low degree and varieties with finite automorphism group.
Mark Gross

Spring Quarter, 2011

Date Speaker Title & Abstract Host
April 15 at 4pm in AP&M 6402 Bernd Sturmfels
(UC Berkeley)
Title:The central curve in linear programming
Abstract:The central curve of a linear program is an algebraic curve specified by linear and quadratic constraints arising from complementary slackness. It is the union of the various central paths for minimizing or maximizing the cost function over any region in the associated hyperplane arrangement. In this lecture we present joint work with Daniel Plaumann and Cynthia Vinzant on the geometry of central curves. We determine the degree, arithmetic genus and defining prime ideal of the central curve, thereby answering a question of Bayer and Lagarias. These invariants, along with the degree of the Gauss image of the curve, are expressed in terms of the matroid of the input matrix. Extending work of Dedieu, Malajovich and Shub, this yields an instance-specific bound on the total curvature of the central path, a quantity relevant for interior point methods.
Jiawang Nie & Bill Helton
May 23 at 2pm in AP&M 6402 George Casella
(University of Florida)
Title:New Findings from Terrorism Data: Dirichlet Process Random Effects Models for Latent Groups
Abstract:Data obtained describing terrorist events are particularly difficult to analyze, due to the many problems associated with the both the data collection process, the inherent variability in the data itself, and the usually poor level of measurement coming from observing political actors that seek not to provide reliable data on their activities. Thus, there is a need for sophisticated modeling to obtain reasonable inferences from these data. Here we develop a logistic random effects specification using a Dirichlet process to model the random effects. We first look at how such a model can best be implemented, and then we use the model to analyze terrorism data. We see that the richer Dirichlet process random effects model, as compared to a normal random effects model, is able to remove more of the underlying variability from the data, uncovering latent information that would not otherwise have been revealed.
Dimitris Politis
May 26 Mu-Tao Wang
(Columbia University)
Title:On the notion of quasilocal mass in general relativity
Abstract:One of the greatest accomplishments of the theory of general relativity in the past century is the proof of the positive mass/energy theorem for asymptotically flat spacetime. This provides the theoretical foundation for the stability of an isolated gravitating system. However, the concept of mass/energy remains a challenging problem because of the lack of a quasilocal description. Most observable physical models are finitely extended spatial regions and measurement of mass/energy on such a region is essential in many fundamental issues. In fact, among Penrose's list of major unsolved problems in classical general relativity, the first one is ``Find a suitable quasi-local definition of energy-momentum in general relativity". In this talk, I shall describe a new proposal of quasi-local mass/energy by Shing-Tung Yau of Harvard University and myself.
Lei Ni