UCSD Math Department Colloqium, 2012 - 2013.

2012 - 2013 UCSD Department of Mathematics Colloquium

Thursdays, 4:00 pm - 5:00 pm, AP&M 6402 (Unless Otherwise Stated)

Coordinator: Bo Li

Spring quarter, 2013

### Host

April 4 Shi Jin
Title. Semiclassical Computation of High Frequency Wave in Heterogeneous Media

Abstract. We introduce semiclassical Eulerian methods that are efficient in computing high frequency waves through heterogeneous media. The method is based on the classical Liouville equation in phase space, with discontinous Hamiltonians due to the barriers or material interfaces. We provide physically relevant interface conditions consistent with the correct transmissions and reflections, and then build the interface conditions into the numerical fluxes. This method allows the resolution of high frequency waves without numerically resolving the small wave lengths, and capture the correct transmissions and reflections at the interface. This method can also be extended to deal with diffraction and quantum barriers.

Bo Li
April 11 Jeong Han Kim (Korea Inst. for Advanced Study (KIAS))
Title. A tale of two models for random graphs

Abstract. Since Erdos-Renyi introduced random graphs in 1959, two closely related models for random graphs have been extensively studied. In the G(n,m) model, a graph is chosen uniformly at random from the collection of all graphs that have n vertices and m edges. In the G(n,p) model, a graph is constructed by connecting each pair of two vertices randomly. Each edge is included in the graph G(n,p) with probability p independently of all other edges.
Researchers have studied when the random graph G(n,m) (or G(n,p), resp.) satisfies certain properties in terms of $n$ and m (or n and p, resp.). If G(n,m) (or G(n,p), resp.) satisfies a property with probability close to 1, then one may say that a typical graph" with m edges (or expected edge density p, resp.) on n vertices has the property. Random graphs and their variants are also widely used to prove the existence of graphs with certain properties. In this talk, a well-known problem for each of these categories will be discussed.
First, a new approach will be introduced for the problem of the emergence of a giant component of G(n,p), which was first considered by Erdos-Renyi in 1960. Second, a variant of the graph process G(n,1), G(n,2), ..., G(n,m), ... will be considered to find a tight lower bound for Ramsey number R(3,t) up to a constant factor.
No prior knowledge of graph theory is needed in this talk.

Jacques Verstraete
May 16 Benny Sudakov
(UCLA)
Title. Induced Matchings, Arithmetic Progressions and Communication

Abstract. Extremal Combinatorics is one of the central branches of discrete mathematics which deals with the problem of estimating the maximum possible size of a combinatorial structure which satisfies certain restrictions. Often, such problems have also applications to other areas including Theoretical Computer Science, Additive Number Theory and Information Theory. In this talk we will illustrate this fact by several closely related examples focusing on a recent work with Alon and Moitra.

Jacques Verstraete
June 10 Jean Bernard Lasserre
Title.

Abstract.

Bill Helton & Jiawang Nie

Winter quarter, 2013

### Host

Jan. 10 Rayan Saab
Title. Near-optimal quantization and encoding for oversampled signals

Abstract. Analog-to-digital (A/D) conversion is the process by which signals (e.g., bandlimited functions or finite dimensional vectors) are replaced by bit streams to allow digital storage, transmission, and processing. Typically, A/D conversion is thought of as being composed of sampling and quantization. Sampling consists of collecting inner products of the signal with appropriate (deterministic or random) vectors. Quantization consists of replacing these inner products with elements from a finite set. A good A/D scheme allows for accurate reconstruction of the original object from its quantized samples. In this talk we investigate the reconstruction error as a function of the bit-rate, of Sigma-Delta quantization, a class of quantization algorithms used in the oversampled regime. We propose an encoding of the Sigma-Delta bit-stream and prove that it yields near-optimal error rates when coupled with a suitable reconstruction algorithm. This is true both in the finite dimensional setting and for bandlimited functions. In particular, in the finite dimensional setting the near-optimality of Sigma-Delta encoding applies to measurement vectors from a large class that includes certain deterministic and sub-Gaussian random vectors. Time permitting, we discuss implications for quantization of compressed sensing measurements.

Ery Arias-Castro
Jan. 15
(Tuesday)
Claus Sorensen
(Princeton Univ.)
Title.
Abstract.
Cristian Popescu
Jan. 17 Sergey Kitaev
(Strathclyde Univ.)
Title. Two involutions on description trees and their applications

Abstract. Description trees were introduced by Cori, Jacquard and Schaeffer in 1997 to give a general framework for the recursive decompositions of several families of planar maps studied by Tutte in a series of papers in the 1960s. We are interested in two classes of planar maps which can be thought as connected planar graphs embedded in the plane or the sphere with a directed edge distinguished as the root. These classes are rooted non-separable (or, 2-connected) and bicubic planar maps, and the corresponding to them trees are called, respectively, $\beta(1,0)$-trees and $\beta(0,1)$-trees.
Using different ways to generate these trees we define two endofunctions on them that turned out to be involutions. These involutions are not only interesting in their own right, in particular, from counting fixed points point of view, but also they were used to obtain non-trivial equidistribution results on planar maps, certain pattern avoiding permutations, and objects counted by the Catalan numbers. The results to be presented in this talk are obtained in a series of papers in collaboration with several researchers.

Jeff Remmel
3:00 pm, Jan. 22
(Special time and date)
Alex Lubotzky
Title. Sieve Methods in Group Theory

Abstract. The sieve methods are classical methods in number theory. Inspired by the 'affine sieve method' developed by Sarnak, Bourgain, Gamburd and others, as well as by works of Rivin and Kowalsky, we develop in a systemtic way a 'sieve method' for group theory. This method is especially useful for groups with 'property tau'. Hence the recent results of Breuillard-Green-Tao, Pyber-Szabo, Varju and Salehi-Golsefidy are very useful and enables one to apply them for linear groups. We will present the method and some of its applications to linear groups and to the mapping class groups. (Joint work with Chen Meiri).

Efim Zelmanov
Feb. 7 Daniel Tartaru
(UC Berkeley)
Title. The two dimensional water wave equation

Abstract. The aim of this talk is to provide an overview of recent developments concerning the motion of a two dimensional incompressible and irrotational fluid with a free surface. The emphasis will be on the case when gravity is present, but surface tension is absent. This is joint work with John Hunter, Mihaela Ifrim and Tak Wong.

Ioan Bejenaru & Jacob Sterbenz
Feb. 21
(SUNY)
Title. Cohomological invariants of projective varieties in positive characteristic

Abstract. Let X be a projective smooth variety over an algebraically closed field k. If k has characteristic zero, then the singular (Betti) cohomology groups of X are finitely generated abelian groups and therefore all the invariants associated to them are discrete and in fact do not change under good deformations. If k has positive characteristic, then the crystalline cohomology groups of X have a much richer structure and are called F-crystals over k. In particular, one can associate to them many subtle invariants which vary a lot under good deformations and which could be of either discrete or continuous nature. We present an accessible survey of the classification of F-crystals over k via subtle invariants with an emphasis on the recent results obtain by us, by Gabber and us, and by Lau, Nicole, and us.

Cristian Popescu
March 7 Stefaan Vaes
(KU Leuven, Belgium)
Title. Von Neumann algebras with a unique Cartan decomposition

Abstract. The subject of this talk is at the crossroads of functional analysis, ergodic theory and group theory. Using a construction by Murray and von Neumann (1943), ergodic actions of countable groups on probability spaces give rise to algebras of operators on a Hilbert space, called von Neumann algebras. In a joint work with Sorin Popa, we proved that such crossed product von Neumann algebras by free groups or, more generally, by hyperbolic groups have a unique Cartan subalgebra. I will explain this result and its consequences for the classification of crossed products by free groups.

March 14 Iskander Taimanov (Russian Acad. Sci.)
The Moutard transformation: an algebraic formalism and applications
Justin Roberts and Efim Zelmanov
March 21 Marston Conder (Auckland Univ., New Zealand)
Discrete objects with maximum possible symmetry

Abstract. Symmetry is pervasive in both nature and human culture. The notion of chirality (or handedness') is similarly pervasive, but less well understood. In this lecture, I will talk about a number of situations involving discrete objects that have maximum possible symmetry in their class, or maximum possible rotational symmetry while being chiral. Examples include geometric solids, combinatorial graphs (networks), maps on surfaces, dessins d'enfants, abstract polytopes, and even compact Riemann surfaces (from a certain perspective). I will describe some recent discoveries about such objects with maximum symmetry, illustrated by pictures as much as possible.

Efim Zelmanov

Fall quarter, 2012

### Host

Oct. 18 Todd Kemp
(Math, UCSD)
Title. Liberating Random Projections

Abstract. Consider two random subspaces of a finite-dimensional vector space -- i.e. two random projection matrices P and Q. What is the dimension of their intersection? This (random) integer is almost surely equal to its minimal possible value, which corresponds to the subspaces being in general position. Many more delicate questions about the geometry of the configuration are encoded by the principal angles between the subspaces, which are determined by the eigenvalues of the operator-valued angle matrix PQP.
The situation is much more complicated in infinite-dimensions. Even the question of whether two random projections are likely to be in general position is difficult to make sense of, let alone answer. Nevertheless, understanding the operator-valued angle in an infinite-dimensional setting is of critical importance to the biggest open problem in free probability theory -- the so-called Unification Conjecture'' -- with ramifications for operator algebras, information theory, and random matrices.
In this talk, I will discuss recent and ongoing joint work with Benoit Collins, addressing the configuration of random subspaces in an infinite-dimensional context. Using a mixture of techniques from stochastic processes, PDEs, and complex analysis, we prove the general position claim and give a complete understanding of the associated geometry. This work proves an important special case of the Unification Conjecture, and has interesting implications for the original finite-dimensional setting as well.
Bruce Driver
(Math, UCSD)
Title. Classification and rigidity for von Neumann algebras

Abstract. I will survey some recent progress on the classification of von Neumann algebras arising from countable groups and their measure preserving actions on probability spaces. This includes the finding of the first classes of (superrigid) groups and actions that can be entirely reconstructed from their von Neumann algebras.
Bruce Driver
Nov. 1
(Joint CCR & UCSD Math Colloquium)

### CANCELLED

Charlie Fefferman
(Princeton Univ.)
Title. Extension and interpolation problems

Abstract. Let X be your favorite space of continuous functions on R^n. How can one decide whether a given function f:E->R, defined on a given (arbitrary) subset E of R^n, extends to a function F in X? The question goes back to Whitney 1934. The answers make contact with algebraic geometry and computer science.
Joe Buhler & Peter Ebenfelt
Nov. 8 Brendon Rhoades (Math, UCSD) Title. Meet the New Faculty: Parking Spaces

Abstract. A sequence $(a_1, \dots, a_n)$ of positive integers is a {\it parking function} if its nondecreasing rearrangement $(b_1 \leq \dots \leq b_n)$ satisfies $b_i < i+1$ for all $i$. Parking functions were introduced by Konheim and Weiss to study a hashing problem in computer science, but have since received a great deal of attention in algebraic combinatorics. We will define two new objects attached to any (finite, real, irreducible) reflection group which generalize parking functions and deserve to be called parking spaces. We present a conjecture (proved in some cases) which asserts a deep relationship between these constructions. This is joint work with Drew Armstrong at the University of Miami and Vic Reiner at the University of Minnesota.
Peter Ebenfelt
Nov. 15 Ioan Bejenaru (Math, UCSD) Title. Meet the New Faculty: Dispersive Equations

Abstract. This talk will cover some of the main problems in the field of nonlinear dispersive equations. I will discuss the stability, instability and blow-up for some simpler models such as the cubic Nonlinear Schr\"odinger equations, as well as for some more delicate geometric equations.
Peter Ebenfelt
Nov. 22 No colloquium. Thanksgiving Holiday.
Nov. 29 Mitchell Luskin
(Univ. Minnesota/
IPAM, UCLA)
Title. Atomistic-to-Continuum Coupling Methods

Abstract. Many materials problems require the accuracy of atomistic modeling in small regions, such as the neighborhood of a crack tip. However, these localized defects typically interact through long ranged elastic fields with a much larger region that cannot be computed atomistically. Many methods have recently been proposed to compute solutions to these multiscale problems by coupling atomistic models near a localized defect with continuum models where the deformation is nearly uniform. During the past several years, we have given a theoretical structure to the description and formulation of atomistic-to-continuum coupling that has clarified the relation between the various methods and the sources of error. Our theoretical analysis and benchmark simulations have guided the development of optimally accurate and efficient coupling methods.
Randy Bank & Bo Li
Nov. 30
(Friday)
Otmar Venjakob
(Univ. Heidelburg, Germany)
Title. Are zeta-functions able to solve Diophantine equations?

Abstract. Motivated by the question whether (some) Diophantine equations are related to special values of $\zeta$- or $L$-functions we first describe the origin of classical Iwasawa theory. Then we give a survey on generalizations of these ideas to non-commutative Iwasawa theory, a topic which has been developed in recent years by several mathematicians, including the author.
Cristian Popescu
Dec. 6 Herbert Heyer
(Univ. Tuebingen, Germany)
Title. Arithmetic properties of the semigroup of probability measures

Abstract. There are two basic theorems on arithmetic properties of probability measures on Euclidean space: the Levy decomposition of infinitely divisible probability measures as convolutions of Poisson and Gaussian measures, and the Khintchine factorization of arbitrary probability measures in terms of indecomposable measures and measures without indecomposable factors. Both theorems have been generalized by K. R. Parthasarathy to measures on an Abelian locally compact group. Within this framework the role of Gaussian factors will be discussed. Moreover, characterizations of Gaussian measures (in the sense of Cramer and Bernstein) will be presented whose validity depends on the structure of the underlying group.
Patrick Fitzsimmons

Last updated by Bo Li on April 28, 2013.