Mathematics Colloquia

Talks are at 3:30 p.m. in South Hall, Room 6635, on Thursdays
Colloquium Committee: Xianzhe Dai

## Feature Event, Spring 2015

 Thursday May 28, 2015 3:30-4:30pm, SH 6635 Yitang Zhang, University of New Hampshire Title: Small gaps between primes Abstract: The twin prime conjecture states that there are infitely many pairs of distinct primes which differ by 2.  Until recently this conjecture had seemed to be out of reach with current techniques. However, in 2013, the author proved that there are infitely many pairs of distinct primes which differ by no more than B with B= 7*10^7.  The value of B has been considerably improved by Polymath8 (a cooperative team) and Maynard.  In this talk we shall describe the basic ideas which lead to the proofs of the above results. In particular,  a breakthrough on the distribution of primes in arithmetic progressions will be introduced. Click here Contact person: Dave Morrison

## Winter and Spring 2015

 Thursday March 5, 2015 3:30-4:30pm, SH 6635 James Zhang, University of Washington, Seattle Title:  The Tits Alternative Abstract: In 1972, J. Tits proved the following dichotomy: every subgroup of the linear automorphism group of a finite dimensional vector space is either virtually solvable or contains a free subgroup of rank two. Automorphism groups of deformations of a polynomial ring have been studied extensively by many mathematicians, for example, J. Alev, J. Dixmier, M. Kontsevich, T. Lenagan, M. Yakimov and others. In this talk, we will explain why the discriminant controls some global structures of a family of automorphism groups. By using the discriminant, a new version of the Tits alternative can be proved. This talk will be suitable for a general audience. Contact person: Ken Goodearl
 Thursday April 14, 2015 3:30-4:30pm, SH 6635 Yuri Tschinkel, Simons Foundation Title:  Contact person: Dave Morrison
 Thursday April 16, 2015 3:30-4:30pm, SH 6635 Simon Rubinstein-Salzedo, Stanford Title: Dessins d'enfants and origamis Abstract:  We begin by looking at branched covers of the projective line with three branch points. Such covers became part of mainstream mathematics thanks to Grothendieck, who was interested in such covers as a way of understanding the absolute Galois group of the rational numbers and hence learning more about the sorts of number fields that exist. In this case, we can explicitly construct some interesting number fields. We then talk about the analogous problem for covers of elliptic curves with one branch point. In this case, the computations are far more difficult, but in theory, they allow us similar opportunities and more. We shall see how one can go about writing down explicit examples of covers of elliptic curves in special cases. We shall also see that both covers of the projective line and covers of elliptic curves have pictorial representations that capture a wealth of combinatorial and topological information, and that are fun to study.                             Contact person: Birge Huisgen-Zimmermann
 Thursday May 7, 2015 3:30-4:30pm, SH 6635 Skip Garibaldi, IPAM Title: Some people have all the luck Abstract: Winning a prize of $600 or more in the lottery is a remarkable thing – probably none of your friends or family have ever won one. But when we investigated records of all claimed lottery prizes in Florida, we discovered that some people had won hundreds of them! Such people seem to be not just lucky, but suspiciously lucky. I will explain what we thought they might have been up to, what mathematics says about it, what further investigations revealed, and the law enforcement actions and state policy changes that occurred as a consequence of our theorems. This talk is about joint work with Richard Arratia, Lawrence Mower, and Philip B. Stark. Contact person: Bill Jacob  Thursday May 21, 2015 3:30-4:30pm, Sh 6635 Chelsea Walton, MIT Title: Quantum symmetry Abstract: Symmetry has long been a crucial notion in mathematics and physics. Groups arose to axiomatize the notion of symmetry; namely, groups are comprised of a set of invertible transformations of an object of interest. But it is common practice to replace the object of study X with an algebra A of functions on X. Symmetries of X are then realized as the set of group actions on A. (Here, the group is a set of automorphisms of A.) So let's kick this up a notch- let's study symmetries of quantum objects. Indeed such objects are impossible to visualize, yet there are natural noncommutative algebras B that arise as 'quantum function algebras' on these objects. We can certainly still consider group actions on B in this setting. But the aim of this talk is to convince you that studying actions of "Hopf algebras" (or of "quantum groups") on B is more appropriate. Classification results (including some from the analytic point-of-view) and lots of examples will be included. Contact person: Ken Goodearl  Thursday May 28, 2015 3:30-4:30pm, Sh 6635 Yitang Zhang, University of New Hampshire Title: Small gaps between primes Abstract: The twin prime conjecture states that there are infitely many pairs of distinct primes which differ by 2. Until recently this conjecture had seemed to be out of reach with current techniques. However, in 2013, the author proved that there are infitely many pairs of distinct primes which differ by no more than B with B= 7*10^7. The value of B has been considerably improved by Polymath8 (a cooperative team) and Maynard. In this talk we shall describe the basic ideas which lead to the proofs of the above results. In particular, a breakthrough on the distribution of primes in arithmetic progressions will be introduced. Click here Contact person: Dave Morrison ## Fall 2014  Thursday October 16, 2014 3:30-4:30pm, SH 6635 Eitan Tadmor, University of Maryland Title: Images, PDEs and critical regularity spaces: Hierarchical construction of their nonlinear solutions Abstract: Edges are noticeable features in images which can be extracted from noisy data using different variational models. The analysis of such variational models leads to the question of representing general images as the of divergence of uniformly bounded vector fields. We construct uniformly bounded solutions of div(U)=f for general f’s in the critical regularity space L^d(R^d). The study of this equation and related problems was motivated by recent results of Bourgain & Brezis. The intriguing aspect here is that although the problems are linear, the construction of their solution is not. These constructions are special cases of a rather general framework for solving linear equations in critical regularity spaces. The solutions are realized in terms of nonlinear hierarchical representations U= \sum_j u_j which we introduced earlier in the context of image processing. The u_j's are constructed recursively as proper minimizers, yielding a multi-scale decomposition of the solutions U. Contact person: Xu Yang  Thursday Nov 20, 2014 3:30-4:30pm, SH 6635 Anna Wienhard, Caltech Title: Abstract: Contact person: Daryl Cooper  Tuesday (Note the unusual date) Dec 2, 2014 3:30-4:30pm, Room TBA Zhen-Su She, Peking University Title: Abstract: Contact person: Bjorn Birnir ## Spring 2014  Thursday April 17, 2014 3:30-4:30pm, SH 6635 Svitlana Mayboroda, University of Minnesota Title: Localization of eigenfunctions and associated free boundary problems Abstract: The phenomenon of wave localization permeates acoustics, quantum physics, elasticity, energy engineering. It was used in construction of the noise abatement walls, LEDs, optical devices. Anderson localization of quantum states of electrons has become one of the prominent subjects in quantum physics, as well as harmonic analysis and probability. Yet, no methods predict specific spatial location of the localized waves. In this talk I will present recent results revealing a universal mechanism of spatial localization of the eigenfunctions of an elliptic operator and emerging operator theory/analysis/geometric measure theory approaches and techniques. We prove that for any operator on any domain there exists a landscape" which splits the domain into disjoint subregions and indicates location, shapes, and frequencies of the localized eigenmodes. In particular, the landscape connects localization to a certain multi-phase free boundary problem, regularity of minimizers, and geometry of free boundaries. This is joint work with D. Arnold, G. David, M. Filoche, and D. Jerison. Contact person: Gustavo Ponce  Thursday May 12-15, 2014 3:30-4:30pm, SH 6635 Distinguished Lectures Alice Chang and Paul Yang, Princeton University Monday, May 12, Dinstinguished Lectures/Distinguished Women in Math Alice Chang Title: On a class of conformal covariant operators and conformal invariants Abstract: In 2005, Graham-Zworski introduced a continuous family of conformal covari- ant operators of high orders via scattering theory on conformal compact Einstein manifolds. This class of operators P° and their associated curvature Q° has played important roles in problems in conformal geometry and in the study of some geometric invariants in the Ads/CFT setting. In the talk, I will survey some of the recent progress in this field, and also report some recent joint work with Jeffrey Case about positivity property of this class of operators under curvature assumptions. Wednesday, May 14, Dinstinguished Lectures Paul Yang Title: CR geometry in 3-D Abstract: In this talk, I report on the Embedding problem for 3-D CR structures. As a consequence of the embedding criteria, we obtain a positive mass theorem. Further application is the analysis of the new operator on the pluriharmonic functions and the associated Q'curvature. The work were joint with Case, Chanillo, Cheng, Chiu, and Malchiodi. Thursday, May 15, Dinstinguished Lectures Paul Yang Title: A fourth order operator in conformal geometry Abstract: In this talk, I report on the Paneitz operator and the Q-curvature equation. We obtain criteria for the sign of the Green's function for this operator, and hence the solvability of the Q-curvature equation in all dimensions. This is a joint work with Fengbo Hang. Contact person: Xianzhe Dai  Thursday June 5, 2014 3:30-4:30pm, SH 6635 Sergey Fomel, University of Texas-Austin Title: Wave Equations and Wave Extrapolations in Seismic Imaging Abstract: Seismic imaging is a multibillion-dollar enterprise aimed at extracting information about the Earth interior from reflected seismic waves. The main task of reflection imaging is to take seismic measurements from multiple experiments with sources and receivers on the surface of the Earth and to extrapolate seismic waves numerically back in time and space to the moment of their reflection. This problem leads to novel partial-differential and pseudo-differential equations describing different parts of the imaging process. A constructive way for numerical wave extrapolation follows from low-rank approximations of Fourier symbols. Contact person: Xu Yang ## Winter 2014  Thursday March 6, 2014 3:30-4:30pm, KITP (note the unusual place) David Gross, KITP Title: Yang-Mills Theory ( Video Link ) Abstract: (from the CMI website)The laws of quantum physics stand to the world of elementary particles in the way that Newton's laws of classical mechanics stand to the macroscopic world. Almost half a century ago, Yang and Mills introduced a remarkable new framework to describe elementary particles using structures that also occur in geometry. Quantum Yang-Mills theory is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories, but its mathematical foundation is still unclear. The successful use of Yang-Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the "mass gap": the quantum particles have positive masses, even though the classical waves travel at the speed of light. This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view. Progress in establishing the existence of the Yang-Mills theory and a mass gap and will require the introduction of fundamental new ideas both in physics and in mathematics. Contact person: Xianzhe Dai ## Fall 2013  Thursday Oct. 17, 2013 3:30-4:30pm, SH 6635 Greg Galloway, University of Miami & MSRI Title: On the topology of black holes and beyond Abstract: In recent years there has been an explosion of interest in black holes in higher dimensional gravity. This, in particular, has led to questions about the topology of black holes in higher dimensions. In this talk we review Hawking's classical theorem on the topology of black holes in 3+1 dimensions (and its connection to black hole uniqueness) and present a generalization of it to higher dimensions. The latter is a geometric result which places restrictions on the topology of black holes in higher dimensions. We shall also discuss recent work on the topology of space exterior to a black hole. This is closely connected to the Principle of Topological Censorship, which roughly asserts that the topology of the region outside of all black holes (and white holes) should be simple. All of the results to be discussed rely on the recently developed theory of marginally outer trapped surfaces, which are natural spacetime analogues of minimal surfaces in Riemannian geometry. This talk is based primarily on joint work with Rick Schoen and with Michael Eichmair and Dan Pollack. Contact person: Guofang Wei  Thursday Oct. 24, 2013 3:30-4:30pm, SH 6635 Mike Freedman, Microsoft & UCSB Title: P vs NP ( Video Link ) Abstract: The title refers to the iconic problem of separating the class of (decision) problems for which we can find solutions quickly from those where it is merely possible to check solutions quickly. For mathematicians this problem asks "Is there really anything to your subject?". For if P=NP it would not be much harder to find proof than to read them. Maybe P vs.NP is undecidable. I'll discuss why some people say the problem is irrelevant. I'll say a little about the little that is known from: diagonalization, oracle reduction, and "natural proofs" and mention a program that exists to solve a related problem ( in algebraic complexity). Then I will tell you my own thoughts, if I have any. Contact person: Xianzhe Dai  Thursday Oct. 31, 2013 3:30-4:30pm, SH 6635 Bisi Agboola, UCSB Title: The Conjecture of Birch and Swinnerton-Dyer Abstract: The problem of finding integer solutions to Diophantine equations is one that has fascinated mathematicians for thousands of years. Although we now know (thanks to the work of Davis, Matiyasevich, Putnam and Robinson resolving Hilbert's 10th problem in the negative) that it is impossible to do this in general, it ought to be possible to say a great deal in special cases. For example, when the equation in question defines an elliptic curve, a remarkable conjecture due to Birch and Swinnerton-Dyer implies that the behaviour of the solutions is governed by the properties of an analytic object (whose very existence is a deep problem in and of itself), namely the L-function attached to the elliptic curve. In this talk, I shall explain some of the ideas that go into the formulation of the Birch and Swinnerton-Dyer conjecture, and I shall discuss some aspects of what is currently known about the conjecture. Contact person: Xianzhe Dai  Thursday Nov. 7, 2013 3:30-4:30pm, SH 6635 Lenny Ng, Duke University Title: Knot invariants via the cotangent bundle Abstract:In recent years, symplectic geometry has emerged as a key tool in the study of low-dimensional topology. One approach, championed by Arnol'd, is to study the topology of a smooth manifold through the symplectic geometry of its cotangent bundle, building on the familiar concept of phase space from classical mechanics. I'll describe how one can use this approach to construct an invariant of knots called "knot contact homology". This invariant is still pretty mysterious 10 years on, but I'll outline some surprising relations to representations of the knot group and to mirror symmetry. Contact person: Zhenghan Wang ## Spring 2013  April 18, 2013 3:30-4:30pm, SH 6635 Jeff Stopple Title: Music of the Primes: from Pythagoras to Riemann Abstract: The Riemann Hypothesis is considered to be one of the most difficult unsolved problems in mathematics. One interpretation, due to physicist Sir Michael Berry, is that “the primes have music in them.” This talk requires only calculus to see what is meant by this. We will hear the (conjectural) music. Contact person: Xianzhe Dai Thursday May 2, 2013 3:30-4:30pm, SH 6635 Bjorn Birnir Title: The Navier-­‐Stokes Millennium Problem: Laminar versus Turbulent Flow Abstract: One of the Millennium Problems is to prove the existence and uniqueness of strong solutions to the initial value problem for the Navier-­‐Stokes equation. This problem is still open. In this talk we will discuss this problem andits relation to the turbulence problem. This is theproblem to find and prove the statistical properties of turbulent flow. We explain how these problems were identified, more than 150 years ago, by O. Reynolds and which problems one has to solve depends on the dimensionless Reynolds number that he defined. We will discuss the recent solution of the turbulence problem and how the techniques developed in its resolution may eventually lead to the solution of the millennium problem. Contact person: Xianzhe Dai  Thursday May 9, 2013 3:30-4:30pm, SH 6635 Jesse Peterson, Vanderbilt Title: Characters and invariant random subgroups for lattices in Lie groups Abstract: A character on a group$G$is a conjugation invariant function$\tau$with$\tau(e) = 1$and such that for$g_1, \ldots, g_n \in G$, the matrix$[\tau(g_j^{-1}g_i)]$is always non-negative definite. For finite groups, the set of extreme points in the space of characters are in one to one correspondence with the set of irreducible representations, and have been extensively studied. The study of characters on infinite groups was initiated in 1964 by Thoma who classified all characters for the group of finite permutations of$\mathbb N\$. In my talk I will discuss the classification of characters on certain lattices in Lie groups, generalizing results of Margulis, and present several applications related to "random subgroups", and rigidity for representations. In contrast to the combinatorial nature of Thoma's result, the techniques involved in studying characters on lattices come from ergodic theory, representation theory, and von Neumann algebras. Contact person: Chuck Akemann
 Thursday May 23, 2013 3:30-4:30pm, SH 6635 Dave Morrison Title: Soap bubbles, the Hodge conjecture, and all that: finding good geometric representatives for topological classes Abstract: Soap bubbles minimize area, and many of the beautiful aspects of minimal surface theory can be illustrated by dipping wire frames into soap solutions and observing the surface which forms.  I will introduce the Hodge conjecture -- one of the Clay Foundation Millenium Prize Problems -- from a somewhat unconventional viewpoint, relating it to a generalization of the minimal surface problem to higher dimensional ambient spaces equipped with interesting geometric structures. Contact person: Xianzhe Dai

Colloquia and Survey Talks are generally preceded by tea at 3 pm in SH 6623.

The COLLOQUIUM SERIES is intended for talks by distinguished visitors from universities other than UCSB. The speaker has the choice of giving a broad survey talk or a talk on a specific research result of major significance. In the former case, the talks will often be directed towards a broad audience that may include scientists and engineers.

SURVEY TALKS are generally by faculty members at UCSB and are specifically designed to be accessible to graduate students. These will often feature topics that graduate students may want to pursue further in graduate courses and seminars.

In addition to the colloquia and survey talks, UCSB Mathematics hosts a wide variety of SEMINARS on special topics at various times throughout the week