# Mathematics Colloquium Fall 2007

For further information please contact the Mathematics Department at 459-2969

**October 2, 2007**

**Symplectic Embeddings**

**Symplectic Embeddings**

**Professor Larry Guth, Stanford University**

We discuss when one open set in R^{2n} may be symplectically embedded into another open set in R^{2n}. The problem has two sides to it. On the one hand, there are some deep theorems that give obstructions to embedding one set into another. On the other hand, there are cases when one set may be symplectically embedded into another, but only by using a tricky mapping. We will focus on the problem of symplectically embedding one 2n-dimensional rectangle into another. Depending on the dimensions of the rectangles, this problem involves both obstructions and tricky mappings.

**October 9, 2007**

**An Introduction to open-closed conformal field theory**

**An Introduction to open-closed conformal field theory**

**Professor Liang Kong, MaxPlanck Institute for Mathematics, Bonn, Germany **

Open-closed conformal field theory describes perturbative open-closed string theory and some critical phenomena in condensed matter physics. It provides a tool to study the still mysterious object called "D-brane". In this talk, I will outline a mathematical study of open-closed conformal field theory based on the theory of vertex operator algebra. In particular, I will give a tensor-categorical formulation of open-closed conformal field theory with boundary condition preserving a rational vertex operator algebra. If time permits, I will briefly discuss what D-branes are in this framework. This talk will be accessible to those who know nothing about conformal field theory.

**October 16, 2007**

**The largest eigenvalues distribution of perturbations of random matrices**

**The largest eigenvalues distribution of perturbations of random matrices**

**Prof. Estelle Basor/Calif Polytechnic State Univ/San Luis Obispo **

This talk will describe the largest eigenvalue distribution for certain perturbations of random matrices. In particular we will consider rank one perturbations of Gaussian Unitary Ensembles,that is, matrices whose entries are Gaussian random variables. If time permits, other ensembles with be discussed. This work was done as part of a summer research project with undergraduates at Cal Poly. The talk will be understandable to students with a knowledge of linear algebra and some basic analysis.

**October 23, 2007**

**Digital Snowflakes**

**Digital Snowflakes**

**Prof. Janko Gravner/UC Davis **

Growth of snow crystals is notoriously difficult to understand, as controlled experiments are impossible, basic physical principles poorly understood and most differential equation models difficult to analyze. We will first discuss a popular class of cellular automata known as Packard Snowflakes, for which a fairly complete mathematical theory exists. Then we will describe a much more realistic mesoscopic snowflake model. The talk will be accompanied by many computer computations and is on joint work with D. Griffeath (Univ. of Wisconsin).

**October 30, 2007**

**Orthogonal polynomials and Markov chains, direct and inverse problems**

**Orthogonal polynomials and Markov chains, direct and inverse problems**

**Professor Alberto Grunbaum, UC Berkeley **

The study of Markov chains with a complicated state space and a fairly general one-step transition probability matrix (aka traffic matrix in network theory)gives rise to interesting nonlinear inverse problems. For some such chains one can solve the direct problem quite explicitly by exploiting connections with (nonabelian) harmonic analysis. Since this can be done in many such "group theoretical" situations one gets a parametric family of models that allow for a solution of the inverse problem. I will try to give an ab-initio presentation of the subject and of some of the results.

**November 6, 2007**

**Regularity theory for elliptic equations**

**Regularity theory for elliptic equations**

**Professor Lihe Wang, University of Iowa **

We will talk about the recent understanding about elliptic, parabolic equations and in particular the recent work about degenerate equations and their applications.

**November 13, 2007**

**Restrictions of Continuous Functions**

**Restrictions of Continuous Functions**

**Professor Yitzhak Katznelson, Stanford University **

Given a continuous real-valued (or R^d-valued) function f on [0,1], and a closed subset E of [0,1], let f|E denote the restriction of f to E. The restriction f|E will typically be "better behaved" than f. It may have bounded variation when f doesn't; it may have a better modulus of continuity than f ; it may be monotone when f is not, etc. The questions that we discuss are about the existence, for every f, or every f in some class, of "substantial" sets E such that f|E has bounded total variation, is monotone, or satisfies a given modulus of continuity. The notion of "substantial" that we use is that of either Hausdorff or Minkowski dimensions.

**November 20, 2007**

**Conformally compact Einstein manifolds**

**Conformally compact Einstein manifolds**

**Professor Sophie Szu-Yu Chen, UC Berkeley**

Conformally compact Einstein manifolds are models of Euclidean formulation of Lorenzian spaces in general relativity. In this talk, we survey the subject from mathematical point of view and discuss some recent progress.

**November 27, 2007**

**Gluing constructions for minimal surfaces and special Lagrangian cones**

**Gluing constructions for minimal surfaces and special Lagrangian cones**

**Nicolaos Kapouleas, Brown University **

We will discuss the current status of three types of constructions:

First, doubling constructions where two parallel copies of a given minimal surface are joined by small catenoidal bridges to produce new minimal surfaces, which tend to a double covering of the original minimal surface, as the mumber of the bridges goes to inifinity. At the moment in collaboration with S.-D. Yang we have such a construction for doubling the Clifford torus where the bridges are centered at the points of a square lattice.

Second, desingularizing constructions where intersection curves of minimal surfaces are replaced by handles modeled after the handles of the singly periodic Scherk surfaces. An application of this is to the combination of doubled Clifford tori, where the corresponding lattices as above are nested.

Third, in collaboration with M. Haskins, gluing constructions for special Lagrangian cones in $C^n$. In these constructions we combine cones whose links (intersections with the unit sphere)are rotationally invariant.In a recent paper we study constructions in the case $n=3$, where we obtain the only known examples whose link is a closed surface of genus $g>1$. More precisely we obtain examples for any odd $g$ or for $g=4$.

We will also discuss ongoing work in the higher dimensional case where $n>3$.

**December 4, 2007**

**From the Mahler conjecture to Gauss linking integrals**

**From the Mahler conjecture to Gauss linking integrals**

**Greg Kuperberg, UC Davis **

The Mahler volume of a centrally symmetric convex body K in n dimensions is defined as the product of the volume of K and the polar body K^o. It is an affinely invariant number associated to a centrally symmetric convex body, or equivalently a basis-independent number associated to a finite-dimensional Banach space. Mahler conjectured that the Mahler volume is maximized by ellipsoids and minimized by cubes. The upper bound was proven long ago by Santalo. Bourgain and Milman showed that the lower bound, known as the Mahler conjecture, is true up to an exponential factor. Their theorem is closely related to other modern results in high-dimensional convex geometry.

I will describe a new proof of the Bourgain-Milman theorem that establishes the Mahler conjecture up to an exponential factor of (pi/4)^n. The proof minimizes a different volume at the opposite end of the space of convex bodies, i.e., at ellipsoids. The minimization argument is based on indefinite inner products and Gauss-type integrals for linking numbers.