# Seminars & Events for 2007-2008

##### New Insights into Semidefinite Programming for Combinatorial Optimization

Beginning with the seminal work of Goemans and Williamson on Max-Cut, semidefinite programming (SDP) has firmly established itself as an important ingredient in the toolkit for designing approximation algorithms for NP-hard problems. Algorithms designed using this approach produce configurations of vectors in high dimensions which are then converted into actual solutions.

##### Non-uniform dependence for the Periodic CH equation

Using approximate solutions we show that the solution map for the periodic Camassa-Holm (CH) equation is not uniformly continuous in Sobolev spaces with exponent greater or equal to one. This extends earlier results to the whole range of Sobolev exponents.

##### Prediction of health care costs via data-mining and algorithmic discovery of medical knowledge

Rising health care costs are one of the world's most important problems. Correspondingly, predicting such costs with accuracy is a significant first step in addressing this problem.

##### Reimagining universal covers and fundamental groups in algebraic and arithmetic geometry

In topology, the notions of the fundamental group and the universal cover are inextricably intertwined. In algebraic geometry, the traditional development of the étale fundamental group is somewhat different, reflecting the perceived lack of a good universal cover.

##### Volume of polytopes, operator analogues, and Arthur's trace formula

There are two ways (among many others) to compute the volume of a (convex) polytope. One using a formula of Brion and another using an argument of P. McMullen and R. Schneider. The ensuing identity suggests a non-commutative generalization which we can currently prove for Coxeter zonotopes (e.g. a permutahedron). This algebraic equality plays a role in Arthur's trace formula.

##### An Outline of the h-Cobordism Theorem

In the first half of the talk, we shall go through the definitions of manifold, cobordism, h-cobordism, Morse functions and gradient-like flows, and demonstrate many of their properties. In the second half of the talk we shall give an outline of the proof. In the process, we shall prove lemmas about commutation and cancellation of critical points, and see the Whitney trick in action.

##### Quenched Central Limit Theorem for Random Toral Automorphism

The statistical properties of the Lorentz gas with periodically positioned obstacles are well understood. The random case, obtained after each of the obstacles undergoes a small i.i.d. displacement, stands as a challenge. The latter can be studied in terms of a random sequence of hyperbolic symplectic (billiard) maps, which however is not i.i.d. due to recollisions. In fact, even the i.i.d.

##### Ramsey numbers of sparse graphs and hypergraphs

The Ramsey number $r(G)$ of a graph $G$ is the minimum $N$ such that every 2-coloring of the edges of the complete graph on $N$ vertices contains a monochromatic copy of $G$. Determining or estimating Ramsey numbers is one of the central problem in Ramsey theory. Besides the complete graph, the next most classical topic in this area concerns the Ramsey number of sparse graphs.

##### Prime Chains and Pratt trees

A sequence of primes $p_1, ..., p_k$ is called a prime chain if $p_j | (p_{j+1}-1)$ for each $j$; e.g. 3, 7, 29, 59. We will discuss problems about counting prime chains with certain properties, and about the existence of prime chains with various properties. The Pratt tree for a prime p is the tree with root node p and below p are the Pratt trees of the odd prime factors of $p_1$.

##### Did the great masters 'cheat' using optics? Computer vision and graphics addresses a bold theory in art history

In 2001, artist David Hockney and scientist Charles Falco stunned the art world with a controversial theory that, if correct, would profoundly alter our view of the development of image making. They claimed that as early as 1420, Renaissance artists employed optical devices such as concave mirrors to project images onto their canvases, which they then traced or painted over.

##### High dimensional volatility models

The conditional standard deviation, or volatility, of asset returns evolves over time; financial volatilities move together over time across assets and markets. For even a handful of assets, the curse of dimensionality quickly makes extimation of most multivariate models impractical.

##### Finite and infinite representations of surface groups and cross ratios

In this talk, I explain how cross ratios—special functions of four arguments—on the boundary at infinity of surfaces groups describe both finite (in SL(nR)) and infinite (in a group related to diffeomorphisms of the circle) dimensional representations of surface groups.

##### Differential Equations and Arithmetic

One of the questions in number theory is to represent integers by quadratic forms. To be more precise, the question asks for a characterization of the integers which can be represented by a given form. In this talk we will deal with representing integers as sums of two squares and if time permits as sums of four squares. The talk will be completely elementary.

##### Exits from an infinite tube

We consider a billiard system in an infinite tube with periodic scatterers. We show that with probability 1 a particle exits from the tube. Surprisingly, the probability that the exit velocity is opposite to the initial one tends to 1 in a limit when the size of scatterers vanishes.

##### Deformation of a hyperbolic 4-orbifold

It is well known that Thurston's beautiful deformation theory of hyperbolic structures is mostly useless in dimensions > 3. Steve Kerckhoff and I have been studying a new example of a hyperbolic deformation in 4 dimensions which produces an infinite number of new hyperbolic 4-orbifolds with interesting properties. The talk will attempt to motivate this work.

##### Integral models of some Shimura varieties

##### Area-dependence in gauged Gromov-Witten theory

I will describe joint work with E. Gonzalez, in which we study the dependence of the moduli space of gauged pseudoholomorphic maps from a surface to a target X as the area form on the surface is varied. As an application, we get some version of the "abelianization" conjecture of Bertram et al relating Gromov Witten theory of symplectic quotients by a group and its maximal torus.

##### Isomorphic uniform convexity in metric spaces

In 1985 Bourgain gave a discrete metric characterization of when a normed space admits an equivalent uniformly convex norm: this happens if and only if the space does not contain arbitrarily large complete binary trees with low distortion.

##### Derivation of the Gross-Pitaevskii equation: the case of strong interaction potential

In this talk I am going to present recent results obtained in collaboration with L. Erdoes and H.-T. Yau concerning the derivation of a certain cubic nonlinear Schroedinger equation, known as the Gross-Pitaevskii equation, from first principle many body quantum dynamics. With respect to previous results, we can now relax the smallness condition on the interaction potential.

##### Nonparametric nonstationary regression

This article studies nonparametric estimation of a regression model for $d_2$ potentially nonstationary regressors. It provides the first nonparametric procedure for a wide and important range of practical problems, for which there has been no applicable nonparametric estimation technique before.