# Seminars & Events for 2008-2009

##### Computational geometry of moduli spaces of curves

A fast algorithm for computing intersection numbers of $\psi$- and $\kappa$-classes on moduli spaces of complex algebraic curves is proposed. As a consequence, the exact large genus asymptotics of these numbers (in particular, Weil-Petersson volumes) is numerically derived.

##### Compactness Properties of the Space of Genus-$g$ Helicoids

I will discuss a recent application of the work of Colding and Minicozzi of structure of embedded minimal surfaces in $\Real3$ to the study of compactness properties of the space of genus-$g$ helicoids. I will introduce the theory of Colding and Minicozzi and then show how it can be used to show (among other results) that the space of genus-one helicoids is compact (modulo symmetries).

##### Unusual Classical Ground States of Matter

A classical ground-state configuration of a system of interacting particles is one that minimizes the system potential energy. In the laboratory, such states are produced by slowly cooling a liquid to a temperature of absolute zero, and usually the ground states are crystal structures. However, our theoretical understanding of ground states is far from complete.

##### Regularity of singular harmonic maps and axially symmetric stationary electrovacuum spacetimes

According to the Ernst-Geroch reduction, to each axially symmetric stationary vacuum/electro-vacuum spacetime, one can associate an axially symmetric harmonic map with singular boundary behavior. This idea has been exploited in the literature to construct asymptotically flat, axially symmetric stationary spacetimes with disconnected horizons, i.e. having multiple black holes.

##### Vojta's conjecture on Blowups and GCD Inequalities

Vojta's conjecture is a deep conjecture in Diophantine geometry, implying for example the Bombieri-Lang conjecture and the abc conjecture. In this talk, I will show some cases of the conjecture for blowup varieties. As a consequence, we derive some interesting inequalities of greatest common divisors.

##### Universality at the spectrum edge for random matrices with independent entries: Soshnikov's theorems and some extensions

We shall discuss the distribution of extreme eigenvalues for several classes of random matrices with independent entries. In particular, we shall discuss the results of Soshnikov and some of their recent extensions, and the combinatorial questions that appear in the proofs. (Based on joint work with Ohad Feldheim).

##### Mathematical Questions Arising from Bose-Einstein Condensation

Bose-Einstein condensation was predicted by Einstein in 1925 and was experimentally discovered 70 years later. This discovery was followed by a flurry of activity in the physics community with many new experiments and with attempts to construct a theory of the newly discovered state of matter. In this talk I will review some recent rigorous results in the subject and outline open problems.

##### Bend and break

Our goal is to explain the following theorem due to Mori: Given a compact complex manifold $X$ whose tangent bundle has lots of (holomorphic) determinantal sections (a Fano manifold), any pair of points on it lie in the image of a holomorphic map from the Riemann sphere $P^1$.

##### The decay of Fourier modes of solutions of 2-D Navier-Stokes system

Also:

Numerical results related to the talk by D. Li

presented by Nikolai I. Chernov, University of Alabama

##### Randomness extractors - applications and constructions

We will investigate the minimal requirements from a random source under which it is potentially useful for generating perfect randomness. The efficient procedures which utilize such sources, "randomness extractors", turn out to have amazing pseudorandomness properties and are useful in numerous contexts. We'll demonstrate applications in complexity theory, error correction and network design.

##### Two generator subgroups of the pure braid group

A group satisfies the "Tits alternative" if every subgroup is either virtually solvable or contains a nonabelian free group. This is named after J. Tits who proved that all finitely generated linear groups enjoy this property. The Tits alternative was established for braid groups by Ivanov and McCarthy, but now also follows from linearity (due to Bigelow-Krammer).

##### Potential automorphy for certain Galois representations to GL(n)

I will describe recent generalizations of mine to a theorem of Harris, Shepherd-Barron, and Taylor, showing that have certain Galois representations become automorphic after one makes a suitably large totally-real extension to the base field.

##### Geometric flows with rough initial data

In a recent joint work with Herbert Koch (University of Bonn) we showed the existence of a global unique and analytic solution for the mean curvature flow (in arbitrary codimensions) and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm.

##### Classical convolution inequalities and Boltzmann equations for integrable angular section

We study the integrability properties of the gain part of the Boltzmann collision operator using radial symmetrization techniques from harmonic analysis to show Young's inequality in the case of hard potentials and Hardy-Littlewood-Sobolev inequality for soft potentials. The contacts are given by exact formulas depending on the angular cross section.

##### Localization bounds for multiparticle systems

We discuss the spectral and dynamical properties of quantum systems of N particles on the lattice of arbitrary dimension, with a Hamiltonian which in addition to the kinetic term includes a random potential with parameters of the model are the strength of the disorder and the strength of the interparticle interaction.

##### Large N limit of random matrices, free probability and the graded algebra of a planar algebra

##### Models and Fields: A delicate Passage to Characteristic p

If a polynomial map from $C^n$ to itself is injective, then it is also surjective. The first proof of this elegant result actually used a passage to characteristic p.

##### Finding Lovasz's Needle in an Exponential Haystack

The Lovasz Local Lemma is a subtle and far-reaching probability lemma. Roughly, given a large number of bad events which are "mostly" independent it sieves out an instance when none of the bad events occur. As an illustrative example, consider a huge family of 10-element sets, where each set overlaps at most 100 others.

##### Limit lognormal process, Selberg integral as Mellin transform, and intermittency differentiation.

The limit lognormal process is a multifractal stochastic process with the remarkable property that its positive integral moments are given by the celebrated Selberg integral. We will give an overview of the limit lognormal construction followed by a summary of our results on functional Feynman-Kac equations and resulting intermittency expansions that govern its distribution.

##### Topologically invariant Chern numbers of projective varieties

In 1954 Hirzebruch asked which linear combinations of Chern numbers are topological invariants of smooth complex projective varieties. Until recently, this problem was wide open, with few non-trivial results. We give a complete solution in arbitrary dimensions.