## Hodge Structures in Symplectic Geometry

Tony Pantev
University of Pennsylvania
October 21, 2011 - 4:30pm

I will explain how essential information about the structure of symplectic manifolds is captured by algebraic data, and specifically by the non-commutative (mixed) Hodge structure on the cohomology of the Fukaya category. I will discuss how mirror symmetry leads to Hodge theoretic symplectic invariants arising from twist functors, and from geometric extensions. I will also explain how the instanton-corrected Chern-Simons theory fits in the framework of normal functions in non-commutative Hodge theory and will give applications to explicit descriptions of quantum Lagrangian branes.

## On the Instability for 2D Fluids

Serguei Denissov
University of Wisconsin; Institute for Advanced Study
October 21, 2011 - 3:00pm

For 2D Euler equation, we prove a double exponential lower bound on the vorticity gradient. We will also discus some further results on the singularity formation for other models.

## Riemannian Exponential Map on the Group of Volume-Preserving Diffeomorphisms

Gerard Misiolek
University of Notre Dame; Institute for Advanced Study
October 19, 2011 - 4:00pm

In 1966 V. Arnold showed how solutions of the Euler equations of hydrodynamics can be viewed as geodesics in the group of volume-preserving diffeomorphisms. This provided a motivation to study the geometry of this group equipped with the $$L^2$$ metric. I will describe some recent work on the structure of singularities of the associated exponential map and related results

## On the Long-Time Behavior of 2-D Flows

Alexander Shnirelman
Concordia University; Institute for Advanced Study
October 19, 2011 - 2:00pm

## The Universal Relation Between Exponents in First-Passage Percolation

Sourav Chatterjee
Courant Institute; NYU
October 18, 2011 - 4:30pm

It has been conjectured in numerous physics papers that in ordinary first-passage percolation on integer lattices, the fluctuation exponent \chi and the wandering exponent \xi are related through the universal relation \chi=2\xi -1, irrespective of the dimension. This is sometimes called the KPZ relation between the two exponents. I will give a rigorous proof of this conjecture assuming that the exponents exist in a certain sense.

## Rigidity of 3-Colorings of the d-Dimensional Discrete Torus

Tel Aviv University
October 18, 2011 - 10:30am

We prove that a uniformly chosen proper coloring of Z_{2n}^d with 3 colors has a very rigid structure when the dimension d is sufficiently high. The coloring takes one color on almost all of either the even or the odd sub-lattice. In particular, one color appears on nearly half of the lattice sites. This model is the zero temperature case of the 3-states anti-ferromagnetic Potts model, which has been studied extensively in statistical mechanics. The proof involves results about graph homomorphisms and various combinatorial methods, and follows a topological intuition.

## How to Construct Topological Invariants via Decompositions and the Symplectic Category

Katrin Wehrheim
Massachusetts Institute of Technology; Institute for Advanced Study
October 17, 2011 - 2:00pm

A Lagrangian correspondence is a Lagrangian submanifold in the product of two symplectic manifolds. This generalizes the notion of a symplectomorphism and was introduced by Weinstein in an attempt to build a symplectic category. In joint work with Chris Woodward we define such a category in which all Lagrangian correspondences are composable morphisms. We extend it to a 2-category by extending Floer homology to cyclic sequences of Lagrangian correspondences.

## On the Number of Hamilton Cycles in Psdueo-Random Graphs

Michael Krivelevich
Tel Aviv University
October 17, 2011 - 11:15am

A pseudo-random graph is a graph G resembling a typical random graph of the same edge density. Pseudo-random graphs are expected naturally to share many properties of their random counterparts. In particular, many of their enumerative properties should be similar to those of random graphs.

In this work we study the number of Hamilton cycles in pseudo-random graphs. We use the so called (n,d,\lambda)-graphs as a model of pseudo-random graphs, these are d-regular graphs on n vertices, all of whose non-trivial eigenvalues are at most \lambda in their absolute values.

## On Real Zeros of Holomorphic Hecke Cusp Forms and Sieving Short Intervals

Kaisa MatomÃ¤ki
University of Turku, Finland
October 13, 2011 - 4:30pm

A. Ghosh and P. Sarnak have recently initiated the study of so-called real zeros of holomorphic Hecke cusp forms, that is zeros on certain geodesic segments on which the cusp form (or a multiple of it) takes real values. In the talk I'll first introduce the problem and outline their argument that many such zeros exist if many short intervals contain numbers whose all prime factors belong to a certain subset of primes. Then I'll speak about new results on this sieving problem which lead to improved lower bounds for the number of real zeros.

## Limit Theories and Higher Order Fourier Analysis

Balazs Szegedy
University of Toronto; Member, School of Mathematics
October 11, 2011 - 10:30am

We present a unified approach to various topics in mathematics including: Ergodic theory, graph limit theory, hypergraph regularity, and Higher order Fourier analysis. The main theme is that very large complicated structures can be treated as approximations of infinite measurable and topological objects. In the limit interesting algebraic structures and new concepts arise which are hard to capture in the finite language but they govern the behavior of the finite objects. A prominent example is the inverse theorem for the Gowers norms on arbitrary abelian groups.