# Seminars & Events for 2011-2012

##### Global existence for solutions of the energy critical nonlinear Schrodinger equation in the Torus

I will present a joint work with A. Ionescu proving that smooth solutions to the energy critical nonlinear Schrodinger equation on T3 remain global in time. One interest of this is that it clarifies the possible obstructions to global existence of the energy critical equation on manifolds due to trapped geodesics and finite volume.

##### Optimization of Polynomial Roots, Eigenvalues and Pseudospectra

The root radius and root abscissa of a monic polynomial are respectively the maximum modulus and the maximum real part of its roots; both these functions are nonconvex and are non-Lipschitz near polynomials with multiple roots.

##### Secant varieties of Segre-Veronese varieties

Secant varieties of Segre and Veronese varieties are classical objects that go back to the Italian school in the nineteen century. Surprisingly, very little is known about their equations.

##### The universal relation between exponents in first-passage percolation

Nonequilibrium statistical mechanics close to equilibrium is a physically satisfactory theory centered on the linear response formula of Green-Kubo. This formula results from a formal first order perturbation calculation without rigorous justification. A rigorous derivation of Fourier's law for heat conduction from the laws of mechanics remains thus a major unsolved problem.

##### Entangled protons in ice

Quantum fluctuations can drive phase transitions in ice. This happens when the protons tunnel between the two equivalent sites on a hydrogen bond. The corresponding dynamics is collective and dominated by strong local correlations originating from the ice rules.

##### On the growth of Betti numbers of arithmetic groups

We study the asymptotic behavior of the Betti numbers of higher rank locally symmetric manifolds as their volumes tend to infinity, and prove a uniform version of the Lueck Approximation Theorem, which is much stronger than the linear upper bounds proved by Gromov.

##### Subgraph densities in signed graphs and local extremal graph problems

Let G be a graph whose edges are signed by +1 or -1. We can "count" labeled copies of a graph F in G by multiplying the edge signatures and summing over all copies of F. The density of F in G is obtained by dividing by the total number of maps from V(F) to V(G). There are interesting inequalities between the densities of various subgraphs in signed graphs.

##### Combinatorial Heegaard Floer homology

After quickly reviewing the construction beyond the topological/combinatorial version of (stable) Heegaard Floer homology, we show how to define the theory over the integers, and how to get the $spin^c$ refined theory. This is a joint work with Peter Ozsvath and Zoltan Szabo.

##### Derived functors in unstable homotopy theory

The talk will be about a functorial approach to the problem of computation of homology of Eilenberg-MacLane spaces, homotopy groups of suspensions of classifying spaces, etc.

##### Hodge correlators, Hodge symmetries, and Rankin-Selberg integrals

Rankin-Selberg integrals, among many other things they do, are the only way to prove that special values $L(f, n)$ of L-functions of weight $k$ modular forms on $GL_2(\mathbb{Q}), n\geq k$, are periods. They pave the road to Beilinson's motivic $\zeta$-elements, organized by Kato into an Euler system.

##### Sharp constants in inequalities on the Heisenberg group

We derive the sharp constants for the inequalities on the Heisenberg group whose analogues on Euclidean space are the well known Hardy-Littlewood-Sobolev inequalities. From these inequalities we obtain the sharp constants for their duals, which are the Sobolev inequalities for the Laplacian and conformally invariant fractional Laplacians.

##### Hodge Structures in Symplectic Geometry

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.

##### On queues and numbers

We will show that certain symmetries which have traditionally played an important role in number theory are also important for analyzing certain simple queueing systems. This connection between number theory and queueing theory leads to some interesting questions in number theory and also helps understand results from several queueing theory papers.

##### Existence and regularity for a class of degenerate diffusions arising in population genetics

Joint PACM Colloquium and Analysis Seminar

##### Fractal iso-contours of passive scalar in smooth random flows

We consider a passive scalar field under the action of pumping, diffusion and advection by a smooth flow with a Lagrangian chaos. We present theoretical arguments showing that scalar statistics is not conformal invariant and formulate new effective semi-analytic algorithm to model the scalar turbulence.

##### The Nash conjecture for surfaces

The space of arcs through the singular set of an algebraic variety has a infinite dimensional scheme structure. In the late sixties Nash proved that it has finitely many irreducible components. He defined a natural mapping from this set of irreducible components to the set of essential divisors of a resolution of singularities.

##### The renormalisation group and recent applications

I will review a method to control irrelevant terms in the renormalisation group and describe an application to self-repelling walk in four dimensions.

##### Littlewood and large forests

Motivated by a classical result of Sz.-Nagy in functional analysis, Dixmier asked in 1950 which group representations can be made unitary. This question is still open, but I will report on some recent progress. We approach the question with ideas borrowed from XIXth century electricity theory as well as from contemporary percolation theory.

##### Beyond Total Unimodularity

A matrix is TOTALLY UNIMODULAR if the determinant of each square submatrix is in {-1, 0, 1}. Such matrices are a cornerstone of the theory of integer programming. The deepest result on such matrices is Seymour's decomposition theorem. The only known way to test efficiently whether a matrix is totally unimodular makes use of this theorem.

##### Feynman categories

There is a plethora of operad type structures and constructions which arise naturally in classical and quantum contexts such as operations on cochains, string topology or Gromov-Witten invariants. We give a novel categorical framework which allows us to handle all these different beasts in one simple fashion. In this context, many of the relevant constructions are simply Kan extensions.