# Seminars & Events for 2010-2011

##### Rounding of 1st Order Quantum Phase Transitions in Low- Dimensional Systems with Quenched Disorder

The addition of quenched disorder has a rounding effect on 1st order phase transition in systems of sufficiently low dimension (d=2, and up to d=4 in case of continuous symmetry).

##### Linear PDEs in critical regularity spaces: Hierarchical construction of their nonlinear solutions

We construct uniformly bounded solutions of the equations $div(U)=f$ and $curl(U)=f$, for general $f$ in the critical regularity spaces $L^d(R^d)$ and, respectively, $L3(R3)$. The study of these equations was motivated by recent results of Bourgain & Brezis. The equations are linear but construction of their solutions is not.

##### Some Inverse problems on Riemann surfaces

We show how to identify a potential $V$ or a connection $\nabla^X=d+iX$ up to gauge on a complex vector bundle from boundary measurements (Cauchy data on the boundary) on a fixed Riemann surface with boundary.

##### The tautological ring of $M_g$

I will talk about an approach to the ring generated by the kappa classes via the moduli space of stable quotients. The main new result (with A. Pixton) is a proof of a conjecture by Faber and Zagier of an elegant set of relations. Whether these are all the relations is an interesting question. I will discuss the data on both sides.

##### Hidden Symmetries at the Percolation Point in Two Dimensions

Percolation is perhaps the simplest non-trivial model in statistical mechanics, but has remained under active study for more than forty years. In 2-D it exhibits a second-order phase transition, at which a number of interesting and little-understood symmetries manifest themselves.

##### Natural maps old and new

In 1995, G. Courtois, S. Gallot and myself constructed a family of maps with very good properties regarding volume elements between certain manifolds. We used it to give an alternative proof of Mostow's rigidity for rank one closed symmetric spaces as well as a rigidity result for their geodesic flow, conjectured by A. Katok.

##### A conjecture of Arnold

The *chord conjecture* of Vladimir Arnold is a contact-geometry analogue of his well-known Lagrangian intersections conjecture in symplectic geometry. It proposes that, for each Legendrian submanifold of a contact form on a compact manifold, there should be a integral curve of the Reeb vector field which crosses the Legendrian submanifold at least twice. I will present the

##### Higher-order Fourier analysis of $F_p^n$ and the complexity of systems of linear forms

We study the density of small linear structures (e.g. arithmetic progressions) in subsets $A$ of the group $F_p^n$. It is possible to express these densities as certain analytic averages involving $1_A$, the indicator function of $A$.

##### Two results on rigidity of commutative actions by toral automorphisms

In 1983 Berend proved rigidity of higher-rank commutative actions by toral automorphisms under some hyperbolicity and irreduciblity assumptions. We will present two rigidity results that respectively extend Berend's theorem to certain non-hyperbolic and reducible cases. We will also discuss some counterexamples of non-homogeneous orbit closures. This is joint work with Elon Lindenstrauss.

##### The EHP sequence and the Goodwillie tower

The EHP sequence and the Goodwillie tower of the identity give two different spectral sequences for computing the unstable homotopy groups of spheres. I will explain how the two can be mixed, so that each provides information about the differentials in the other.

##### Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities.

Consider a planar, bounded, $m$-connected region $\Omega$, and let $\partial\Omega$ be its boundary. Let $\mathcal{T}$ be a cellular decomposition of $\Omega\cup\partial\Omega$, where each 2-cell is either a triangle or a quadrilateral.

##### The Iwasawa Main Conjectures for Modular Forms

##### Differentiable rigidity with Ricci bounded below

We consider a closed hyperbolic manifold $(N,h)$ of dimension $n\geq 3$ and a manifold $(M,g)$ with a degre one map $f:M \to N$. We will show that if $Ricci_g \geq -(n-1)$ and $Vol (M,g) \leq (1+\epsilon) Vol (N,h)$, then the manifolds $M$ and $N$ are diffeomorphic. The proof relies on Cheeger-Colding theory of limits of Riemannian manifolds under lower Ricci curvature bound.

##### Diffusions Interacting Through Their Ranks, and the Stability of Large Equity Markets

We introduce and study ergodic multidimensional diusion processes interacting through their ranks. These interactions give rise to invariant measures which are in broad agreement with stability properties observed in large equity markets over long time-periods.

##### Periodic DNLS: weighted Wiener measures, gauge transformation and almost global well-posedness

##### Nonequilibrium: Thermostats, BBGKY Hierarchy, Fourier's Equation

Review of rigorous results on thermostats. Families of exact formal solutions of the BBGKY hierarchy for hard sphere systems with free boundary conditions at collisions and Fourier equation emergence, to first order in the temperature difference, after boundary conditions are imposed.

##### Aspects of stringy global quotients, de Rham, singularities and gerbes

We discuss stringy functors from the pull back point of view of Jarvis-K-Kimura and the push forward point of view of our orbifold Milnor ring constructions. We show how these approaches merge to give a de Rham theory and apply back to singularity theory. If time allows, we also give results on global gerbe twists and the Drinfel'd Double.

##### On the distribution of gaps for saddle connection directions

In joint work with J. Chaika, we prove results on the distribution of gaps of angles between saddle connections on flat surfaces. Our techniques draw on the work of Marklof-Strombergsson on the periodic Lorentz gas and that of Eskin-Masur on flat surfaces. We describe some applications to billiards in polygons.

##### Proving the Lovász-Plummer conjecture

In the 1970s, Lovász and Plummer conjectured that every cubic bridgeless graph has exponentially many perfect matchings with respect to the number of vertices. The conjecture was proven by Voorhoeve for bipartite graphs and by Chudnovsky and Seymour for planar graphs.

##### Cohomology of graph products of infinite groups with group ring coefficients

I will explain a computation of the cohomology of any graph product of infinite groups in terms of the factor groups. For example, this gives a calculation for right-angled Artin groups, which are, by definition, graph products of copies of the infinite cyclic group. The method of proof is a simple spectral sequence argument which I don't think has been used previously.