# Seminars & Events for 2010-2011

##### Nondistortedsubgroups of $Out(F_n)$, via Lipschitz retraction in spaces of trees (joint work with M. Handel)

We prove that various subgroups of $Out(F_n)$ — the outer automorphism group of a free group of rank $n$ — such as the stabilizer of the conjugacy class of a rank $n-1$ free factor, are undistorted in $Out(F_n)$.

##### Periods of quaternionic Shimura varieties

In the early 80's, Shimura made a precise conjecture relating Peterssoninner products of arithmetic automorphicforms on quaternion algebras over totally real fields, up to algebraic factors. This conjecture (which is a consequence of the Tate conjecture on algebraic cycles) was proved a few years later by Michael Harris.

##### Geometry of quantum response in open systems

I shall describe a theory of adiabatic response for controlled open systems governed by Lindblad evolutions. The theory gives quantum response a geometric interpretation induced from the geometry of Hilbert space projections. For a two level system the metric turns out to be the Fubini-Study metric and the symplectic form the adiabatic curvature.

##### An Aronsson type approach to extremal quasiconformal mappings

Quasiconformal mappings $u:\Omega\to \Omega'$ between open domains in $R^n$, are $ W^{1,n}$ homeomorphisms whose dilation $K=|du|/ (det du)^1/n$ is in $L^\infty$. A classical problem in geometric function theory consists in finding QC minimizers for the dilation within a given homotopy class or with prescribed boundary data. In a joint work with A.

##### Generalized Markov models in population genetics

Population geneticists study the dynamics of alternative genetic types in a replicating population. Most theoretical works rests on a simple Markov chain, called the Wright-Fisher model, to describe how an allele's frequency changes from one generation to the next.

##### Cluster Algebras and Quiver Grassmannians

A cluster algebra, which was introduced by Fomin and Zelevinsky, is a commutative algebra with a family of distinguished generators (the cluster variables) grouped into overlapping subsets (the clusters) which are constructed by mutations. A quiver Grassmannian is a projective variety parametrizing subrepresentations of a quiver representation with a given dimension vector.

##### The spectrum of non-normal random matrices

We will discuss the asymptotics of the spectrum of non-normal random matrices with size going to infinity, and in particular the single ring phenomenon observed for unitary invariant models.

##### Topological expansion for random matrices

Department Colloquium (joint with ORFE)

##### Linear systems modulo composites

Computation modulo composite numbers is much less understood than computation over primes; and sometimes surprisingly much more powerful. A positive example is that the best construction of locally decodable codes known today heavily relies on computations modulo composites.

##### Nonimmersions of real projective spaces and tmf

We use the spectrum tmf to obtain new nonimmersion results for many real projective spaces RP^n for n as small as 113. The only new ingredient is some new calculations of tmf-cohomology groups. We present an expanded table of nonimmersion results. We also present several questions about tmf.

##### Unlink detection and the Khovanov module

Kronheimer and Mrowka recently showed that Khovanov homology detects the unknot. Their proof does not obviously extend to show that Khovanov homology detects unlinks of more than one component, and one could reasonably question whether it actually does (the Jones polynomial, for instance, does not detect unlinks with multiple components).

##### Affine sieve and expanders

I will talk about the fundamental theorem of affine sieve (joint with Sarnak). The main black box in the proof of this result will be also explained. It is a theorem on a necessary and sufficient condition for a finitely generated subgroup of SL(n,Q) under which the Cayley graphs of such a group modulo square free integers form a family of expanders (joint with Varju).

##### Diffusion in a periodic Lorentz gas with narrow tunnels (P. Balint, N. Chernov, and D. Dolgopyat)

In a periodic Lorentz gas a particle moves bouncing off a regular array of fixed convex obstacles (scatterers), like in a pinball machine. When the horizon is finite, one observes a classical diffusion law. When the obstacles are so large that the tunnels between them become narrow (of width $\epsilon \to 0$) then the diffusion matrix scales with $\epsilon$.

##### CR moduli spaces on a contact 3-manifold

We study low-dimensional problems in topology and geometry via a study of contact and Cauchy-Riemann ($CR$) structures. In particular, we consider various $CR$ moduli spaces on a contact 3-manifold. A contact structure is called spherical if it admits a compatible spherical $CR$ structure.

##### Niebur Integrals and Mock Automorphic Forms

Among the bounty of brilliancies bequeathed to humanity by Srinivasa Ramanujan, the circle method and the notion of mock theta functions strike wonder and spark intrigue in number theorists fresh and seasoned alike. The former creation was honed to perfection for its original purpose of counting partitions by Hans Rademacher.

##### Brother, can you spare a compacton?

Unlike certain personal or national tragedies which may extend indefinitely, patterns observed in nature are of finite extent. Yet, as a rule, the solitary patterns predicted by almost all existing mathematical models extend indefinitely with their tails being a by product of their analytical nature.

##### Pseudo-Anosov flows in Seifert fibered and solvable 3-manifolds

We discuss the following rigidity results: 1) A pseudo-Anosov flow in a Seifert fibered manifold is up to finite covers topologically conjugate to a geodesic flow; 2) A pseudo-Anosov flow in a solv manifold is topologically conjugate to a suspension Anosov flow. The proofs use the structure of the fundamental groups in these manifolds and the topological theory of pseudo-Anosov flows.

##### Massey Triple Products

A technique will be discussed to control the indeterminacy in cohomology Massey triple products. A variety of non-vanishing and vanishing results for Massey triple products are proved using this technique. Here are three examples. Many authors have noticed that non-trivial triple products in a submanifold produce non-trivial triple products in the blowup along the submanifold.

##### Ergodic properties of infinite extensions of area-preserving flows

We consider infinite volume preserving flows that are obtained as extensions of flows on surfaces. Consider a smooth area-preserving flow on a surface $S$ given by a vector field $X$ and consider a real valued function $f$ on $X$.

##### Equivariant cohomology and orbit structure

Let T be a torus, and let X be a T-space. In this talk I will relate algebraic properties of the equivariant (co)homology of X to the structure of the T-orbits in X. This generalizes a result of Atiyah and Bredon.