The moduli space of abelian differentials carries a natural Lebesgue measure class, and, by the Theorem of H.Masur and W.Veech, the
Teichmueller flow on the moduli space of abelian differentials
preserves a finite ergodic measure in the Lebesgue measure
class. The entropy of the flow with respect to the absolutely
continuous measure has been computed by Veech in 1986.

The main result of this talk, obtained by B.M. Gurevich and the
speaker, is that the absolutely continuous measure is the
unique measure of maximal entropy for the Teichmueller flow.

The first step of the proof is an observation that the absolutely
continuous measure has the Margulis property of uniform expansion
on unstable leaves. After that, the argument proceeds in Veech's
space of zippered rectangles. The flow is represented as a
symbolic flow over a countable topological Bernoulli chain
and with a Hoelder roof function depending only on the future.
Following the method of Gurevich, the flow is then approximated
by a sequence of flows whose suspension functions depend on only
one coordinate in the sequence space. For these, conditions for
existence and uniqueness of the measure of maximal entropy are
known by theorems of Gurevich and Savchenko. Since the roof function
of our initial flow is Hoelder, the approximation is rapid enough
and yields maximality of entropy for the smooth measure as well
as the uniqueness of the measure of maximal entropy.

We investigate the system of partial differential equations used in
resistive magnetohydrodynamic modeling of fusion plasmas. This system
couples the Euler and Maxwell equations for evolution of a charged
fluid
in an electromagnetic field, hence the magnetic field in the resulting
PDE system must evolve on a divergence-free constraint manifold. As
traditional numerical solution approaches often violate these
constraints, we investigate a reformulation of the resistive MHD
system
to allow for accurate evolution of the continuum-level equations,
while
simultaneously ensuring that the solution satisfies the solenoidal
constraint.

Convergence of excursion measures is discussed by means of time change of the Brownian excursion measure. As an application an invariance principle of meanders of positive recurrent diffusion process is obtained. (Based on a joint work with P. J. Fitzsimmons.)

String topology (defined by Chas and Sullivan) is the study of
the topology of the space of loops (or strings) in a manifold. Chas and
Sullivan's work, as well as recent work by Cohen, Jones, Godin, and others
focuses on defining various algebraic operations on the space of loops (or
its homology). One can phrase many of these constructions in the language
of field theories used by physicists (though our approach will be purely
mathematical). I'll give an introduction to these sort of field theories
from the point of view of algebraic topology, and explain how various
flavors of string topology fit into this framework.

We give a geometric analog of the Weil representation of the metaplectic
group, placing it in the framework of the geometric Langlands program.
For a smooth projective curve $X$ we introduce an algebraic stack
Bun$\backsim$G of metaplectic bundles on $X$. We give a
tannakian description of the Langlands dual to the metaplectic group.
Namely, we introduce a geometric version Sph of the (nonramified)
Hecke algebra of the metaplectic group and describe it as a tensor
category. The tensor category Sph acts on the derived category
D(Bun$\backsim$G) by Hecke operators.
Further, we construct a perverse sheaf on Bun$\backsim$G corresponding
to the Weil representation and show that it is a Hecke eigensheaf with
respect to Sph.

The classical Langlands correspondence relates representations
of a reductive algebraic group over a local non-archimedian field $F$ and
representations of the Galois group of $F$. If we replace $F$ by the field
$C((t))$ of complex Laurent power series, then the corresponding group
becomes the (formal) loop group. It is natural to ask: is there an
analogue of the Langlands correspondence in this case? It turns out that
the answer is affirmative, and there is an interesting theory which may be
viewed as both "geometrization" and "categorification" of the classical
theory. I will explain the general set-up for this new theory and give
some examples using representations of affine Kac-Moody algebras.

The central topic of the talk is the class of Markov processes associated with the random walk on the group of the affine
transformations of the real line. The corresponding generators L have a form of the functional-differential operators with
the linearly transformed argument. These or similar operators were studied by Poincare, Birkhoff, Kato and other classics from the pure analytical point of view. We will present the complete analysis of the bounded L-harmonic functions (i.e. Martin
boundary of our Markov processes).

This talk will explain one of the basic building blocks of cryptography, cryptographic hash functions, and
relate them to another beautiful mathematical object: expander graphs.

While he was studying partial differential equations, Sophus Lie came up with the idea of trying to solve them by using their symmetry group. His idea was to apply Galois Theory to differential equations instead of polynomials. Lie's key observation was that these symmetry groups are locally determined by their Lie algebras.

Normally Lie groups of differential equations are only locally defined , i.e. they are only defined in a neighborhood of the identity element. However if we enlarge the manifold where the group is acting we can find a globally defined group action whose restriction to the original manifold is the original action.

In this talk we will calculate the symmetry group of the line equation $y''=0$ and see that, despite the simplicity of this equation, the symmetry group is not globally defined! However, the action can be enlarged to a well defined action on $RP^2$. We will do the same with Maxwell's equations obtaining, in this way, a conformal model of the universe where the symmetry group of Maxwell's equations is well defined.

We will discuss the non-vanishing problem of global theta
lifts from even orthogonal groups to symplectic groups, especially
focusing on the first occurrence conjecture of the global theta lifts.

As with many areas of topology and geometry, a starting point in the study of 3-manifolds is to try to understand codimension one objects in them, namely embedded surfaces. A particularly useful class of surfaces are the "incompressible" ones which are topologically essential; a 3-manifold containing such a surface is called a Haken manifold. There are many 3-manifolds which are not Haken, but if we ask about immersed, rather than embedded, surfaces the situation becomes much more mysterious. A closely related question is this: Suppose M is a 3-manifold with infinite fundamental group, does M have a finite cover which is Haken? The Virtual Haken Conjecture posits that the answer to this question is yes.

This talk will survey some recent results in this area, focusing on my work with (variously) William Thurston, Dylan Thurston, and Frank Calegari. From the point of view of Thurston's Geometrization Conjecture, this is really a question about hyperbolic 3-manifolds, that is, lattices in PSL(2, C). This opens the door to a rich array of tools that might seem quite surprising in light of the purely topological description of the problem above. Indeed, some unlikely-sounding terms that I will probably mention in my talk are "the Classification of Finite Simple Groups" and "the Langlands Conjecture", as well as such topological oddities as "random 3-manifolds"!