All groups under our consideration are finitely generated. Asymptotic cones (AC) of groups were
introduced by M.Gromov in 1981. He used them for the description of groups with polynomial
growth. AC of groups are homogeneous geodesic, metric spaces. There exists a group having
non-homeomorphic cones. All AC of G are R-trees iff the group G is hyperbolic. In a recent joint
paper with D.Osin and M.Sapir, we called a group G lacunary hyperbolic (LH) if at least one AC
of G is an R-tree. We characterize LH groups as direct limits of hyperbolic groups satisfying
certain restrictions on the hyperbolicity constants and injectivity radii. We show that the class
of LH groups is very large. Many group-theoretical couner-examples (E.G., some Tarski
monsters) are LH groups. Among new examples, we construct a group having an AC with
a non-trivial countable fundamental group. This solves Gromov's problem of 1993.

I'm going to explain the theorem of Hopkins and Miller (and partly Goerss) that gives a version of Lubin-Tate deformation theory in the context of algebraic topology. More concretely, the theorem says that there is a functor $(k,\Gamma) \mapsto E_{(k,\Gamma)}$ from formal groups laws over perfect fields of characteristic $p>0$ to a very nice category of commutative ring spectra, namely $E_{\infty}$-ring spectra. It has the property that the formal group law of the cohomology theory associated with the ring spectrum $E_{(k,\Gamma)}$ is the universal deformation of $(k,\Gamma)$. By functoriality, we obtain an action of the (extended) Morava stabilizer group on the spectrum $E_{(\mathbb{F}_{p^n},H_n)}$, where $H_n$ denotes the Honda formal group law of height $n$. One application is the construction of the higher real $K$-theories $EO_n$ as the homotopy fixed point spectra obtained from the action of finite index subgroups of the Morava stabilizer subgroup.

Markov models on trees arise naturally in many fields, notably in molecular
biology - as models of evolution; in statistical physics - as models of
spin systems; and in networking - as models of broadcasting. In this talk,
I will discuss various inference problems motivated especially by
applications in statistical phylogenetics, i.e. the reconstruction of
evolutionary histories of organisms from their molecular sequences. In
particular, I will consider the "root reconstruction" problem: how
accurately can one guess the value at the root of the tree, given the state
at the leaves? I will focus on recent work establishing new conditions for
the impossibility of such reconstruction. I will also discuss the related
"phylogenetic reconstruction" problem: given enough samples at the leaves,
can one reconstruct the tree that generated this data and, if so, how
efficiently? I will present a recent result on a sharp transition in the
number of samples required to recover the tree topology, using a connection
to the root reconstruction problem above. Time permitting, I will describe
briefly connections to computational learning theory and network tomography
as well. This is joint work with S. Bhamidi, C. Borgs, J. Chayes, C.
Daskalakis, E. Mossel, and R. Rajagopal.

In this talk, I will talk about an incredibly wonderful theorem of Charles
Fefferman about biholomorphisms of strictly pseudoconvex domains. I will
also talk about a certain kernel named after a famous movie actress. You
will find out what is a biholomorphic mapping, strictly pseudoconvex domain,
kernel, etc... I might also tell you some historical background, where I
will make up the bits that I don't actually know.

I will present some recent results on the Serrin-type conditional regularity of the Navier-Stokes equations. Basically, if one component of the weak solution of the Navier-Stokes equation belongs to a Serrin type space of regularity then the weak solution is regular and is unique. The second part of the talk is devoted to the primitive equations of the ocean with the Dirichlet boundary condition for which we prove the global regularity. This is a joint work with I. Kukavica.