In this talk I will show how understanding of the possible limiting measures of translations of a measure can help us to deal with certain counting problems. Then I talk about the limiting measures of translations of a horospherical measure. Finally I discuss how one can use this result to count the number rational points in a flag variety with respect to any line-bundle, reproving a result of Franke-Manin-Tschinkel (anti-canonial line-bundle) and Batyrev-Tschinkel (arbitrary line-bundle). (Joint with A. Mohammadi)

Must the Fourier series of an $L^2$ function converge pointwise almost everywhere? In the 1960's, Carleson answered this question in the affirmative, by studying a particular type of maximal singular integral operator, which has since become known as a Carleson operator. In the past 40 years, a number of important results have been proved for generalizations of the original Carleson operator. In this talk we will introduce the Carleson operator and survey several generalizations, and then describe new joint work with Po Lam Yung that introduces curved structure to the setting of Carleson operators.

Over the last hundred years, the circle method has become one of the
most important tools of analytic number theory. This talk (on joint work
with Roger Heath-Brown) will describe a new application of the circle
method to pairs of quadratic forms, via a novel two-dimensional analogue
of Kloosterman's version of the circle method. As a result, we prove
(under a mild geometric constraint) that any two quadratic forms with
integer coefficients, in 5 variables or more, simultaneously attain
prime values infinitely often.

I will begin with a friendly introduction to the deformation theory of
Galois representations and its role in modularity lifting, focusing on
the case of elliptic curves over Q. This will motivate the study of
local deformation rings and more specifically flat deformations. Next,
we will discuss Kisin’s resolution of the flat deformation ring at l = p
and describe conceptually the importance of local models of Shimura
varieties in analyzing its geometry. In the remaining time, we will
address the title of the talk; the additional structure we consider
could be a symplectic form, an orthogonal form, or more generally any
reductive subgroup G of $GL_N$. I will describe briefly the role that
recent advances in p-adic Hodge theory and local models of Shimura
varieties play in this situation.

An important open problem in quantum information concerns with the question whether entanglement between signal states can help in sending classical information over a quantum channel. Recently, Hastings proved that entanglement does help by finding a counter example for the long standing additivity conjecture that the minimum entropy output of a quantum channel is additive under taking tensor products. In this talk I will show that the minimum entropy output of a quantum channel is locally additive. Hastings' counter example for the global additivity conjecture makes this result somewhat surprising. In particular, it indicates that the non-additivity of the minimum entropy output is related to a global effect of quantum channels. I will end with a few related open problems.

We consider a classical continuum model which allows to describe essential electrokinetichenomena such as electro-phoresis and -osmosis. Applications and correspondingheory range from design of micro fluidic devices, energy storage devices,emiconductors to emulating communication in biological cells by synthetic nanopores.

Based on this classical formulation, we derive effective macroscopic equationshich describe binary symmetric electrolytes in porous media. Theeterogeneous materials naturally induce corrected transport parameters which weall "material tensors". A better understanding of the influence ofeterogeneous media on ionic transport is expected by the new formulation.he new equations provide also an essential computational advantage by reliablyeducing the degrees of freedom required to resolve the microstructure.

The presented results are gained by asymptotic multi-scale expansions.his formal procedure is then made rigorous by the derivation of error boundsetween the exact microscopic solution and the new upscaled macroscopic approximation.

A \emph{mixed graph} is a graph with directed edges, called
arcs, and undirected edges. A $k$-coloring of the vertices is
\emph{proper} if colors $1,2,\ldots,k$ are assigned to each vertex such
that vertices $u$ and $v$ have different colors if $uv$ is an edge and
the color of $u$ is less than or equal to (resp. strictly less than) the
color of $v$ if $uv$ is an arc. The \emph{weak (resp. strong) chromatic
polynomial} of a mixed graph is a counting function that counts the
number of proper $k$-colorings. This talk will discuss previous work on
reciprocity theorems for other types of chromatic polynomials, and our
reciprocity theorem for weak chromatic polynomials which uses partially
ordered sets and order polynomials. This is joint work with Matthias
Beck, Daniel Blado, Joseph Crawford, and Taina Jean-Louis.