For the density of the occupation time of general one-dimensional diffusion processes, continuity and its asymptotic behaviors at extremal points are studied. \\
(Based on a joint work with S. Watanabe and Y. Yano.)

A period of an automorphic form on a reductive group $G$ over anumber fieldis defined by its integral over a subgroup $H$ of $G$. Suchperiods are often related to special values of automorphic $L$-functions.In this talk, we present a conjecture in the case of special orthogonalgroups, which can be regarded as a refinement of the global Gross-Prasadconjecture about the restriction of automorphic representations of $SO(n+1$) to $SO(n)$. If time permits, we also discuss a relation of ourconjecture to Arthur's conjecture on the multiplicity of representationsin the space of automorphic forms. This is a joint work with Tamotsu Ikeda.

Ramanujan's legacy to mathematics is well documented with connections to some of the deepest subjects in modern number theory: Deligne's proof of the Weil Conjectures, the proof of Fermat's Last Theorem, the birth of probabilistic number theory, the introduction of the "circle method" among others. Although most of Ramanujan's mathematics is now well understood, one baffling enigma remained. In his last letter to Hardy (written on his death bed), Ramanujan gave 17 examples of functions he referred to as "mock $\theta$ functions". Over the next fifty years, ad hoc works by many number theorists (such as Andrews, Atkin, Cohen, Dyson, Selberg, Swinnerton-Dyer, Watson...) clearly pointed to the importance of these strange functions. Their work motivated Freeman Dyson to proclaim:

"Mock $\theta$-functions give us tantalizing hints of a grand synthesis still to be discovered. Somehow it should be possible to build them into a coherent group-theoretical structure ... This remains a challenge for the future." -- Freeman Dyson, 1987

Over the last year, Kathrin Bringmann and I have written a series of three papers on this enigma. Extending recent work of Zwegers, we solve Dyson's challenge in terms of harmonic Maass forms. We fully developed the arithmetic (i.e. p-adic and Galois theoretic) and analytic properties of all such forms, and we have applied these results to solve open problems in additive number theory.

We will see how a problem in genome rearrangement leads to describe a new
kind of phase transition for random walks on graphs. This phase transition
is related to the well-known Erdos-Renyi double jump phenomenon for random
graphs. I will particularly try to describe the effect that the scale of
mutations may have on the analysis of the problem, and will outline two
possible approaches: one borrowing ideas from hyperbolic geometry and the
other based on cheating and using branching random walks.

We are motivated by a question arising in population genetics, and
try to describe the effect of migratory fluxes and spatial
structure on the genealogy of a population. This leads to the
study of systems of particles performing simple random walk on a
given graph, and where particles coalescence according to a
certain mechanism (typically, Kingman's coalescent) when they are
on the same site. We obtain various asymptotic results for this
process, at both small and large time scales, which are of
intrerest to population genetics. We will also discuss some
related conjectures.

Let $p$ be a prime number, $K$ a $CM$ field containing a primitive $p^(n)th$ root of unity and k the maximal real subfield of $K$. We show that
a special case of a conjecture of Solomon gives rise to a (conjectural)
congruence mod $p^n$, relating certain $S$-units of $k$ to the principal
semi-local units of $K$, via the use of local Hilbert symbols.
We will discuss the progress made in proving this conjecture and (briefly) some computational verifications.

Covering a graph with subgraphs of certain type has a long history in graph theory and it goes back to Boole, Ore and M.Hall. The problem of covering a graph by cliques or bicliques has been studied by many researchers including Erdos, Goodman, Posa, Chung, Graham, Pollak, Alon etc. Few of the results obtained for graphs extend to hypergraphs. In this talk, I will present some of these results and show some new results
regarding the covers of an $r$-uniform hypergraph by $k$-cuts such that the total size of the cuts (the sum of the number of edges of
all cuts) is minimum. This is joint work with Andr$\rm\acute e$ K$\rm\ddot u$ndgen
(Cal.State San Marcos).

I will discuss three classes examples in which the geometry of an
algebraic variety illuminates a certain combinatorial object. The first
example will involve the classical relationship between toric varieties and
polytopes. The next two examples will deal with a relatively new class of
spaces called hypertoric varieties. These may be thought of as
quaternionifications of toric varieties, and they interact richly with the
combinatorics of matroids, or of finite collections of hyperplanes in a
vector space.