Single particle electron microscopy is a powerful method that biophysicists employ to learn
about the structure of biological macromolecules. In contrast to the more traditional crystallographic
methods, this method images "unconstrained" particles, thus posing a variety of statistical problems. We
formulate and study such a problem, one that is essentially of a random tomographic nature. Although
unidentifiable (ill-posed), this problem can be seen to be amenable to a statistical solution, once strict
parametric assumptions are imposed.It can also be seen to present challenges from a data analysis point of
view (e.g. uncertainty estimation and presentation). In addition, motivated by the "physics" involved in
the data-collection process, we define and explore a new type of diffusions in D.G. Kendall's shape space.
These diffusions arise as the stochastic evolution of the orbits (under a group action) of a projected
motion.

Recently, a lot of papers have been devoted to the analysis of lamplighter random walks, in particular when the underlying graph is the infinite path $\mathbb{Z}$. In the present talk, we develop a spectral analysis for lamplighter random walks on finite graphs. In the general case, we use the $C_2$-symmetry to reduce the spectral computations to a series of eigenvalue problems on the underlying graph.
In the case the graph has a transitive isometry group $G$, we also describe the spectral analysis in terms of the representation theory of the wreath product $C_2\wr G$. We apply our theory to the lamplighter random walks on the complete graph and on the discrete circle. These examples were already studied by Haggstrom and Jonasson by probabilistic methods.

This talk includes three parts. In Part I, I will briefly
introduce a number of imaging techniques for 3D structure
reconstructions at different scales (including tissue, cellular,
subnanometer, and atomic levels). In Part II, a couple of image
processing algorithms will be presented, in order to extract features
of interest for geometric modeling. Part III is mesh generation, an
essential step for finite element method. I will introduce two
approaches for mesh generation: Delaunay triangulation and
Octree-based method.

A graph $G=(V,E)$ is representable if there exists a word $W$ over
the alphabet $V$ such that letters $x$ and $y$ alternate in $W$ if
and only if $(x,y) \in E$ for each $x \neq y$. If $W$ is $k$-uniform
(each letter of $W$ occurs exactly $k$ times in it) then $G$ is
called $k$-representable.

The notion of representable graphs appears in study, by Seif and
later by Kitaev and Seif, of the Perkins semigroup which has played
central role in semigroup theory since 1960. Thus, the objects in
question being on border between combinatorics on words and graph
theory, come from needs in algebra.

In my talk, I will discuss some properties of representable graphs.
In particular, I will give examples of non-representable graphs and
will discuss certain wide classes of graphs that are 2- and
3-representable. I will conclude with a number of open problems in
this direction of research.

This is a joint work with Artem Pyatkin.

The $P-$Eulerian polynomial $W_P(x)$ of a finite naturally
labelled poset $P$ is the generating function for linear extensions of $P$ by
number of descents. We will discuss some recent work related to the
question of whether the coefficients of $W_P(x)$ are unimodal and
whether $W_P(x)$ has real zeros. In particular we will explain the
beautiful work of Branden on sign-graded posets.

Many random media models involve random variables attached to sites in a state space
such as $Z^d$ or $ R^d$. In these models, the sums of values of these random variables
(or integrals) along paths through the state space are are of great interest. In many cases,
the supremum or infimum of these sums are physically relevant. Examples of this are
first passage percolation and the parabolic Anderson model.
The growth of these extrema are generally linear in the path length and satisfy a law of
large numbers. In this talk we examine deviations above the mean and below the mean for
a variety of models and show they are generally very unbalanced.

This talk is an introduction to analytic number theory via two very classical results. We will start with a discussion of the Riemann zeta function, including a quick proof of the functional equation and some discussion of the location of the zeros. Then we will go through Dirichlet's proof that there are infinitely many primes congruent to a modulo $q$ when $(a, q) = 1$.

To keep the talk at a survey level, some results will be stated without proof. (Translation: I'm going to avoid long, techincal calculations as much as possible.)

Based on the recent progress in the Langlands functoriality
for classical groups, we establish the relation between the central
value of the Rankin-Selberg L-functions for GL(n) x GL(m) and certain
periods of automorphic forms on classical groups. Such a relation is
a natural generalization of Gross-Prasad conjecture made about 10 years ago. The lecture is based on the joint work with Ginzburg and Rallis.

The Langlands functoriality conjecture is one of the basic problems in the theory of automorphic forms and representations. In the past four decades, there was essentially no progress made until a few years ago when the Langlands functoriality was established for generic cuspidal automorphic representations from classical groups
to general linear groups in a series of papers by Cogdell$-$Kim$-$Piatetski$-$Shapiro$-$Shahidi, Ginzburg$-$Rallis$-$Soudry,
and Jiang-Soudry. Many important applications were found afterwards. The problem of establishing the theory for general cuspidal automorphic representations remains a big open problem. In this talk, I will report on some recent progress on this problem, based on my work and my work with Soudry