fusion categories are generalizations of the representation categories of finite groups. one source of new fusion categories are subfactors, inlusions of von neumann algebras with trivial center. the search for exotic subfactors led to new interesting fusion categories. one can study chiral conformal field theory via so-called conformal nets. i will explain how conformal nets give rise to fusion categories via its (higher) representation theory. it is an open question if all unitary fusion categories come from conformal nets. i will give examples of families of fusion categories for which one can reconstruct a conformal net.

in this talk, we consider a class of orthogonal constrained optimization problems, the feasible region of which is called the stiefel manifold. our new proposed framework combines a function value reduction step with a multiplier correction step. different with the existing approaches, the function value reduction is conducted in the euclidean space instead of the stiefel manifold or its tangent space. we construct two types of algorithms based on! this new framework. the first type is gradient reduction based algorithms which consists of gradient reflection (gr) and gradient projection (gp) two implementations. the other one adopts a column-wise block coordinate descent (cbcd) scheme with a novel idea for solving the corresponding cbcd subproblem inexactly. theoretically, we can prove that both gr/gp with a fixed stepsize and cbcd belong to our framework, and any clustering point of the iterates generated by the proposed framework is a first-order stationary point. preliminary experiments illustrate that our new framework is of great potential.

we construct a pair of related diffusions on a space of partitions of the unit interval whose stationary distributions are the complements of the zero sets of brownian motion and brownian bridge respectively. our methods can be extended to construct a class of partition-valued diffusions obtained by decorating the jumps of a spectrally positive levy process with independent squared bessel excursions. the processes of ranked interval lengths of our partition-valued diffusions are members of a two parameter family of infinitely many neutral allele diffusion models introduced by ethier and kurtz (1981) and petrov (2009). our construction is a step towards describing a diffusion on the space of real trees, stationary with respect to the law of the brownian crt, whose existence has been conjectured by aldous. based on joint work with n. forman, s. pal, and m. winkel.

the riemann mapping theorem (rmt) is a staple in complex analysis in one variable: {\it any simply connected domain in the plane (other than the plane itself) is biholomorphically equivalent to the unit disk.} a direct analog is not true in two dimensions and higher. as discovered by poincar\'e, the unit ball in $c^2$ is not biholomorphic to the bidisk. the reason is that in higher dimensions the boundary of a domain inherits a non-trivial structure---a cr structure--- from the ambient complex structure. we will discuss how one can formulate a version of the rtm that holds in higher dimensions as well. after this introduction, we shall mention some current fundamental problem in this area.

the l-adic cohomology of rapoport-zink spaces is expected to realize local langlands correspondences in many cases. along this line is a conjecture by harris, which roughly says that when the underlying rapoport-zink space is not basic, the l-adic cohomology of the space is parabolically induced. in this talk, we will discuss a result on this conjecture when the rapoport-zink space is of hodge type and ``hodge-newton reducible''. the main strategy is to embed our rapoport-zink space to an appropriate space of el type, for which the conjecture is already known to hold. if time permits, we will also discuss other applications of this strategy.

this is not a mathematics talk but it is a talk for mathematicians. too often, we think of historical mathematicians as only names assigned to theorems. with vignettes and anecdotes, i'll convince you they were also human beings and that, as the chinese say, ``may you live in interesting times'' really is a curse.

let $f: c' -> c$ be a cyclic cover of smooth projective curves.
its prym variety is by definition the complement of the pullback of the
jacobian of $c$ in the jacobian of $c'$. it is an abelian variety with a
polarization depending on the genus of $c$, the degree of $f$ and the
ramification type of the covering $f$. this gives a map from the moduli
space of coverings of this type into the moduli space of abelian varieties
of the corresponding type with endomorphism structure induced by the
automorphism given by $f$, called prym map. in many cases the prym map is
generically injective. particularly interesting are the cases where the
prym map is finite and dominant. in this talk these cases will be worked
out for covers of degree a prime number and twice an odd prime. in some
cases the degree of the prym map is determined. this is joint work with
angela ortega.