It is a long-standing problem whether the Jones polynomial detects
the unknot, and it has been known that the Jones polynomial does not
detect unlinks. In the knot homology world, Kronheimer and Mrowka
proved that Khovanov homology, the categorification of Jones
polynomial, detects the unknot. On the other hand, the question
whether Khovanov homology detects unlinks remains open. In this talk,
we will show that Khovanov homology with an additional natural module
structure detects unlinks. This is joint work with Matt Hedden.

It is hard to find analogues of MA in which aleph_1 is replaced by the successor of a singular cardinal because
a) The consequences of MA-like axioms have large consistency strength
b) There is no satisfactory analogue of finite support ccc iteration

Dzamonja and Shelah found an ingenious approach to proving results of this general kind. I will outline their work and then describe some recent joint work with Dzamonja and Morgan, aimed at bringing results of this kind down to aleph_{omega+1}

In the early 1970s Drinfeld introduced a family of rigid analytic
spaces parameterizing deformations of certain formal groups with level
structure. This family is called the Lubin-Tate tower. He found an
open affinoid in the first level of this tower whose reduction is
isomorphic to what is now known as a Deligne-Lusztig variety for GL_n
over a finite field. This established a link between depth 0
supercuspidal representations of GL_n(K) (where K is a local field)
and cuspidal representations of GL_n(F_q) (where F_q is the residue
field of K). I will explain a similar construction at a higher level
of the tower, which leads to an analogue of the Deligne-Lusztig theory
for a class of unipotent groups over finite fields. This approach
yields a geometric construction of explicit local Langlands
correspondence for a certain class of positive depth supercuspidal
representations of GL_n(K). The talk is based on joint work with Jared
Weinstein (Boston University). A large portion of the talk will be
very elementary and will require no background apart from some
familiarity with algebraic groups over finite fields.

Let $C$ be a curve over a local field whose reduction is totally
degenerate. We discuss the related problems of 1) determining the
group structure of the torsion subgroup of the Jacobian of $C$, and 2)
determining if a given line bundle on $C$ is divisible by a given
integer $r$. Under certain hypotheses on the reduction of $C$, we
exhibit explicit algorithms for answering these two questions.

In this talk, we will introduce some advances on non-analytic quasi-periodic cocycles, including Schrodinger and non-Schrodinger situations. Moreover, we will discuss some conjectures in this area.