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.

Talk Abstract:
In geophysics, multilayer models are derived under the assumption that
the fluid consists of a finite number of homogeneous layers of
distinct densities. We introduce a two-layer model that was derived to
study the perturbation about a vertical shear flow. We show that the
model is linearly unstable, however the solutions of the nonlinear
model are bounded in time. We prove the existence of finite
dimensional compact attractor and derive upper bounds on its
dimension.

In plasma physics, the 3D Hasegawa-Mima equation is one of the most
fundamental models that describe the electrostatic drift waves. In the
context of geophysical fluid dynamics, the 3D Hasegawa- Mima equation
appears as a simplified model of a reduced Rayleigh-Bénard convection
model that describes the motion of a fluid heated from below.
Investigating the 3D Hasegawa-Mima model is challenging even though
the equations look simpler than the 3D Euler equations. Inspired by
these models, we introduce and study a simplified mathematical model
that has a nicer mathematical structure. We prove the global existence
and uniqueness of solutions of the 3D simplified model as well as a
continuous dependence on the initial data result. These results are
one of the first results related to the 3D Hasegawa-Mima equation.

Basic theory of modified Prikry forcing will be presented. In particular, the proof of Prikry lemma for modified Prikry forcing will be discussed.