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.

The almost Matthieu operator arises as a model for Bloch electrons in a magnetic field. Aubry and Andre famously made a conjecture about the spectral properties of this operator more than thirty years ago. In the process of its study, variations of the conjecture arose naturally. Although the original conjecture was recently settled, new problems remain unsettled. We discuss some of these open problems and possible methods (and their shortcomings) to their solution.

We investigate a series of related problems in the area of incomplete Weil sums where the sum is run over a set of points that produces the image of the polynomial. We establish a bound for such sums, and establish some numerical evidence for a conjecture that this sum can be bounded in a way similar to Weil's bounding theorem.

To aide in the average case, we investigate the problem of the cardinality of the value set of a positive degree polynomial (degree $d > 0$) over a finite field with $p^m$ elements. We show a connection between this cardinality and the number of points on a family of varieties in affine space. We couple this with Lauder and Wan's $p$-adic point counting algorithm, resulting in a non-trivial algorithm for calculating this cardinality in the instance that $p$ is sufficiently small.

This study is important in understanding the mechanism and dynamics of some biological problems such as tumor invasion and wound healing. Firstly, we describe microscopically the model and we derive the corresponding mesoscopic approximation, via the mean field assumption. In the following, we upscale our model providing a PDE which serves as
a macroscopic manifestation of the underlying cellular interactions. We focus on investigating the speed and the structure of the invasion
front, using the above mentioned approximations, as functions of the underling cell phenotypes and microenvironmental factors (i.e. nutrients).