One of the most difficult problems in mathematics and physics is to
find an accurate, practical description of turbulent flows. Turbulence
is ubiquitous in nature, occurring in very diverse physical settings,
such as aerodynamics, geophysics, weather and climate modeling, ocean
and atmospheric flows, star formation, blood flow in the heart, and
many others. This problem is not only untenable by current
mathematical tools, but direct numerical simulation of detailed
turbulent flows has proven to be computationally prohibitive, even
using the most powerful state-of-the-art computers. A major piece of
the puzzle of understanding these phenomena is widely believed to lie
in a system of nonlinear PDEs known as the Navier-Stokes equations,
which are the subject of one of the seven $1,000,000 Clay Millennium
Prize problems. I will discuss give an introduction to the
Navier-Stokes equations and discuss their relationship to turbulence
and the Millennium problem.

We consider three constitutive relations to describe tumor growth: Darcys law, Stokes law, and the combined Darcy-Stokes law. Darcys law is used to describe fluid flow in a porous medium. Stokes law describes the flow of a viscous fluid. In this talk, we will discuss using linear theories to study tumor shape stability (the ability of the tumor to return to being spherical or exhibit protrusions) described by the three physical relations and to evaluate the consistency between theoretical model predictions and experimental data. The motivation behind this work is that shape instabilities (growing protrusions) are associated with local invasiveness, which is often a precursor to tumor metastasis (infiltration of the distant organs). We will discuss the results and further show that it is feasible to extract parameter values from a limited set of data and create a self-consistent modeling framework that can be extended to the multiscale study of cancer. Numerical methods are used to simulate the nonlinear effects of stress on solid tumor growth and invasiveness.

Extended Harper's model arises in a quantum description of a 2d- crystal layer subjected to an external magnetic field. As a first step towards the spectral analysis we shall introduce the Lyapunov exponent and present a
method of computation valid for any analytic cocycle with possible singularities. This enables us to give a description of the metal-insulator properties for extended Harper's model, which so far did not even exist on a heuristic level in physics literature. We finish the
talk with some results on the spectral analysis of the model.