The sequencing of the human genome and the development of new methods for acquiring biological data have changed the face of biomedical research. The use of mathematical and computational approaches is taking advantage of the availability of these data to develop new methods with the promise of improved understanding and treatment of disease.

I will describe some of these approaches as well as our recent work on a Bayesian method for integrating high-level clinical and genomic features to stratify pediatric brain tumor patients into groups with high and low risk of relapse after treatment. The approach provides a more comprehensive, accurate, and biologically interpretable model than the currently used clinical schema, and highlights possible future drug targets.

How to extract trend from highly nonlinear and nonstationary data is an important problem that has many practical applications ranging from bio-medical signal analysis to econometrics, finance, and geophysical fluid dynamics. We review some exisiting methodologies in defining trend and instantaneous frequency in data analysis. Many of these methods use pre-determined basis and is not completely adaptive. They tend to introduce artificial harmonics in the decomposion of the data. Various attempts to preserve the temportal locality property of the data introduce problems of their own. Here we discuss how adaptive data analysis can be formulated as a nonlinear optimization problem in which we look for a sparse representation of data in some unknown basis which is derived from the physical data. We will show that this formulation has some beautiful mathematical structure and can be considered as a nonlinear version of compressed sensing.

A dynamical system is chaotic if its behavior is sensitive to a change in the initial data. This is usually associated with instability of trajectories. The hyperbolic theory of dynamical systems provides a mathematical foundation for the paradigm that is widely known as "deterministic chaos" -- the appearance of irregular chaotic motions in purely deterministic dynamical systems. This phenomenon is considered as one of the most fundamental discoveries in the theory of dynamical systems in the second part of the last century. The hyperbolic behavior can be interpreted in various ways and the weakest one is associated with dynamical systems with nonzero Lyapunov exponents.

I will describe main types of hyperbolicity and the still-open problem of whether dynamical systems with nonzero Lyapunov exponents are "typical" in a sense. I will outline some recent results in this direction and relations between this problem and two other important problems in dynamics: whether systems with nonzero Lyapunov exponents exist on any phase space and whether nonzero exponents can coexist with zero exponents in a robust way.

Variational principles are at the core of the formulation of mechanical problems. What happens in the presence of symmetry when variables can be eliminated? I will discuss the geometry underlying this reduction process and present the induced constrained variational principle and the associated Euler-Lagrange equations. The rigid body and the Euler equations for ideal fluids are examples of such reduced Euler-Lagrange equations in convective and spatial representations, respectively. This geometric structure permits the introduction of a new class of optimal control problems that have the remarkable property that the control satisfies precisely these reduced Euler-Lagrange equations. As an example, it is shown that geodesic motion for the normal metric can be controlled by geodesics on the symmetry group. In the case of fluids, these optimal control problems yield the classical Clebsch variables and singular solutions for the Camassa-Holm equation. Relaxing the constraint to a quadratic penalty yields associated optimization problems. Time permitting, the equations of metamorphosis dynamics in imaging will be deduced from this optimization problem.

We shall revisit the Boltzmann equation for rarefied non-linear particle dynamics, of conservative or dissipative nature, and on the stochastic N-particle model, introduced by M. Kac.
Related to this equation, we consider a a probabilistic dynamics from generalizations to N-particle model which includes multi-particle interactions. From basic symmetries and invariances for a general class of stochastic interactions, we show existence and uniqueness of states and recover the longtime dynamics and decay rates approaching stable laws characterized by self-similar rescaling, with finite or infinity energy initial data. We classify the moments integrability and see that broad tails (Pareto type) attractors are possible.

There is a large class of applications to these models including classical elastic or inelastic Maxwell type interactions with or without a thermostat, and social dynamics such as information percolation models, or wealth distributions models with Pareto tail formation.

This is work in collaboration with A. Bobylev, C. Cercignani and H. Tharkabhushanam.