Finite-time Lyapunov exponents and vectors are used to define and diagnose boundary-layer type, two-timescale behavior and to determine the associated manifold structure in the flow. Two-timescale behavior is characterized by a slow-fast splitting of the tangent bundle for a state space region. The slow-fast splitting, defined by finite-time Lyapunov exponents and vectors, is interpreted in relation to the asymptotic theory of partially hyperbolic sets. The finite-time Lyapunov approach relies more heavily on the Lyapunov vectors due to their relatively fast convergence compared to that of the corresponding exponents. Examples of determining slow manifolds and solving Hamiltonian boundary-value problems associated with optimal control are described.

The aim of this talk is to introduce the subject of spin glasses,
and more generally the statistical mechanics of quenched disorder,
as a problem of general interest to physicists from multiple disciplines and
backgrounds. Despite years of study, the physics of quenched
disorder remains poorly understood, and represents a major gap in our
understanding of the condensed state of matter. While there are many
active areas of investigation in this field, I will narrow the focus of this
talk to our current level of understanding of the low-temperature
equilibrium structure of
realistic (i.e., finite-dimensional) spin glasses.

I will begin with a brief survey of why the subject is of interest not only
to physicists,
but also mathematicians, computer scientists, and scientists working in
other areas. A brief review of the basic features of spin glasses and what
is
known experimentally will follow. I will then turn to the problem of
understanding the nature of the spin glass phase --- if it exists.
The central question to be addressed is the nature of broken symmetry in
these systems. Parisi's replica symmetry breaking approach,
now mostly verified for mean field spin glasses, attracted great excitement
and interest as a novel and exotic form of symmetry breaking. But does it
hold also for real spin glasses in finite dimensions? This has been a
subject of intense controversy, and although the issues surrounding it have
become more sharply defined
in recent years, it remains an open question. I will explore this problem,
introducing new mathematical constructs such as the metastate along the way.
The talk will conclude with an examination of how and in which respects the
statistical mechanics of disordered systems might differ from that of
homogeneous systems.

In this talk, I will talk about one approach to study the Diophantine equation f(x)=g(y), which combines the tools from Galois theory, algebraic geometry and group theory.
In particular, I will explain how the methods are used in the joint work with Mike Zieve on the equation ax^m+bx^n+c=dy^p+ey^q.

The ideas and methods above are also used in a recent theorem
of Carney-Hortsch-Zieve, which says that for any polynomial f(x) in Q[x], the map f: Q -> Q, a -> f(a) is at most 6-to-1 off a finite subset of Q. I will state a much more general conjecture on uniform boundedness of rational preimages of rational functions on number fields, of which a quite special case implies the theorems of Mazur and Merel on rational torsion points of elliptic curves.