The classic Linear Preserver Problem asks to determine, for a polynomial function f on a vector space V, the linear transformations g of V such that fg = f. In case f is invariant under a simple algebraic group G acting irreducibly on V, we prove that the subgroup of GL(V) stabilizing f often has identity component G and we give applications realizing various groups, including the largest exceptional group E8, as automorphism groups of polynomials and algebras. We show that starting with a simple group G and an irreducible representation V, one can almost always find an f whose stabilizer has identity component G and that no such f exists in the short list of excluded cases. The main results are new even in the special case where the field is the complex numbers, and have implications for Hasse principles for polynomials over number fields.  This talk is about joint work with Bob Guralnick. 

Up until the mid 70s the kind of spectra most people had in mind in the
context of theory of Schrodinger operators were spectra occurring for
periodic potentials and for atomic and molecular Hamiltonians. Then
evidence started to build up that "exotic" spectral phenomena such as
singular continuous, Cantor, and dense point spectrum do occur in
mathematical models that are of substantial interest to theoretical
physics. One area where such exotic phenomena are particularly abundant is
quasiperiodic operators. They feature a competition between
randomness (ergodicity) and order (periodicity), which is often resolved
at a deep arithmetic level. Mathematically, the methods involved include a
mixture of ergodic theory, dynamical systems, probability, functional and
harmonic analysis and analytic number theory. The interest in those models was enhanced by strong
connections with some major discoveries in physics, such as integer
quantum Hall effect, experimental quasicrystals, and quantum chaos theory,
in all of which quasiperiodic operators provide central or important
models.

We will give a general overview concentrating on aspects where the
competition and/or collaboration between order and chaos plays an
important role

We will talk about a paper by A. Moreira and J.C. Yoccoz, where they proved a conjecture by Palis according to which the arithmetic sums of generic pairs of regular Cantor sets on the line either has zero Lebesgue measure or contains an interval.

For a sparse polynomial f(x) of high degree and few terms
over a non-algebraically closed field F, the number of F-rational
roots is often very small. In the case F is the real numbers, this
is the famous Descartes's rule. In the case that F is a finite field,
the situation is much more complicated. In this lecture, we discuss
some recent results and conjectures in this direction, both
theoretical and numerical.

Abstract: The dynamics of complex biological systems is often driven by multiscale, rare but important events. In this talk, I will first introduce the numerical methods for computing transition states, in particular, the Optimization-based Shrinking Dimer (OSD) method we recently proposed. Then I will give two applications of rare events and transition states in biology: boundary sharpening in zebrafish hindbrain and neuroblast delamination in Drosophila. The joint work with Qiang Du (Columbia), Qing Nie (UC Irvine), Yan Yan (HKUST).