For a closed Riemann surface X and complex reductive Lie
group G, the moduli space of G-Higgs bundles on X
is a hyperkaehler algebraic completely integrable system
that plays an important role in moduli space theory,
representations of surface groups, and supersymmetric gauge
theories.  The uniformization of X and the choice of a principal SL2 in G
give rise to a distinguished point in the moduli space called the
Fuchsian point.  In this talk I will discuss the first order
behavior of certain geometric and dynamical quantities at the
Fuchsian point. These may be regarded as "higher" analogs of
results in Teichmueller theory and for complex projective
structures.  This is joint work with Francois Labourie.

We develop new results on global existence and convergence
of solutions to the gradient flow equation for the Yang-Mills energy
functional on a principal bundle, with compact Lie structure group, over
a closed, four-dimensional, Riemannian, smooth manifold, including the
following. If the initial connection is close enough to a minimum of the
Yang-Mills energy functional, in a norm or energy sense, then the
Yang-Mills gradient flow exists for all time and converges to a
Yang-Mills connection. If the initial connection is allowed to have
arbitrary energy but we restrict to the setting of a Hermitian vector
bundle over a compact, complex, Hermitian (but not necessarily Kaehler)
surface and the initial connection has curvature of type (1,1), then the
Yang-Mills gradient flow exists for all time, though bubble
singularities may (and in certain cases must) occur in the limit as time
tends to infinity. The Lojasiewicz-Simon gradient inequality plays a crucial role in our approach and we develop two versions of that inequality for the
Yang-Mills energy functional.

A real number x is called diophantine if its distance to rationals p/q is large relative to q -- more precisely, if for every d > 0 there is a positive C such that for every reduced rational p/q, we have |x - p/q| > Cq^{-2-d}, or equivalently |qx-p| > Cq^{-1-d}. Almost all reals have this property. Furthermore, almost every pair (x_1, x_2) has the property that for every d > 0 there is a C such that |q_1x_1+q_2x_2 -p| > C||q||^{-2(1+d)} for all p, q_1, q_2. In this talk, we discuss a noncommutative analog of this property for elements of SO(3). Namely, a pair (A,B) is called diophantine if there exists a constant D such that for every positive integer n and every reduced word W of length n in A, B, A^{-1}, B^{-1}, we have ||W - E|| > D^{-n}, where E is the identity matrix. It is conjectured that almost every such pair (in the sense of Haar measure) is diophantine. We will present a paper of Kaloshin and Rodnianski, in which the weaker bound D^{-n^2} is obtained.

Phyllotaxis, the arrangement of phylla (leaves, bracts, seeds) near the shoot apical meristems of plants has intrigued and mystified natural scientists for over two thousand years. It is surprising that only within the last two decades have quantitative explanations emerged that describe the wonderful architectures which are observed. I will give an overview of two types of explanation, teleological and mechanistic, one based on rules which posit that each new phyllo be placed according to some optimal packing principle and the other which uses plain old biophysics and biochemistry to build mechanistic models which lead to pattern forming pde's. One of the stunning new results is that, while the latter is richer, both approaches lead to completely consistent results. This may well have broader ramifications in that it suggests that nature may use instability driven patterns to achieve optimal outcomes.

The talk should be accessible to students and colleagues in other disciplines.