Talk Schedule

2:00 - 2:50 PM     Karen Uhlenbeck (UT Austin)

3:10 - 4:00 PM     Gudlaugur Thorbergsson (Cologne)

4:00 - 4:30 PM     Reception

4:30 - 5:20 PM     Robert Bryant (Duke)

We prove that Alexandrov's conjecture relating the area and diameter of a convex surface holds for the surface of a general ellipsoid. This is a direct consequence of a more general result which estimates the deviation from the optimal conjectured bound in terms of the length of  the cut locus of a point on the surface. We also prove that the natural extension of the conjecture to general dimension holds among closed convex spherically symmetric Riemannian manifolds. Our results are based on a new symmetrisation procedure which we believe to be interesting in its own right.

This is joint work with Pedro Freitas accepted for publication in the Tohoku Mathematical Journal, preprint on http://arxiv.org/abs/1406.0811.

We will explain a systematic way to study a type of curve flows in R^{n+1, n}, whose geometric invariants are solutions of some integrable systems. In detail, we will construct a hierarchy of isotropic curve flows in R^{2, 1}, construct explicit solutions from Backlund transformation, and study its Hamiltonian formulation.

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.