titles and abstracts

Robert Bryant: Riemannian Submersions as PDE

The problem of determining the Riemannian submersions

f: (Q,g) →(M, g‾))

with a given source Riemannian manifold (Q,g) will be discussed as a PDE problem. When the dimension of the target M is greater than 1, this is an overdetermined PDE system whose local nature is not well-understood. I will describe what is known and give some new results that classify such submersions when (Q,g) is a Riemannian space form of low dimension.

Eleny Ionel: TBA

Claude LeBrun: Einstein Metrics on Complex Surfaces

Which smooth compact 4-manifolds admit Einstein metrics?

The answer to this fundamental question still eludes us in general, but we can say a great deal if extra hypotheses are imposed. For example, if M is the underlying smooth 4-manifold of a compact complex surface, we now have a complete answer when the Einstein constant is non-negative. The negative case is more complicated, but we can still say some interesting things. This lecture will attempt to combine a general overview of the problem with a technical discussion of some of the most recent results.

Melissa Liu: Instanton Counting on Toric Surfaces

We will survey the relationships among counting ASD instantons on toric surfaces, period integrals of Riemann surfaces, counting holomoprhic curves in toric threefolds, and invariants of knots and links in the three sphere.

Ken-Ichi Yoshikawa: Analytic torsion and automorphic forms

Some years ago, we introduced an invariant of K3 surfaces with involution, which we constructed using equivariant analytic torsion. This invariant, which is a function on the moduli space, is expressed as the Petersson norm of an automorphic form. We would like to talk about the structure of this automorphic form. In many cases, this automorphic form is expressed as the tensor product of a Borcherds lift and Igusa's modular form.

If time permits, we will talk about the relation between this automorphic form and the BCOV invariant of some Borcea-Voisin threefolds. The BCOV conjecture suggests that the elliptic modular form appearing in the Borcherds lift should be equivalent to the elliptic Gromov-Witten invariants of some Calabi-Yau threefolds.

Wolfgang Ziller: Manifolds with positive curvature

Manifolds with positive sectional curvature have been studied frequently since the beginning of global Riemannian geometry. But there is still
little known about which manifolds can admit such metrics. On the other hand, much progress has been made over the last 10 years if one
assumes in addition the existence of a large isometry group. We will discuss these developments and some recent interesting candidates that are closely connected to self dual Einstein orbifolds.