"Toric Kahler metrics in symplectic coordinates"

Abstract: I will motivate and describe a parametrization for toric Kahler metrics on symplectic 2n-manifolds (and orbifolds) equipped with a Hamiltonian n-torus action, using symplectic coordinates and potentials (instead of the more common complex analogues). The relevant properties of this approach will be discussed and illustrated with several explicit examples.     I will then concentrate on the constant and extremal (in the sense of Calabi) scalar curvature equation, motivating its special form in this setting, describing explicit interesting solutions and discussing some of its analytical properties.     Almost all of what will be presented in these two talks is the result of separate work by Guillemin, Abreu and Donaldson.

"Some geometric PDEs arising from calibrations"

Abstract: In the first talk, I will discuss mean curvature flows for 2-dimensional symplectic surface in a Kaehler-Eistein surface and Lagangian submanifolds in a Calabi-Yau n-fold. In the second talk, I will discuss maps between hyperkaehler manifolds, which satisfy a first order system and are related to harmonic maps and minimal surfaces.

"Minimal Submanifolds I: Lagrangian volume minimization and special lagrangian submanifolds"

Abstract: In high codimension there are special classes of volume minimizing submanifolds which one would like to construct in general situations. These include special lagrangian submanifolds. In this talk we will describe this class of submanifolds and survey the progress on existence questions via variational methods, mean curvature flow, and gluing methods. We will discuss some of the many open problems in this area.

"Minimal Submanifolds II: On the isoperimetric inequality for minimal surfaces"

Abstract: A longstanding conjecture concerning the geometry of minimal surfaces is that the sharp isoperimetric inequality (relating the area of the surface and its boundary length) should hold in general. In this talk we will survey this problem and present some new methods of attack which solve the problem in general situations. The methods involve a careful analysis of flat cone metrics which are related in a natural way to the minimal surface.

" Ricci flow I: A selected survey."

Abstract: I'll describe what the Ricci flow is, and some of the basic theory, including a few ideas of Perelman which are interesting from a geometric analysis point of view. (Aimed at a general audience who know a bit of Riemannian geometry.)

"Ricci flow II: Singular initial configurations."

Abstract: I'll show how one of the ideas from the 1st talk can be used to prove a compactness theorem for Ricci flows, which gives us some unconventional existence theorems.

"Harmonic maps into complex Finsler manifolds"

"Function theory and its applications"

Abstract: Function theory has been successfully applied to derive various geometric and topological information of Riemannian manifolds. In the two talks, I plan to explain some of these aspects and mention a few open problems.

"On an isoperimetric inequality for mapping with remainder term"