

"Toric Kahler metrics in symplectic coordinates"

Abstract: I will motivate and describe a parametrization for toric Kahler metrics
on symplectic 2nmanifolds (and orbifolds) equipped with a
Hamiltonian ntorus 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 2dimensional
symplectic surface in a KaehlerEistein surface and Lagangian
submanifolds in a CalabiYau nfold. 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"
