Fabio Podestà:

Tight Lagrangian Homology spheres in homogeneous Kaehler manifolds.


Abstract:

I will talk about the full classification of tight Lagrangian submanifolds L of compact homogeneous Kaehler manifolds, when L is supposed to have the Z_2-homology of a sphere.

Yoshihiro Ohnita:

Geometry of Lagrangian submanifolds related to isoparametric hypersurfaces


Abstract:

In this talk I shall provide a survey of my recent works and their environs on differential

geometry of Lagrangian submanifolds in specific Kaehler manifolds, such as complex projective spaces, complex space forms, Hermitian symmetric spaces and so on.  I shall emphasis on the relationship between certain minimal Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres.  I shall discuss their properties and related problems

such as classification, Hamiltonian stability and Lagrangian intersection theory for the Gauss

images of isoparametric hypersurfaces. This talk is mainly based on my joint work with Hui Ma (Tsinghua University, Beijing).

Marco Radeschi:

Mean curvature flow and generalized isoparametric foliations.


Abstract:

We study the mean curvature flow evolution of a leaf in a generalized isoparametric foliation. We prove that the the flow evolves through leaves of the foliation, and if the flow has a finite time singularity then this must be of of type I. This answers a question of Terng and Liu, for a more general setting than the original one.

Tommaso Pacini:

From Lagrangian to totally real geometry


Abstract:

The totally real condition is an obvious relaxation of the Lagrangian condition, involving a change of direction from symplectic to complex geometry. I will present some recent results, joint with Jason Lotay (UCL), on the geometry of the space of totally real submanifolds. The overall picture is surprisingly rich, considering that it is an open subset of the space of all submanifolds. Specifically, we define a symmetric space structure, geodesics and a convex functional analogous to those found by Donaldson, Mabuchi et al. on the space of Kahler potentials. I will also present some preliminary results on the existence of geodesics and on possible connections to the ideas of Thomas-Yau concerning Lagrangian MCF and the search for special Lagrangians.

Leonardo Biliotti:

Invariant convex sets in polar representations


Abstract:

In this talk we study a compact invariant convex set E in a polar  representation of a compact Lie group. Polar   representations are given by the adjoint action of K on p, where K is  a maximal compact subgroup of a real semisimple Lie group G  with Lie algebra g =k + p. If a is a maximal abelian subalgebra of p, then the intersection of E with a, that we denote by P, is a  Weyl-invariant convex set in a.  We prove that up to conjugacy the face structure of E is completely determined by that of P and that a face of E is exposed if and only if the corresponding face of P is exposed.  We apply these results to the convex hull of the image of a restricted momentum map. We will also present some preliminary results on the action of  a compact Lie group, which acts in Hamiltonian fashion on a Kahler compact manifold M, on the space of the measure on M. This talk is based on my joint works with Alessandro Ghigi (University of Milano Bicocca) and Peter Heinzner (University of Bochum).

Miguel Abreu:

Toric constructions of monotone Lagrangian submanifolds in CP^2 and CP^1 x CP^1


Abstract:

Motivated by a toric model of the Chekanov-Schlenk exotic Lagrangian torus in CP^2, I will present a toric construction of certain Lagrangian submanifolds and apply it to obtain some old and new monotone examples in CP^2 and CP^1 x CP^1.

This is joint work with Agnès Gadbled.

Jake Solomon:

Holomorphic disks and special Lagrangians

 

Abstract:

Special Lagrangians in Calabi-Yau manifolds are expected to be plentiful. However, in practice, it is difficult to find special Lagrangian submanifolds in compact Calabi Yau manifolds except in two special cases: fixed points of anti-symplectic involutions and holomorphic Lagrangians in hyper-Kahler manifolds. Thus it is natural to look for a modified special Lagrangian condition that reduces to the standard one in these two cases. I will describe such a modification of the special Lagrangian condition. It arises as the Euler-Lagrange equation of a convex functional modified by contributions from holomorphic disks. This is joint work with G. Tian.