Jerusalem

A Lagrangian submanifold of a Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. A Hamiltonian isotopy class of positive Lagrangian submanifolds admits a Riemannian metric with non-positive curvature. Its universal cover

admits a functional, with critical points special Lagrangians, that is strictly convex with respect to the metric. If time permits, I'll explain

how mirror symmetry relates the metric and functional to the infinite dimensional symplectic reduction picture of Atiyah, Bott, and Donaldson in

the context of the Kobayashi-Hitchin correspondence.

In recent years there has been extensive interest in word maps on groups, and various results were obtained, with emphasis on simple groups. We shall focus on some new results on word maps for more general families of finite and infinite groups.

Word maps on groups were studied extensively in the past few years, in connection to various conjectures on profinite groups, finite groups, finite simple groups, etc. I will provide background, as well as very recent works (joint with Larsen, Larsen-Tiep,

Liebeck-O'Brien-Tiep) on word maps with relations to representations (e.g. Gowers' method and character ratios), geometry and probability.

Recent applications, e.g. to subgroup growth and representation varieties, will also be described.

I will conclude with a list of problems and conjectures which are still very much open.  The talk should be accessible to a wide audience.

(IN: LADY MARGARET HALL)

As part of the Conference on Geometric Model Theory in honour of Professor Boris Zilber