The Langlands conjecture originated as a highly non-trivial generalization of the reciprocity laws in number theory. In my talk, I explain how after certain `geometrization', it becomes a statement about sets (`moduli spaces') of vector bundles on a Riemann surface. The result is a kind of Fourier transform relating sets of vector bundles and local systems on a Riemann surface.

This `geometric Langlands transform' can be used to motivate theorems and conjectures in such diverse areas of mathematics (and physics) as theory of Painleve equations, representation theory of loop groups, autoduality of Jacobians, and mirror symmetry. Some of the relations will be explained in the talk.

In this talk, I will report a recent joint work with Mu-Tao Wang. We construct examples of shrinkers and expanders for Lagrangian mean curvature flows. These examples are Hamiltonian stationary and asymptotic to the union of two Hamiltonian stationary cones found by Schoen and Wolfson. The Schoen-Wolfson cones are obstructions to the existence problems of special Lagragians or Lagrangian minimal surfaces in the variational approach. It is known that these cone singularities cannot be resolved by any smooth Lagrangian submanifolds. The shrinkers and expanders that we found can be glued together to yield solutions of the Brakke motion-a weak formulation of the mean curvature flow, and thus provide a canonical way to resolve the union of two such cone singularities. Our theorem is analogus to the Feldman-Ilmanen-Knopf gluing construction for the K\"ahler-Ricci flows.

I'll discuss the papers "Berger's isoperimetric problem and minimal
immersions of surfaces" by Nadirashvili, "Extremal metric for the first
eigenvalue on a Klein Bottle" by Jakobson, Nadirashvili and Polterovich
and "How large can the first eigenvalue be on a surface of genus two?" by
Jakobson, Levitin, Nadirashvili, Nigam and Polterovich. The authors find
among metrics of fixed area one that maximizes the fundamental frequency
of the torus and Klein Bottle.