Blumenthal Lectures 2022-23
Grigory Mikhalkin (University of Geneva)
Lecture I: Wednesday, Nov. 9, 14:10 (Geometry and Dynamics Seminar)
Room: Schreiber 309
Title: Toric geometry and tropical trigonometry
Abstract:
Toric varieties were constructed as algebraic varieties about 50 years ago, and also as symplectic varieties about 40 years ago. The two constructions are dual to each other, but are based on the same geometry in R^n. Symmetries in this geometry are linear transformations given by invertible n-by-n matrices with integer coefficients, as well as all translations. This makes the notion of a tangent integer vector as well as a notion of tropical curve well-defined. The talk will review basic constructions with a focus on tropical triangles that underlie some recent progress in symplectic embedding problems.
Lecture II: Thursday, Nov. 10, 16:15 (Seminar on Real and Complex Geometry)
Room: Orenstein 111
Title: Tropical, real and symplectic geometry
Abstract:
This lecture will focus on the way how tropical curves appear in symplectic geometry settings. On one hand, tropical curves can be lifted as Lagrangian submanifolds in the ambient toric variety. On the other hand, they can be lifted as holomorphic curves. The two lifts use two different tropical structures on the same space, related by a certain potential function. We pay special attention to correspondence theorems between tropical curves and real curves, i.e. holomorphic curves invariant with respect to an antiholomorphic involution. The resulting real curves produce, in their turn, holomorphic membranes for tropical Lagrangian submanifolds.