The Laplace Transform and Mirror Symmetry

Spring Quarter 2011. CRN #39708

Tuesdays and Thursdays, 10:00 - 12:00

2240 MSB

Organizer: Motohico Mulase

Mirror symmetry is a physics idea that tells us that there are two different geometries to describe our universe: complex analytic geometry (including algebraic geometry) and symplectic geometry. By some miraculous reasons, these two geometries give the same answer, reflecting the fact that there is only one universe.

On the symplectic geometry side, many branches of mathematics, including combinatorics, discrete mathematics, and number theory, appear in the theory. On the complex geometry side, complex analysis, algebraic geometry, and integrable systems appear. The Laplace transform seems to be a mechanism that connects these two worlds. The interplay of the two geometries gives us topological information of the universe.

The whole picture is big. It includes a large portion of all branches of mathematics. But we will be interested in working on a particular and computable example of the theory to see all these links. Through our experience of discovering new phenomena, we wish to learn all the interesting mathematics.

It is not geometry, not analysis, not combinatorics, not topology, not algebra, not number theory, not anything. It is everything!

Since Professor Kanehisa Takasaki of Kyoto University, a world class expert of integrable systems, is visiting us this week, I will ask him to give us an introductory lecture on integrable systems on Tuesday and Thursday. I think it provides a good opportunity for you to learn something extremely important, yet you never have a chance to learn in our courses.

March 29. Motohico Mulase: "Introductory Remarks."

March 29. Professor Kanehisa Takasaki (Kyoto University): "Introduction to Integrable Systems."

In this talk KdV, KP, and Toda lattice equations will be introduced in connection with the Gromov-Witten theory.

March 31. Professor Kanehisa Takasaki (Kyoto University): "Determinants and counting problems in combinatorics."

We will go through an example of tree-counting problem using the method of determinant. These counting problems are deeply related to integrable systems of nonlinear partial differential equations.

April 5 and April 7. Motohico Mulase: "Homology, homotopy, and de Rham cohomology theories."

We review necessary accounts from the homology theory, homotopy theory, and the de Rham theorem.

April 12 and April 14. Professor Michael Penkava (University of Wisconsin and UC Davis): "Introduction to the Infinity Algebras."

We review necessary accounts from the theory of A-infinity and L-infinity algebras.