Title

Inspirations from Mathematics
Lecture 1: "From Zeta(3) to Mirror Symmetry"
Lecture 2: "Discontinuous and Biholomorphic?"

Date

May 15 & May 29 (Wed), 10:00--12:00, 2024

Place

May 15: Room 110, May 29: Room 111 RIMS

Speaker

Motohico Mulase (UC Davis)

Abstract

These are the lectures aimed at a wider audience, including graduate students and undergraduate students with a strong curiosity and background of modern mathematics, to display frontiers of mathematical research in the scope of "Development in Algebraic Geometry related to Integrable Systems and Mathematical Physics."

The goal is to present *inspirations from mathematics*. The lectures will not be anything like "an introduction to xyz" type talks. Key terminologies may be used even without definition, if it is easily available in books and trusted online resources (excluding ChatGPT). I will present an inspiration from mathematics of the past, and inspire the audience toward the frontiers in mathematics of today by this inspiration as a guide. A large part of the topics is based on my own research, both current interest and past accomplishments, but the most recent materials are not mine.

Each talk will be 75-minute long, and open discussions follow after a short break.

Lecture 1: "From Zeta(3) to Mirror Symmetry"
Wednesday, May 15. 10:00--12:00

Abstract: The Riemann Zeta function is the most mysterious function in mathematics. This talk focuses on its special values. In the first part, I will explain my own unexpected encounter with some special values of Zeta. Topological recursion and moduli spaces of curves are behind the scene, which gives a new understanding of the Kontsevich proof of the Witten conjecture. Then I will present recent discoveries associated with Zeta(3) in the context of algebraic geometry and differential equations. Apéry's irrationality proof of Zeta(3) is the source of our inspiration. Apéry discovered a mysterious integer sequence in his proof. Later it was noticed that these numbers have direct relevance to mirror symmetry of a particular Fano 3-fold and its mirror Landau-Ginzburg model. I will report what has been proven in this direction. In the discussion part, I will formulate what seems to be true. Still we do not know the whole story.

Lecture 2: "Discontinuous and Biholomorphic?"
Wednesday, May 29. 10:00--12:00

Abstract: How can we define a global higher order differential operator on a compact Riemann surface? This naïve question leads us to encountering the half-canonical sheaf and the concept of *opers*. They are connections in holomorphic vector bundles, but form only a very thin slice of the moduli space of connections. This slice forms a holomorphic Lagrangian subvariety of the moduli space, which is a holomorphic symplectic manifold. Are there other Lagrangians in this symplectic space, and if so, can we realize the moduli space as the total space of an analytic family of disjoint Lagrangians? This is the Lagrangian foliation conjecture of Carols Simpson. Very recently, an amazing proof was discovered for the case of SL(2) connections by a starting postdoctoral scholar. I will present several exciting moments of discoveries of the key facts appearing in this new result.