Title

Prill's problem

Date

September 2 (Monday), 10:00--11:30, 2024

Place

Room 110 RIMS, Kyoto University

Speaker

Aaron Landesman (Harvard University)

Abstract

An open question of David Prill from the 1970's, popularized in ACGH, asks whether there is a cover of curves $f: X \to Y$ with $g(Y) \geq 2$ so that every fiber moves in a pencil. We discuss joint work with Daniel Litt, answering this question. In order to do so, we describe its connection to a conjecture in geometric topology, the Putman-Wieland conjecture, by studying the derivative of an associated period map via Hodge theory. In case you happened to be at the conference last week, the methods and results involved in this talk were instrumental in proving the results discussed there.

Title

Intersection of holonomy varieties of complex projective structures on Riemann surfaces.

Date

July 3 (Wed), 10:00--12:00, 2024

Place

Room 206 RIMS

Speaker

Shinpei Baba (Osaka University)

Abstract

In this talk, we discuss the intersection of certain analytic Lagrangians in the PSL(2, C)-character variety of a surface group.

A holomorphic quadratic differential on a Riemann surface corresponds to a complex projective structure and also a PSL(2, C)-oper. Note that the set of holomorphic quadratic differentials on a Riemann surface from a complex vector space.

Then, by the holonomy map, this vector space properly maps onto an analytic Lagrangian subvariety of the space of representations of the surface group into PSL(2, C). Given two (marked) Riemann surface structures of the same topological type, we show that their corresponding Lagrangians intersect in a discrete set.

If time permits, applying this discreteness, we explain a new proof of Bers’ simultaneous uniformization.

Title

Interplay between Higgs bundles, opers and quantum curves

Date

June 12 (Wed), 10:00--12:00, 2024

Place

Room 206 RIMS

Speaker

Olivia Dumitrescu (University of North Carolina, Chapel Hill)

Abstract

The rainbow is one of the most beautiful phenomena in nature. It has inspired art, mythology, and has been a pleasure and challenge to the mathematical physicists for centuries. You might have wondered what awaited you if you went over the rainbow. Is the world on the other side of the rainbow the same as what we know? Sir George Airy discovered the rainbow integral and explained the classical analysis of rainbows, 150 years later, Kontsevich related it to intersections numbers on moduli spaces of punctured Riemann surfaces. These stories are a simple example of a mathematical theory of "quantum curves." I will further continue the exposition and I will present a general framework of quantum curves and I will relate it to topological recursion and the Gaiotto conformal limits that appeared in the previous talks. I will illustrate main differences between the two diffeomorphic moduli spaces, the Hitchin and the deRham moduli spaces, in terms of lagrangians filling up the enti! re space in rank 2 and rank 1.

Title

Cones of Curves Stratification

Date

June 19 (Wed), 10:00--12:00, 2024

Place

Room 206 RIMS

Speaker

Olivia Dumitrescu (University of North Carolina, Chapel Hill)

Abstract

The study of curves in projective space is a well-known problem in algebraic geometry, that goes back centuries. The minimal model program in birational geometry has been formulated via the theory of divisors, and it is an interesting question to understand it via the theory of curves.

In this talk, we discuss the polyhedrality of the cones of divisors ample in codimension k on a Mori dream space and the duality between such cones and the cones of k-moving curves by means of the Mori chamber decomposition of the former. This is based on joint work with Chiara Brambilla, Elisa Postinghel and Luis Santana Sanchez.

Title

On the Semiclassical limit of Opers and Complex Lagrangians

Date

June 5 (Wed), 10:00--12:00, 2024

Place

Room 206 RIMS

Speaker

Olivia Dumitrescu (University of North Carolina, Chapel Hill)

Abstract

In this talk we will understand the Beilinson-Drinfeld oper as a quantum curve associated to an SL(r, C) Higgs bundle and I will present an algebraic geometry description for SL_n(C) oper case. We prove that semiclassical limit of opers recovers the spectral curves of the associated to meromorphic Higgs bundles. In rank 2, I will relate this algebraic construction with the holomorphic lagrangian foliation conjecture of Simpson. This talk is based on joint work with Motohico Mulase.

Title

Higgs bundles and the Hitchin fibration, old and new

Date

May 22 (Wed), 10:00--12:00, 2024

Place

Room 206 RIMS

Speaker

Laura Schaposnik Massolo (University of Illinois Chicago)

Abstract

During the first half of the talk we will introduce Higgs bundles and their integrable system, focusing on how they can both be described in terms of spectral data. After describing some dualities they satisfy (not only from mirror symmetry but also via other correspondences such as low-rank isogenies), we will then focus on different methods to understand the Hitchin fibration and specially its singular fibres (monodromy, isogenies, cayley correspondences).

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.