Wednesday Seminar

Date: May 15 (Wed),  10:00--12:00

Venue: Room 110,  RIMS, Kyoto University

Speaker: Motohico Mulase (UC Davis, RIMS Kyoto University)

Title: "From Zeta(3) to Mirror Symmetry" 

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. 

Date: May 29 (Wed),  10:00--12:00

Venue: Room 111,  RIMS, Kyoto University

Speaker: Motohico Mulase (UC Davis, RIMS Kyoto University)

Title: "Discontinuous and Biholomorphic?" " 

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. 

Date: June  12 (Wed),  10:00--12:00

Venue: Room 206,  RIMS, Kyoto University

Speaker: Olivia Dumitrescu (University of North Carolina at Chapel Hill) 

Title: "Interplay between Higgs bundles, opers and quantum curves " 

 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 de Rham moduli spaces, in terms of lagrangians filling up the entire space in rank 2 and rank 1.

Date: July  3 (Wed),  10:00--12:00

Venue: Room 206,  RIMS, Kyoto University

Speaker: Shinpei Baba  (Osaka University)

Title: "Intersection of holonomy varieties of complex projective structures on Riemann surfaces.  "

 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.