Workshop on:

Mirror symmetry and differential equations

Supported in part by JSPS grant (A) 21244004 "Exploitation of new relations between differential geometry and quantum cohomology in the context of integrable systems"

PROGRAM

10:30: Noriko Yui (Queen's University, Canada), "The modularity of certain K3-fibered Calabi-Yau threefolds over Q"

Abstract: We consider certain K3-fibered Calabi-Yau threefolds definedover $\mathbf{Q}$. These Calabi-Yau threefolds are constructed using the method of Voisin and Borcea, and are smooth resolutions of quotients of $S\times E$ by some involution. (Here $S$ is an algebraic $K3$ surface and $E$ is an elliptic curve.)

We will discuss the modularity (automorphicity) of these Calabi-Yau threefolds over $\mathbf{Q}$. Some of the Calabi--Yau threefolds considered here are shown to be of CM type. We establish the modularity of such Calabi-Yau threefolds, and also their mirror partners (if they exist) in the sense of Arthur and Clozel. Several explicit examples are discussed.

This reports on a joint work in progress with Y. Goto (Hakodate) and R. Livne (Jerusalem).

13:00: Sergey Galkin (IPMU, Japan), "(Mirror) modularity of Fano threefolds"

Abstract: Minimal Fano threefolds are known to be (mirror) modular i.e. their regularized I-series can be expressed as a weight 2 newform expanded in terms of a Hauptmodul of the same level. It turns out similar phenomena holds also for some non-minimal Fano threefolds - we'll review these results and discuss approaches to predict (mirror) modular Fano varieties.

14:30: Claus Hertling (Mannheim University, Germany), "Stokes data, Fourier-Laplace transformation and non-commutative Hodge structures"

Abstract: The notion of non-commutative Hodge structures (ncHS) is involved. But a semisimple ncHS is controlled by rather elementary data, a set of complex numbers, a complex number of absolute value 1 and a real (or integer) upper triangular matrix with 1's on the diagonal. These are called Stokes data.

The talk will discuss the geometry in these Stokes data, in the ncHS and in its Fourier-Laplace transformation, which is a holomorphic bundle on P^1 with a connection with regular singularities. It will present examples and results on the problem when the ncHS is pure and polarized.

16:00 Yifan Yang (National Chiao Tung University, Taiwan), "Apery limits and special values of L-functions"

Abstract: In this talk, we will discuss Apery limits of differential equations that have modular-form origin.

17:30 Chang-Shou Lin (National Taiwan University and TIMS, Taiwan), "Nonlinear PDE aspects of the tt* equations of Cecotti and Vafa"

Abstract: We shall discuss some features of solutions to the tt* equations, from the p.d.e. point of view.

19:00 dinner

FOR FURTHER INFORMATION PLEASE CONTACT:

Martin Guest martin.comp.metro-u.ac.jp (change first dot to at)