OCAMI-TIMS@WASEDA workshop on

Integrable systems, modular forms, and related applications Waseda University (Nishi-Waseda Campus, for rooms see below)

PART I: 31 May 2013
PART II: 7 June 2013

 

Links between integrable systems, partial differential equations, and geometry are increasingly important sources of research problems in mathematics and mathematical physics. The workshop will focus on topics related to modular forms and related special functions. This is an area which has very classical roots, but which is related to very modern developments such as the KZ equations and mirror symmetry.


Nishi-Waseda Campus access map

Map showing buildings 55 and 63


SCHEDULE OF TALKS:

Friday 31 May 2013 in BUILDING 55N, 1st floor (ground floor), 2nd conference room

10:30-12:00 Chang-Shou Lin (National Taiwan University), "Mean field equations, hyper-elliptic curves and modular forms 1"

13:30-14:30 Todor Milanov (Kavli IPMU, Tokyo University), "Gromov-Witten invariants of elliptic orbifolds and quasi-modular forms 1"

14:30-15:30 Todor Milanov (Kavli IPMU, Tokyo University), "Gromov-Witten invariants of elliptic orbifolds and quasi-modular forms 2"

16:00-17:30 Chang-Shou Lin (National Taiwan University), "Mean field equations, hyper-elliptic curves and modular forms 2"

 

Friday 7 June 2013 in BUILDING 63, 1st floor (ground floor), mathematics department conference room

10:30-12:00 Chang-Shou Lin (National Taiwan University), "Mean field equations, hyper-elliptic curves and modular forms 3"

13:30-14:30 Kimio Ueno (Waseda University), "KZ equation on the moduli space $\cM_{0.n}$, and the Riemann-Hilbert problem I"

14:30-15:30 Shu Oi (Waseda University), "KZ equation on the moduli space $\cM_{0.n}$, and the Riemann-Hilbert problem II"

16:00-17:30 Chang-Shou Lin (National Taiwan University), "Mean field equations, hyper-elliptic curves and modular forms 4"


ABSTRACTS:

Chang-Shou Lin: "Mean field equations, hyper-elliptic curves and modular forms" In this series of lectures, I will talk about the mean field equations with singular sources in a flat torus. This type of equation has arisen from different research areas in mathematics. In conformal geometry, it is related to existence of positive constant curvature metrics with conic singularity. In mathematical physics, the equation is a limiting equation of nonlinear PDEs arising in different gauge theories. Although this equation has been studied for more than several decades, recently it still attract a lot of attention. One issue is about bubbling solutions near or at the critical parameters. The locations of blowup points of a sequence of bubbling solutions are determined by some equations involving the Green function. We can show that those equations describe a hyper-elliptic curve. Interestingly, this hyper-elliptic curve has appeared in the literature in different contexts, for example in the Lame equation and KdV hierarchy, where it is known as the spectral curve. Although it has been known for more than one hundred years, there are very few studies about the geometry of this curve. The connection with the mean field equation motivates the study of geometric problems related to the curve. I will talk about this and some open problems. The mean field equation has been known as a completely integrable system since Liouville. From the Liouville theorem, a developing map is associated with each solution of the mean field equation. We shall show how modular forms naturally arise from developing maps. We shall give applications of the modular form to the mean field equation and conversely apply analytic results on the mean field equation to determine the zeros of some modular forms.

Todor Milanov: "Gromov-Witten invariants of elliptic orbifolds and quasi-modular forms" An elliptic orbifold is an orbifold obtained by taking a finite group quotient of an elliptic curve. In a joint work with Y. Ruan and Y. Shen we prove that the Gromov-Witten invariants of a certain class of elliptic orbifolds are quasi-modular forms. Our approach is based on mirror symmetry: we prove that the Gromov-Witten invariants can be expressed in terms of the period integrals of an appropriate simple elliptic singularity. The quasi-modularity of the latter is easy to establish. My goal is to explain in more details our construction as well as to give the necessary background.


Organizing committee: Martin Guest (Waseda University), Ping Li (Tongji University/Waseda University), Takashi Otofuji (Nihon University)

Sponsors: Waseda University, Osaka City University Advanced Mathematical Institute (OCAMI), National Taiwan University - Taida Institute for Mathematical Sciences (TIMS)


Past OCAMI-TIMS events:

Workshop on Integrable Systems and Geometric PDEs at TIMS 20-22 October 2012

5th OCAMI-TIMS Joint International Workshop on Differential Geometry and Geometric Analysis 25-27 March 2013

4th TIMS-OCAMI Joint International Workshop on Differential Geometry and Geometric Analysis 17-19 March 2012

3rd OCAMI-TIMS Joint International Workshop on Differential Geometry and Geometric Analysis 13-15 March 2011

2nd TIMS-OCAMI Joint International Workshop on Differential Geometry and Geometric Analysis 21-23 March 2010

1st OCAMI-TIMS Joint International Workshop on Differential Geometry and Geometric Analysis 9-10 March 2009