
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.
NishiWaseda 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
10:3012:00 ChangShou Lin (National Taiwan University), "Mean field equations, hyperelliptic curves and modular forms 1"
13:3014:30 Todor Milanov (Kavli IPMU, Tokyo University), "GromovWitten invariants of elliptic orbifolds and quasimodular forms 1"
14:3015:30 Todor Milanov (Kavli IPMU, Tokyo University), "GromovWitten invariants of elliptic orbifolds and quasimodular forms 2"
16:0017:30 ChangShou Lin (National Taiwan University), "Mean field equations, hyperelliptic curves and modular forms 2"
Friday 7 June 2013 in BUILDING 63, 1st floor (ground floor), mathematics department conference room
10:3012:00 ChangShou Lin (National Taiwan University), "Mean field equations, hyperelliptic curves and modular forms 3"
13:3014:30 Kimio Ueno (Waseda University), "KZ equation on the moduli space $\cM_{0.n}$, and the RiemannHilbert problem I"
14:3015:30 Shu Oi (Waseda University), "KZ equation on the moduli space $\cM_{0.n}$, and the RiemannHilbert problem II"
16:0017:30 ChangShou Lin (National Taiwan University), "Mean field equations, hyperelliptic curves and modular forms 4"
ABSTRACTS:
ChangShou Lin: "Mean field equations, hyperelliptic 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 hyperelliptic curve. Interestingly, this hyperelliptic 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: "GromovWitten invariants of elliptic orbifolds and quasimodular 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 GromovWitten invariants of a certain class of elliptic orbifolds are quasimodular forms. Our approach is based on mirror symmetry: we prove that the GromovWitten invariants can be expressed in terms of the period integrals of an appropriate simple elliptic singularity. The quasimodularity 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 OCAMITIMS events:
Workshop on Integrable Systems and Geometric PDEs at TIMS 2022 October 2012
5th OCAMITIMS Joint International Workshop on Differential Geometry and Geometric Analysis 2527 March 2013
4th TIMSOCAMI Joint International Workshop on Differential Geometry and Geometric Analysis 1719 March 2012
3rd OCAMITIMS Joint International Workshop on Differential Geometry and Geometric Analysis 1315 March 2011
2nd TIMSOCAMI Joint International Workshop on Differential Geometry and Geometric Analysis 2123 March 2010
1st OCAMITIMS Joint International Workshop on Differential Geometry and Geometric Analysis 910 March 2009