From Picard-Lefschetz to quantum singularity theory

Guangbo Xu , Princeton University
Fine Hall 314

Picard-Lefschetz theory can be viewed as a holomorphic analogue of Morse theory, which studies isolated singularities of holomorphic functions $W: {\mathbb C}^n \to {\mathbb C}$. In this talk, I will discuss the topological aspects about isolated singularities, and discuss the ADE-classification of simple elliptic singularities.   Witten made a remarkable conjecture about intersection numbers on moduli space of curves which was proved by Kontsevich. Indeed this is only a case of a more general conjecture specialized to the $A_1$-singularity. The $A_n$-case was proved by Faber-Shadrin-Zvonkine. More recently Fan-Jarvis-Ruan developed the their quantum singularity theory (or FJRW-theory) based on Witten's idea, and proved the corresponding conjecture for type D and type E singularities. If time permits, I will discuss their results very briefly.