Bi-Flat $F$-manifolds, Miura invariants and integrable systems of conservation laws
Speaker: Alessandro Arsie, University of Toledo
Abstract: In this talk, I will present present a survey of the theory of bi-flat $F$-manifolds, introduced few years ago by Paolo Lorenzoni (Universita' di Milano - Bicocca) and myself. These structures generalize the notion of Frobenius manifold introduced by Boris Dubrovin few years ago and they are somehow more versatile to capture integrability phenomena that are not associated (at least a priori) to Gromov-Witten invariants. In particular, I will focus on integrable systems of conservations laws that are associate to these structure and to a new set of invariants that we call Miura invariants. We conjecture that these invariants form a complete set of invariants for integrable systems of conservation laws. Furthermore, focusing on one-parameter families of dispersionless systems of integrable conservation laws associated to Coxeter groups of rank $2$, I will discuss the corresponding integrable deformations up to order $2$ in the deformation parameter $\epsilon$. Finally I will provide preliminary results on the existence of integrable deformations to all order in the deformation parameter $\epsilon$ for the aforementioned one-parameter families associated to Coxeter groups of rank $2$, based on the notion of partial cohomological field theory (this is a research project in collaboration with Alexander Buryak, Paolo Lorenzoni and Paolo Rossi).
Room Reservation Information
Room Number: 106 McAllister
Time: 2:30pm - 3:30pm