Quantum curves, topological recursion, and Hitchin systemsÂ
Lecture 1. Quantum Curves I: Examples from classical analysis and simplest Gromov-Witten invariants.
In this talk the notion of quantum curves will be introduced through the most typical examples. The interplay between Gromov-Witten invariants, algebraic geometry of Higgs bundles and Hitchin systems, classical analysis, and quantum mechanics techniques such as the WKB analysis, will be presented.
Lecture 2. Quantum Curves II: A mathematical theory.
A mathematical definition of quantum curves using the notion of Rees D-modules will be presented in the framework of Hitchin's work on Higgs bundles. It will be explained how certain Gromov-Witten (and other quantum) invariants provide the exact WKB analysis of differential equations. The idea of topological recursion, originally due to Eynard and Orantin, and extended to the Hitchin framework by Dumitrescu and myself, will be introduced.
Lecture 3. Quantum Curves III: Our current understanding and what is expected in the future.
What we know so far about quantum curves will be presented (this part is based on my joint work with many other collaborators, including Bouchard, Dunin-Barkowski, Hernandez-Serrano, Liu, Norbury, Popolitov, Shadrin, Spitz, and Sulkowski). If time permitting, I will present the most recent work with Dumitrescu on birational geometry of ruled surfaces and quantum curves, and beyond.