Spring Emphasis Semester 2020

Topological and Geometric Recursion in Interaction with Resurgence:

Organizers: Maxime Kontsevich (IHES/Miami) and Jorgen Andersen (Aarhus, Denmark)

Topological recursion is an algebra-geometric construction which takes the spectral curve and makes out of it a recursive definition of infinite sequences of symmetric meromorphic n-forms with poles. It was first discovered for random matrices. A main goal of random matrix theory is to find the large size asymptotic expansion of n-point correlation functions, and in some cases, the asymptotic expansion takes the form of an infinite series. The n-form ω(g,n) is than the gth coefficient in the asymptotic expansion of the n-point correlation function. It was found that the coefficients ω(g,n) always obey the same recursion on 2 g -2 +n. The idea is to consider this universal recursion relation beyond random matrix theory, and to promote it as a definition of algebraic curve invariants, was first introduced by Eynard-Orantin in 2007, where they studied the main properties of these invariants.

An important application of topological recursion is to Gromov-Witten invariants. Marino has conjectured that Gromov-Witten invariants of a toric Calabi-Yau 3-fold X are the topological recursion invariants of a spectral curve that is the mirror of X. More recently, Kontsevich has taken topological recursion to a new level, connecting it with category theory and HMS. The wall crossing interpretation of topological has led to new applications in understanding classical questions in ordinary and partial differential equations, in particular a new approach to Voros and Ecalle resurgence phenomena.

Very recently Andersen in collaboration with Borot and Orantin has created "Geometric Recursion", which is a recursively theory, building a number of different geometric objects associated surfaces. It provides a generalization of Mirzahkani's work and further it offers an effective means to show that various quantities can be computed using topological recursion, such as volumes of moduli spaces, statistics of length spectrums of simple closed geodesics and Masur-Veech volumes

Main Workshops

  1. Homological Mirror Symmetry (01/27/2020-02/01/2020)
  2. Topological and Geometric Recursion (02/03/2020-02/08/2020)
  3. Workshop on Resurgence (03/16/2020-03/21/2020)
  4. Final Workshop (04/22/2020-04/26/2020)

Participants:

Jorgen Andersen, Gaëtan Borot, Francois David, Bertrand Eynard, Seigei Gukov, Rinat Kashaev, Ralph Kauffman, Maxime Kontevich, Grigory Mikhalkin, Marco Mariño, Marcos Maurino, Nicolas Orantin, Tony Pantev, Du Pei, Sylvain Ribault, David Sauzin, Ricardo Schiappa, Yan Soibelman, Yakov Soibelman, Nikita Nekrasov, Smason Shatashvilli, Ludmil Katzarkov, Bertrand Eynard, Nivolas Orantin, postdoctoral associates from Mexico and Brazil TBA

There will be additional short courses (Andersen, Eynard, Kontevich, Soibelman) and seminars.