Spring Emphasis Semester 2020

Main Workshops Registration Links

  1. Homological Mirror Symmetry and Topological Recursion (01/27/2020-02/01/2020)
  2. Topological and Geometric Recursion (02/03/2020-02/07/2020)
  3. Workshop on Resurgence and Quantum Invariants (03/10/2020-03/13/2020)
  4. School on Resurgence (03/16/2020-03/20/2020)
  5. Quantum Toric Geometry (03/20/2020-04/02/2020)
  6. TBA (06/11/2020-6/14/2020)

Homological Mirror Symmetry and Topological Recursion

This event is partially supported by the Simons Collaboration on Homological Mirror Symmetry, with the assistance of the University of Miami Department of Mathematics. The programme will feature 3 mini-courses and a range of research talks in various areas of homological mirror symmetry and related topics.

Topological and Geometric Recursion in Interaction with Resurgence

Organizers: Maxime Kontsevich (IHES/Miami) and Jorgen Ellegaard Andersen (QM, SDU, 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. The School on Resurgence is exactly dedicated to this subject.

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.

The theory of resurgence also has applications in quantum topology. This was pioneered by Garoufalidis and Witten. The main idea in Witten's work is to formally apply Pham-Picard-Lefschetz theory to the partition function of Chern-Simons theory. In the work of Gukov-Marino-Putrov Witten's idea was used to argue that the WRT invariant of a closed homology 3-sphere can be related via resurgence to the q-series invariant Z-hat introduced by Gukov-Pei-Putrov-Vafa, something which has very recently been establish by Andersen and Mistegaard for Seifert fibered homology 3-sphere. Further the work of Gukov and Manulescu has open the door on understanding how Z-hat might be extended to a TQFT. 

The workshop on Resurgence on Quantum Invariants is dedicated to this purpose.

Participants:

Jorgen Ellegaard Andersen, Gaëtan Borot, Francois David, Bertrand Eynard, Seigei Gukov, Rinat Kashaev, Ralph Kauffman, Maxime Kontevich, Ernesto Lupercio, 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.

Quantum Toric Geometry

The purpose of this conference is to explore the state of the art in toric geometry stressing its current generalizations, and in particular Quantum Toric Geometry (arXiv:2002.03876). Classical toric geometry has been a key element in the solution of truly diverse problems in mathematics spanning from combinatorics to algebraic and symplectic geometry to mathematical physics. In classical toric geometry a combinatorial and rational object is assigned to a transcendental geometric object. In various recent generalizations the rationality condition can be dropped. In Quantum Toric Geometry a quantization of the classical systems appearing in toric geometry allow both, the relaxation of such condition and the appearance of a new moduli space of toric varieties. One of the main points of this conference is to explore this new kinds of toric geometry. 


Workshop Programs

  1. Homological Mirror Symmetry and Topological Recursion 
  2. Topological and Geometric Recursion
  3. Workshop on Resurgence and Quantum Invariants (Coming Soon) 
  4. School on Resurgence
  5. Quantum Toric Geometry (Comming Soon)
  6. TBA (Coming Soon)