KMI-Math School

About: 

The aim of KMI-Math School is to introduce fascinating interactions between mathematics and physics, and to invite students and non-experts to such exciting fields. Both mathematicians and physicists are more than welcome to participate in, and the speakers will co-ordinate their lectures to clearly address the main idea. Our ultimate goal is to break the boundaries between mathematics and physics, and trigger new collaborations across the two.

This year, our focus is on topological recursion and related subjects. Topological recursion is a universal recursive structure that appears in many fields in mathematics and physics. We will start with a definition of topological recursion, and study how it is related to e.g. moduli spaces of curves and SYK models. If you are interested in some of the keywords below, KMI-Math school 2026 would be a great opportunity for you:

Keywords:

• Moduli space of curves, cohomological field theories

• Topological recursion, Airy structures

• Large-N expansion, SYK models, JT gravity

Here is a drawing of the topological recursion formula. You will study what this picture tells us in mathematics and physics at the school :)

Speakers:

Registration:

Registration is free but mandatory. Please register through this link.

Deadline for Gong-Show is: December 15, 2025

Registration itself is open until January 31, 2026. In order to have a good estimate of the number of participants, however, we would greatly appreciate it if you register before Debember 15, 2025

Unfortunately, no financial support is available for participants.

Lecture Materials:

Title: Introduction to topological recursion (by Chidambaram)

Abstract: Topological recursion, discovered by Eynard and Orantin, is a phenomenon that appears in various contexts in enumerative geometry and physics — examples include matrix models, moduli space of curves, hyperbolic geometry, gauge theories, Hurwitz theory, and Gromov-Witten theory. We will begin by introducing an algebraic framework known as Airy structures, as defined by Kontsevich and Soibelman, that underlies topological recursion. Then we will introduce the topological recursion formalism itself, which is purely complex geometric, and discuss various properties, generalizations, and interesting examples.

References: Lecture notes

Title: Moduli spaces of Riemann surfaces (by Giacchetto)

Abstract: We provide an introduction to the moduli space of Riemann surfaces, a fundamental concept in the theories of 2D quantum gravity, topological string theory, and matrix models. We begin by reviewing some basic results concerning the recursive boundary structure of the moduli space and the associated cohomology theory. We then present Witten's celebrated conjecture and its generalisation, framing it as a recursive computation of cohomological field theory correlators via topological recursion. Time permitting, we touch on JT gravity in relation to hyperbolic geometry and topological strings. 

References: Lecture notes

Title: A First Glimpse at Large N Matrices (by Mazenc)

Abstract: We will focus our lectures on three beautiful results in the field of random matrix theory. First, we will derive Wigner's famous semi-circle law describing the eigenvalue distribution of hermitian matrices whose entries are i.i.d. Gaussian-distributed variables. We will see how many things simplify in the limit of large rank N, and why that is analogous to a „classical limit" in physical theories. We then introduce the basics of a diagrammatic calculus known as Feynman diagrams, allowing us to explore large N matrices beyond the Gaussian regime. It will provide a first glimpse into why theoretical physicists care about large N gauge theories, and we will briefly sketch their connection to string theory. Finally, we review recent developments in two-dimensional quantum gravity, relating matrix integrals to JT-gravity and the Weil-Petersson volumes of the moduli space of Riemann surfaces.  

References: Paper1, Paper2

Schedule: 

All lectures will be at ES635, KMI. 

• Feb. 9 (Mon)
  
  10:00 - 11:30: Mazenc 
  
  13:30 - 15:00: Giacchetto
  
  15:30 - 17:00: Chidambaram
  
  

• Feb. 10 (Tue)
  
  10:00 - 11:30: Mazenc 
  
  13:30 - 15:00: Giacchetto
  
  15:30 - 17:00: Chidambaram
  
  

• Feb. 11 (Wed)
  
  10:00 - 11:30: Mazenc 
  
  13:30 - 15:00: Giacchetto
  
  15:30 - 17:00: Chidambaram
  
  

• Feb. 12 (Thu)
  
  10:00 - 11:30: Mazenc 
  
  13:30 - 15:00: Giacchetto
  
  15:30 - 17:00: Chidambaram

Contact:

If you have any questions, please contact Kento Osuga, the organiser of KMI-Math School.

Acknowledgements:

This event is supported in part by the Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI/Flap Proposals by Young Researchers), and also in part by JSPS KAKENHI Grant-in-Aid for Early-Career Scientists (23K12968)