Integrability, Enumerative Geometry and Quantization: August 22nd – September 23rd, 2022

Participant ListApply Now

Organized by:

Gaëtan Borot (Humboldt University of Berlin, Germany),

Alexandr Buryak (Higher School of Economics, Russia), 

Chiu-Chu Melissa Liu (Columbia University),

Nikita Nekrasov (SCGP),

Paul Norbury (University of Melbourne, Australia),

Paolo Rossi (University of Padua, Italy)

Talk Schedule

Time Title Speaker Location
Monday August 22
10:45am

GW/Hurwitz correspondence for curves

Abstract

Alexei Oblomkov SCGP 313
4:00pm

Counting curves in 1, 3, and 5 dimensions

Notes

Andrei Okounkov SCGP 313
Tuesday August 23
10:45am

Equivariant GW theory of P^1

Abstract

Alexei Oblomkov SCGP 313
4:00pm

Counting curves in 1, 3, and 5 dimensions

Notes

Andrei Okounkov SCGP 313
Wednesday August 24
10:45am

Equivariant GW theory of P^1

Abstract

Alexei Oblomkov SCGP 313
2:00pm

Counting curves in 1, 3, and 5 dimensions

Notes

Andrei Okounkov SCGP 313
4:00pm Discussion Session 1 Henry Liu SCGP 313
Thursday August 25
10:45am

Descendent GW/PT correspondence via vertex operators

Abstract

Alexei Oblomkov SCGP 313
4:00pm

Counting curves in 1, 3, and 5 dimensions

Notes

Andrei Okounkov SCGP 313
Friday August 26
10:45am Discussion Session 2 Henry Liu SCGP 313
2:00pm

Counting curves in 1, 3, and 5 dimensions

Notes

Andrei Okounkov SCGP 313

Time Title Speaker Location
Monday August 29
9:30am

Examples of relative quantum cohomology

Abstract

Solomon SCGP 102
10:30am Coffee Break   SCGP Cafe
11:00am

Quantum spectrum and Gamma structures for FJRW theory of general type

Abstract

Shen SCGP 102
12:00pm Lunch   SCGP Cafe
2:30pm

Symplectic groupoid and cluster algebra description of closed Riemann surfaces

Abstract

Chekhov SCGP 102
Tuesday August 30
9:30am

KP integrability of triple Hodge integrals

Abstract

Slides

Alexandrov SCGP 102
10:30am Coffee Break   SCGP Cafe
11:00am

Logarithmic double ramification cycles

Abstract

Slides

Schmitt SCGP 102
12:00pm Lunch   SCGP Cafe
      SCGP 102
Wednesday August 31
9:30am

Enumeration of hypermaps, rationally constrained KP hierarchies and the Givental-Milanov-Tseng approach to their Hirota equations

Abstract

Carlet SCGP 102
10:30am Coffee Break   SCGP Cafe
11:00am

Symplectic duality for topological recursion
a joint work with Bychkov, Dunin-Barkowski, Kazarian
https://arxiv.org/abs/2206.14792

Abstract

Shadrin SCGP 102
12:00pm Lunch   SCGP Cafe
       
Thursday September 1
9:30am

The amplituhedron and its triangulations

Abstract

Tessler SCGP 102
10:30am Coffee Break   SCGP Cafe
11:00am

Central charges and Higgs-Coulomb correspondence in abelian GLSMs

Abstract

Aleshkin SCGP 102
12:00pm Lunch   SCGP Cafe
2:30pm

Multiplicative vertex algebras and wall-crossing in equivariant K-theory

Abstract

H. Liu SCGP 102
Friday September 2
9:30am

Tautological relations and integrable systems

Abstract

Slides

Buryak SCGP 102
10:30am Coffee Break   SCGP Cafe
11:00am

Polynomial relations among kappa classes on the moduli space of stable curves

Abstract

Norbury SCGP 102
12:00pm Lunch   SCGP Cafe
2:30pm

Bethe ansatz in the geometric setting

Abstract

Zeitlin SCGP 102

Time Title Speaker Location
Tuesday September 6
9:30am

Double ramification cycles and integrable hierarchies, I

Abstract

Paolo Rossi SCGP 102
11:30am

Airy ideals and topological recursion: An investigative tool for enumerative geometry, VOAs, and gauge theories

Abstract

Vincent Bouchard SCGP 102
Wednesday September 7
9:30am

Airy ideals and topological recursion: An investigative tool for enumerative geometry, VOAs, and gauge theories

Abstract

Vincent Bouchard SCGP 102
11:30am

Double ramification cycles and integrable hierarchies, II

Abstract

Paolo Rossi SCGP 102
Thursday September 8
9:30am

Double ramification cycles and integrable hierarchies, III

Abstract

Paolo Rossi SCGP 102
11:30am

Airy ideals and topological recursion: An investigative tool for enumerative geometry, VOAs, and gauge theories

Abstract

Vincent Bouchard SCGP 102
Friday September 9
9:30am

Double ramification cycles and integrable hierarchies, IV

Slides

Sasha Buryak SCGP 102
11:30am

Airy ideals and topological recursion: An investigative tool for enumerative geometry, VOAs, and gauge theories

Abstract

Vincent Bouchard SCGP 102

Additional Talks: 
Tuesday, September 131:00 – 2:30pm, SCGP room 313: David Holmes
Title: Variations on double ramification cycles
Abstract

Tuesday, September 201:00 – 2:30pm, SCGP room 313: Song Yu

Notes by Patrick Leihttps://www.math.columbia.edu/~plei/docs/ISEG.pdf

The role played by integrable systems in enumerative geometry has been first observed by physicists at the beginning of the nineties: the initial observations by Witten on the relation between the partition function of 2d topological gravity, matrix models and intersection theory of the moduli space of curves and the Korteweg-de Vries equation has developed into a rich theory relating $(1+1)$-dimensional classical integrable systems to cohomological field theories on the moduli space of stable curves. This has, in turn, become a powerful tool for probing the topology of spaces of curves and maps to target varieties.

The last few years have seen dramatic development in our understanding of the tautological ring of the moduli space of curves, its intersection theory and the role played by natural geometric cycles therein. The study of cohomological field theories (systems of cohomology classes compatible with the strata structure of the moduli spaces) and double ramification cycles (loci of curves whose marked points support principal divisors) has provided new results both towards describing (conjecturally all) tautological relations and towards constructing and quantizing integrable field theories.

Chekhov-Eynard-Orantin topological recursion has become a unifying tool embracing intersection theory on the moduli space of curves, B-model quantization on Landau-Ginzburg models, integrable systems, where the role of infinite-dimensional symmetries such as W-algebras has been recently clarified, opening the way to understanding better its connection with 4d supersymmetric gauge theories where many of the aforementioned geometric and algebraic structures come into play.

Correspondingly, in the physics community, there has been a renewal of interest for two-dimensional quantum gravity thanks to the Sachdev-Ye-Kitaev model, its conjectured connections to Jackiw–Teitelboim gravity and the connections of the latter to the geometry of the moduli space of curves via the topological recursion and matrix models. Moreover, moduli spaces of super-Riemann surfaces, moduli spaces of Riemann surfaces with boundary and more generally open Gromov-Witten theory, which is being rigorously constructed, also appear to show beautiful connections with integrability and topological recursion.

Some of these connections may be broader than currently known. On the one hand, the appearance of more general quantum algebras (Hecke algebras, Yangians, quantum toroidal algebras, etc.) in 5d gauge theories, in the quantization of character varieties, and in matrix models is well-documented, and it would be desirable to extend the relation to topological recursion and enumerative geometry in this direction. On the other hand, in the quantization of integrable field theories coming from the double ramification cycles, the analog of Virasoro/W-algebras symmetries and the relation to physical theories remain to be explored.

As it often happens when research fields grow and specialize, the communities behind these developments tend to become distinct, especially from the geometry vs. physics viewpoint. The idea of the workshop is to bring together experts from these different communities, including several leaders of the respective fields, to exchange views and ideas and initiate fruitful collaborations.