Participant ListView Videos
Organized by:
- Qile Chen (Boston College)
- Felix Janda (University of Illinois Urbana-Champaign)
- Sheldon Katz (University of Illinois Urbana-Champaign)
- Melissa Liu (Columbia University)
- John Pardon (SCGP)
- Rachel Webb (Cornell University)
Modern curve-counting theories were in part inspired by the work of physicists yet have active lives of their own as interesting and rich mathematical notions with connections to many areas of mathematics. A plethora of enumerative invariants have been developed, including Gromov–Witten (GW) invariants, GLSM invariants, Fan–Jarvis–Ruan–Witten (FJRW) invariants, as well as variants of these curve-counting theories. Many conjectures about enumerative invariants have arisen from physics, providing both deep insight as well as strategies for effective computation. Several of these conjectures have been proven in recent years, sometimes in their original form, and other times after the conjecture has been translated into a mathematically more natural framework. This program will focus on the higher genus curve counts from multiple angles, including geometric, computational and categorical perspectives.
On the geometric side, the program will investigate various moduli spaces recently introduced as tools to understand higher genus invariants, including (but not limited to) desingularizations of moduli of stable maps, moduli of Mixed-Spin-P fields, and logarithmic gauged linear sigma models, as well as recent progress on skein-valued invariant counts of higher genus holomorphic curves with Lagrangian boundary conditions in Calabi-Yau threefolds and higher genus open BPS invariants.
On the computational side, higher genus GW invariants of Calabi–Yau threefolds are expected to satisfy universal properties such as Yamaguchi–Yau finite generation, the BCOV holomorphic anomaly equations, and the Castelnuovo bound, which have been established for quintic Calabi–Yau threefolds. Progress and difficulties in generalizing these results to other Calabi–Yau threefolds, and to more general targets, will be investigated during the program, with the participation of physicists for synergistic effect. Building on these results and other developments, the g=2 Gromov–Witten invariants of the quintic have been rigorously determined. There is much room and hope for further exciting progress, as physicists have a prediction for the Gromov–Witten invariants of the quintic up to genus 64.
Most of the above enumerative invariants are expected to agree with the corresponding enumerative categorical invariants constructed by Caldararu–Tu: the Fukaya category of a symplectic manifold for GW theory; the wrapped Fukaya category of a symplectic Landau-Ginzburg model for FJRW theory; the derived category of a Calabi–Yau manifold for BCOV theory; the category of matrix factorizations for B-model FJRW theory. From this point of view, categorical/homological mirror symmetry (which can be viewed as a version of genus-zero open mirror symmetry) implies enumerative mirror symmetry for all genera. The main challenge lies in identifying categorical invariants with geometric ones, which is one of the main themes of this program.
WEEK 1: JANUARY 6-10, 2025
Topic: Higher genus GW invariants and FJRW invariants
Mini-course #1
Title: Higher genus computations via logarithmic Gauged Linear Sigma Models
Speakers: Qile Chen, Shuai Guo, and Felix Janda
Abstract: Log Gauged Linear Sigma Models (log GLSM) form a new geometric technique for computations in higher genus Gromov-Witten theory.In the first part of the lectures, we discuss the geometry of the moduli spaces that form the foundations of log GLSM, including two structural “tropical decomposition formula” and “localization formula”.
In the second part of the lectures, we discuss how to apply log GLSM to computations in higher genus Gromov-Witten theory, such as to the fundamental structural predictions for higher genus Gromov-Witten invariants of quintic threefolds (including holomorphic anomaly equations) and the LG/CY correspondence.
Mini-Course:
Monday, 1/6/25 9:30am – Seminar Room 313
Speaker: Qile Chen
Title: Higher genus computations via logarithmic Gauged Linear Sigma Models (1)
Monday, 1/6/25 11:30am – Seminar Room 313
Speaker: Patrick Lei
Title: Axiomatic approach to enumerative geometry: Lagrangian cone, CohFTs, and R-matrix actions (1)
Abstract: In these two lectures, I will explain an axiomatic approach to Gromov-Witten type enumerative theories constructed by Kontsevich-Manin, Givental, and others. A key insight is that there is an action of the symplectic loop group on CohFTs such that the effect on generating functions is given by geometric quantization of symplectic matrices, which can be computed combinatorially as a sum over stable graphs.
Mini-Course:
Tuesday, 1/7/25 9:30am – Seminar Room 313
Speaker: Qile Chen
Title: Higher genus computations via logarithmic Gauged Linear Sigma Models (2)
Tuesday, 1/7/25 11:30am – Seminar Room 313
Speaker: Patrick Lei
Title: Axiomatic approach to enumerative geometry: Lagrangian cone, CohFTs, and R-matrix actions (2)
Abstract: In these two lectures, I will explain an axiomatic approach to Gromov-Witten type enumerative theories constructed by Kontsevich-Manin, Givental, and others. A key insight is that there is an action of the symplectic loop group on CohFTs such that the effect on generating functions is given by geometric quantization of symplectic matrices, which can be computed combinatorially as a sum over stable graphs.
Mini-Course:
Wednesday, 1/8/25 9:30am – Seminar Room 313
Speaker: Qile Chen
Title: Higher genus computations via logarithmic Gauged Linear Sigma Models (3)
Thursday, 1/9/25 9:30am – Seminar Room 313
Speaker: Albrecht Klemm
Title: Introduction to the topological B-model: Higher genus calculations, Modularity and Stability considerations and non perturbative completions (1)
Mini-Course:
Thursday, 1/9/25 11:30am – Seminar Room 313
Speaker: Felix Janda
Title: Higher genus computations via logarithmic Gauged Linear Sigma Models (4)
Friday, 1/10/25 9:30am – Seminar Room 313
Speaker: Albrecht Klemm
Title: Introduction to the topological B-model: Higher genus calculations, Modularity and Stability considerations and non perturbative completions (2)
Topic: Higher genus GW invariants and FJRW invariants
Mini-course #1
Title: Higher genus computations via logarithmic Gauged Linear Sigma Models
Speakers: Qile Chen, Shuai Guo, and Felix Janda
Abstract: Log Gauged Linear Sigma Models (log GLSM) form a new geometric technique for computations in higher genus Gromov-Witten theory.In the first part of the lectures, we discuss the geometry of the moduli spaces that form the foundations of log GLSM, including two structural “tropical decomposition formula” and “localization formula”.
In the second part of the lectures, we discuss how to apply log GLSM to computations in higher genus Gromov-Witten theory, such as to the fundamental structural predictions for higher genus Gromov-Witten invariants of quintic threefolds (including holomorphic anomaly equations) and the LG/CY correspondence.
Mini-course #2
Title: See BCOV from A side: Mixed-Spin-P fields
Speakers: Huai-Liang Chang, Shuai Guo, Wei-Ping Li, and Yang Zhou
Mini-course #3
Title: Reduced Gromov-Witten invariants in higher genus via desingularizations of sheaves
Speakers: Alberto Cobos Ranabo, Etienne Mann, Cristina Manolache, and Renata Picciotto
Abstract: The moduli space of stable maps, the basic object of study in Gromov–Witten theory, has a rich geometry, with components parametrizing very degenerate stable maps. The existence of these excess components is a very consequential aspect of the theory, which can be leveraged to obtain recursive formulae for Gromov–Witten invariants.
Reduced Gromov–Witten invariants, which are associated to a moduli space of smoothable stable maps, have been intensely studied since the seminal works of Vakil–Zinger, Li–Zinger and Zinger, who provided a mathematical construction in genus 1 and verified the Bershadsky-Cecotti Ooguri-Vafa (BCOV) conjecture for the quintic threefold.
In this 4 lecture mini-course we will discuss our construction of reduced Gromov–Witten invariants in arbitrary genus. We construct this theory for targets which are complete intersections in projective space, as well as for a large class of GIT quotients. The main tools involved are desingularizations of coherent sheaves. We will also discuss some ideas for further work towards proving recursive formulae relating reduced and absolute GW invariants.
Tuesday, 1/21/25 11:15am – Seminar Room 313
Speaker: Renata Picciotto
Title: Moduli space of stable maps, components and conjectures
Abstract: In this talk, I will briefly review some aspects of the geometry of the moduli space of stable maps and its components. I will mention some of the conjectured recursive formulae involving reduced Gromov–Witten theory,
which will be discussed in more detail in Talk 4. Mathematically, the construction of reduced GW invariants (i.e. invariants capturing only smoothable stable maps) starts with the celebrated Vakil–Zinger desingularization in genus 1 and its study in the algebraic context by Li–Zinger. There has been a rich study of reduced Gromov–Witten invariants in genus 1 and 2, of which I will give a short overview. The starting point for our all-genus construction of reduced GW invariants is to relate this problem to the classical one of flattening of coherent sheaves, which we will discuss in details in Talk 2. I will give an overview of the ideas involved.
Wednesday, 1/22/25 11:15am – Seminar Room 313
Speaker: Yi Hu
Title:Derived and modular resolution of moduli of higher genus stable maps and applications
Thursday, 1/23/25 11:15am – Seminar Room 313
Speaker: Alberto Cobos Rabano
Title: Desingularization of coherent sheaves
Abstract: In this talk, I will explain the main technical results which underlie our definition of reduced Gromov–Witten invariants, namely desingularizations and diagonalizations of coherent sheaves. Their study has appeared
prominently in the context of resolution of singularities, where they take various names. I will start by presenting the geometric construction of Rossi and its algebraic generalizations by Villamayor, Raynaud-Gruson, Raynaud and others. I will explain how, in some contexts, these constructions satisfy a universal property akin to that of the blow-up of an ideal sheaf. This allows us to glue local constructions over certain carefully chosen Artin stacks. All of these constructions take as input a coherent sheaf and produce birational models for their cones (total spaces) in which either the main
component or all of the components are relatively smooth.
Monday, 1/27/25 11:15am – Seminar Room 313
Speaker: Etienne Mann
Title: Reduced Gromov–Witten invariants via desingularization of sheaves
Abstract: In this talk, I will explain how the technical constructions illustrated in Talk 2 allow us to define virtual fundamental classes supported on (unions of) components of various moduli spaces. The class supported on the main component is our proposed definition of reduced Gromov–Witten invariants. I will construct reduced invariants for complete intersections in various GIT quotients. In the case of a complete intersection projective variety, reduced GW invariants satisfy the Quantum–Lefschetz formula of Kim–Kresch–Pantev component-wise. I will discuss to what extent these constructions are independent of the choices involved (e.g. of the desingularization, or of projective embeddings of the target variety). Owing to the universal properties discussed in the previous talk, our constructions satisfy some minimality properties which allow us to relate our definition of reduced GW invariants to previous ones. I will compare the various constructions in genus 1, where the work of Hu–Li gives us explicit local equations for the moduli spaces.
Wednesday, 1/29/25 11:15am – Seminar Room 313
Speaker: Cristina Manolache
Title: Strategy for the absolute vs reduced GW Conjecture for 3-folds
Abstract: n this talk, I will arch back to some of the conjectures on recursion for reduced Gromov–Witten invariants which were mentioned in Talk 1. I will discuss some of the strategies that have been employed to prove them. In particular, I will be discussing the approach taken by Chang–Li in genus 1 and how this may be generalized.
WEEK 5: February 3-7, 2025
Mini-course #4
Title: Skein valued curve counts
Speakers: Tobias Ekholm and Vivek Shende
Abstract: We present a rigorous mathematical formalism for counting holomorphic curves with Lagrangian boundary in Calabi-Yau 3-folds. We will sketch the foundations of the theory and give various extended examples. The four lectures will cover
(1) Skein-valued curve counting and the Ooguri-Vafa conjecture
(2) The topological vertex
(3) Nonabelianization and skein-valued cluster algebra
(4) Foundations
Monday, 2/3/25 11:15am – Seminar Room 313
Speaker: Vivek Shende
Title: Skein valued curve counts (1): Skein-valued curve counting and the Ooguri-Vafa conjecture
Tuesday, 2/4/25 11:15am – Seminar Room 313
Speaker: Tobias Ekholm
Title: Skein valued curve counts (2): The topological vertex
Wednesday, 2/5/25 11:15am – Seminar Room 313
Speaker: Vivek Shende
Title: Skein valued curve counts (3): Nonabelianization and skein-valued cluster algebra
Thursday, 2/6/25 11:15am – Seminar Room 313
Speaker: Tobias Ekholm
Title: Skein valued curve counts (4): Foundations
Thursday, 2/6/25 2:00pm – Seminar Room 313
Speaker: MIngyuan Hu
Title: Skein valued cluster theory and Ekholm-Shende wavefunctions
Abstract: We consider a class of Lagrangians living in $\bC^3$, which are asymptotic fillings of certain Legendrian surfaces, generalizing the Aganagic-Vafa brane. Their Ekholm-Shende wavefunctions satisfy some skein valued equations. We develop a skein valued cluster theory to solve these equations, hence compute the Ekholm-Shende wavefunctions. After a rank one reduction, this skein cluster theory will recover the quantum cluster theory. In the case of Aganagic-Vafa brane, our computation matches up with the topological vertex. We also prove a pentagon relation for the skein dilogarithms. which will imply the 5-term relation of Garsia and Mellit, originally formulated in terms of Macdonald operators. This talk is partially based on joint works with Gus Schrader and Eric Zaslow. Some of the results are also independently obtained by Scharitzer-Shende and Nakamura.
Topic: Workshop – Recent developments in higher genus curve counting: February 10-14, 2025
WEEK 7 & 8: February 17-28, 2025
Topic: Categorical enumerative invariants
Mini-course “Higher genus enumerative invariants from Calabi-Yau categories” by Andrei Căldăraru and Junwu Tu (Week 7)
