# Integrable Systems and Moduli Spaces (13w5064)

Arriving in Banff, Alberta Sunday, August 25 and departing Friday August 30, 2013

## Organizers

Dmitry Korotkin (Concordia University)

Peter Zograf (Steklov Mathematical Institute of the Russian Academy of Sciences)

## Objectives

The main objectives of the conference are:

- To foster interactions among the researchers working in the fields of integrable systems, algebraic geometry of moduli spaces and related areas of mathematics and mathematical physics (such as random matrices, dynamical systems, etc.);

- Based on these interactions, to design new mathematical tools and to achieve new progress in the fields of integrable systems and moduli spaces (in particular to better understand the algebro-geometric aspects of integrability and to apply this knowledge to intersection theory on various moduli spaces associated with complex algebraic curves);

- To use this conference as an opportunity to involve younger researchers into discussions with senior mathematicians, which will be beneficial for their scientific development and may lead to new fresh ideas.

The following senior researchers from the enclosed list have indicated strong support of the idea of the conference and considerable interest to participate in it, if approved. In alphabetical order by the last name, they are:

Arnaud Beauville, University of Nice

Boris Dubrovin, SISSA

Alex Eskin, University of Chicago

Bertrand Eynard, Institut de Physique Th'eorique, CEA-Saclay

John Harnad, Concordia University

Nigel Hitchin, Oxford University

Jacques Hurtubise, McGill University

Gerard van der Geer, University of Amsterdam

Alexander Its, IUPUI

Igor Krichever, Columbia University

Eduard Looijenga, Utrecht University

Maryam Mirzakhani, Stanford University

Motohiko Mulase, UC Davis

Andrei Okounkov, Columbia University

Emma Previato, Boston University

Leon Takhtajan, Stony Brook University

Ravi Vakil, Stanford University

Richard Wentworth, University of Maryland

Roughly speaking, the complete list of proposed participants (see below) can be split into the following (partially overlapping) categories:

- Experts in integrable systems with solid background in algebraic and/or differential geometry (Bertola, Chekhov, Dubrovin, Eynard, Grava, Harnad, Its, Korotkin, Krichever, Mazzocco, Orantin, Smirnov, Takhtajan);

- Algebraic and/or differential geometers working in the area of moduli spaces with strong interest in integrable systems (Arbarello, Beauville, Cornalba, van der Geer, Grushevsky, Hitchin, Hurtubise, de Jong, Kazarian, Kouvidakis, Looijenga, Mulase, Previato, Salvati Manni, Shapiro, Vakil, Wentworth, Zograf, Zvonkine).

- Specialists in dynamical systems with high level of expertise in the geometry of moduli spaces (Eskin, Forni, Mirzakhani, Okounkov, Zorich).

The core topics of the workshop are:

1. Applications of integrable systems to the geometry of moduli spaces.

The moduli spaces of interest at the workshop are mainly associated with complex algebraic curves: first of all, these are moduli spaces of (pointed) curves, moduli spaces of meromorphic functions on curves (Hurwitz spaces and spaces of admissible covers) and, more generally, moduli spaces of stable maps, moduli spaces of holomorphic and meromorphic differentials, moduli spaces of holomorphic vector bundles on curves, moduli of abelian varieties, etc. These moduli spaces naturally arise in the study of various integrable hierarchies, like Korteveg-deVries, Kadomtsev-Petviashvili, Toda, Witham and many other systems of partial differential equations, as well as finite-dimensional integrable systems (Hitchin systems). The machinery of integrable systems often helps to shed a new light on difficult problems in algebraic geometry of moduli spaces. The main emphasis will be made on the intersection theory of moduli spaces: geometric realization of algebraic cycles on moduli spaces, relations in the Chow ring of algebraic cycles, recursive and explicit description of intersection numbers on moduli spaces and their large genus asymptotic. Some of the most important applications of that kind are mentioned in Section 1 of this proposal. An explicit computation by van der Geer-Kouvidakis of the class of the Hurwitz divisor on the moduli space of curves of even genus that utilizes a formula for the tau function divisor by Kokotov-Korotkin-Zograf is yet another recent example of such an interaction.

Let us also mention the Schottky problem of characterizing the Jacobian locus in the moduli space of abelian varieties. The solution of the Schottky problem via the theory of the KP equation (Novikov's conjecture proved by Shiota) is rather implicit. A more explicit approach has been developed recently (Grushevsky-Krichever).

We expect some progress here during the workshop as well.

2. Random matrices and moduli spaces.

Since the pioneering idea of Mumford on the cell decomposition of the moduli space of curves and a computation of its Euler characteristic by Harer-Zagier, the importance of matrix integrals in the theory of moduli spaces became quite transparent. Kontsevich's proof of Witten's conjecture on the intersection numbers of tautological classes on moduli space of curves came as a triumph of matrix integration (an independent subsequent proof by Okounkov-Pandharipande also used a matrix integral, but of a different kind). Starting from random matrix theory, Eynard-Orantin recently proposed a general scheme of obtaining topological recursions for intersection numbers that unifies all previously known cases -- intersection numbers of tautological classes, Hurwitz numbers, Weil-Petersson volumes, and some other Gromov-Witten invariants. Elucidation of the universal role of the topological recursion relations (``loop equations'') in the theory of moduli spaces, their relationships with Virasoro constraints and other computational methods will be one of the primary objectives of the workshop.

Conversely, we also hope that the appearance of various moduli spaces in the study of large N limit of random matrix models can be explained in the framework of recent results by Bertola, Dubrovin, Harnad, Hurtubise and others.

3. Frobenius manifolds and moduli spaces.

Frobenius manifolds were introduced around 1990 by Dubrovin as a geometrization of quantum cohomology that originated from Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) associativity equation in topological field theories. Frobenius manifolds provide a useful link between integrable systems (due to their relationship to isomonodromic deformations and matrix Riemann-Hilbert problem) and geometry. In particular, Frobenius structures naturally arise on various moduli spaces (the simplest examples are Hurwitz spaces of branched covers of a projective line).

This relationship proved to be quite useful in many respects (cf., e.g., the above mentioned application of the isomonodromic tau function to the explicit geometric realization of the Hurwitz divisor on the moduli space of curves). We expect that the interaction between the workshop participants can lead to further progress in understanding the role of Frobenius manifolds in algebraic geometry of various moduli spaces.

4. Dynamical systems and moduli spaces.

Moduli spaces naturally arise in the study of many dynamical systems. Moreover, these two fields experience mutual influence in both directions that often involves integrable systems as well. Say, the dynamics of flat billiards is closely related to the behavior of the Teichm"uller flow on the moduli space of abelian and quadratic differentials on algebraic curves. The sum of Lyapunov exponents for the latter flow has an interpretation as a ratio of two intersection numbers (Kontsevich-Zorich), and the properties of the tau function on the moduli spaces of differentials may help to prove rationality of this ratio (Eskin-Kontsevich-Zorich, Chen, Korotkin-Zograf). A remarkable work of Mirzakhani gives an example of an opposite type. From the behavior of the hyperbolic geodesic flow she derived the Virasoro constraints for the Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces that, in particular, implied Witten's conjecture. We truly believe that the interactions between the experts in dynamics, integrable systems and algebraic geometry may appear quite fruitful.

We hope that the conference centered at these topics will be of high interest and will attract both leading senior specialists and talented young researchers, will stimulate interdisciplinary contacts and will generate new successful ideas and directions of research. In order to stress the anticipated interest in the workshop, we include below a partial list of people who we believe would be much interested in attending and would greatly contribute to the workshop success. We would also like to keep a few spots for postdocs, junior researchers and ''last minute" applicants.

- To foster interactions among the researchers working in the fields of integrable systems, algebraic geometry of moduli spaces and related areas of mathematics and mathematical physics (such as random matrices, dynamical systems, etc.);

- Based on these interactions, to design new mathematical tools and to achieve new progress in the fields of integrable systems and moduli spaces (in particular to better understand the algebro-geometric aspects of integrability and to apply this knowledge to intersection theory on various moduli spaces associated with complex algebraic curves);

- To use this conference as an opportunity to involve younger researchers into discussions with senior mathematicians, which will be beneficial for their scientific development and may lead to new fresh ideas.

The following senior researchers from the enclosed list have indicated strong support of the idea of the conference and considerable interest to participate in it, if approved. In alphabetical order by the last name, they are:

Arnaud Beauville, University of Nice

Boris Dubrovin, SISSA

Alex Eskin, University of Chicago

Bertrand Eynard, Institut de Physique Th'eorique, CEA-Saclay

John Harnad, Concordia University

Nigel Hitchin, Oxford University

Jacques Hurtubise, McGill University

Gerard van der Geer, University of Amsterdam

Alexander Its, IUPUI

Igor Krichever, Columbia University

Eduard Looijenga, Utrecht University

Maryam Mirzakhani, Stanford University

Motohiko Mulase, UC Davis

Andrei Okounkov, Columbia University

Emma Previato, Boston University

Leon Takhtajan, Stony Brook University

Ravi Vakil, Stanford University

Richard Wentworth, University of Maryland

Roughly speaking, the complete list of proposed participants (see below) can be split into the following (partially overlapping) categories:

- Experts in integrable systems with solid background in algebraic and/or differential geometry (Bertola, Chekhov, Dubrovin, Eynard, Grava, Harnad, Its, Korotkin, Krichever, Mazzocco, Orantin, Smirnov, Takhtajan);

- Algebraic and/or differential geometers working in the area of moduli spaces with strong interest in integrable systems (Arbarello, Beauville, Cornalba, van der Geer, Grushevsky, Hitchin, Hurtubise, de Jong, Kazarian, Kouvidakis, Looijenga, Mulase, Previato, Salvati Manni, Shapiro, Vakil, Wentworth, Zograf, Zvonkine).

- Specialists in dynamical systems with high level of expertise in the geometry of moduli spaces (Eskin, Forni, Mirzakhani, Okounkov, Zorich).

The core topics of the workshop are:

1. Applications of integrable systems to the geometry of moduli spaces.

The moduli spaces of interest at the workshop are mainly associated with complex algebraic curves: first of all, these are moduli spaces of (pointed) curves, moduli spaces of meromorphic functions on curves (Hurwitz spaces and spaces of admissible covers) and, more generally, moduli spaces of stable maps, moduli spaces of holomorphic and meromorphic differentials, moduli spaces of holomorphic vector bundles on curves, moduli of abelian varieties, etc. These moduli spaces naturally arise in the study of various integrable hierarchies, like Korteveg-deVries, Kadomtsev-Petviashvili, Toda, Witham and many other systems of partial differential equations, as well as finite-dimensional integrable systems (Hitchin systems). The machinery of integrable systems often helps to shed a new light on difficult problems in algebraic geometry of moduli spaces. The main emphasis will be made on the intersection theory of moduli spaces: geometric realization of algebraic cycles on moduli spaces, relations in the Chow ring of algebraic cycles, recursive and explicit description of intersection numbers on moduli spaces and their large genus asymptotic. Some of the most important applications of that kind are mentioned in Section 1 of this proposal. An explicit computation by van der Geer-Kouvidakis of the class of the Hurwitz divisor on the moduli space of curves of even genus that utilizes a formula for the tau function divisor by Kokotov-Korotkin-Zograf is yet another recent example of such an interaction.

Let us also mention the Schottky problem of characterizing the Jacobian locus in the moduli space of abelian varieties. The solution of the Schottky problem via the theory of the KP equation (Novikov's conjecture proved by Shiota) is rather implicit. A more explicit approach has been developed recently (Grushevsky-Krichever).

We expect some progress here during the workshop as well.

2. Random matrices and moduli spaces.

Since the pioneering idea of Mumford on the cell decomposition of the moduli space of curves and a computation of its Euler characteristic by Harer-Zagier, the importance of matrix integrals in the theory of moduli spaces became quite transparent. Kontsevich's proof of Witten's conjecture on the intersection numbers of tautological classes on moduli space of curves came as a triumph of matrix integration (an independent subsequent proof by Okounkov-Pandharipande also used a matrix integral, but of a different kind). Starting from random matrix theory, Eynard-Orantin recently proposed a general scheme of obtaining topological recursions for intersection numbers that unifies all previously known cases -- intersection numbers of tautological classes, Hurwitz numbers, Weil-Petersson volumes, and some other Gromov-Witten invariants. Elucidation of the universal role of the topological recursion relations (``loop equations'') in the theory of moduli spaces, their relationships with Virasoro constraints and other computational methods will be one of the primary objectives of the workshop.

Conversely, we also hope that the appearance of various moduli spaces in the study of large N limit of random matrix models can be explained in the framework of recent results by Bertola, Dubrovin, Harnad, Hurtubise and others.

3. Frobenius manifolds and moduli spaces.

Frobenius manifolds were introduced around 1990 by Dubrovin as a geometrization of quantum cohomology that originated from Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) associativity equation in topological field theories. Frobenius manifolds provide a useful link between integrable systems (due to their relationship to isomonodromic deformations and matrix Riemann-Hilbert problem) and geometry. In particular, Frobenius structures naturally arise on various moduli spaces (the simplest examples are Hurwitz spaces of branched covers of a projective line).

This relationship proved to be quite useful in many respects (cf., e.g., the above mentioned application of the isomonodromic tau function to the explicit geometric realization of the Hurwitz divisor on the moduli space of curves). We expect that the interaction between the workshop participants can lead to further progress in understanding the role of Frobenius manifolds in algebraic geometry of various moduli spaces.

4. Dynamical systems and moduli spaces.

Moduli spaces naturally arise in the study of many dynamical systems. Moreover, these two fields experience mutual influence in both directions that often involves integrable systems as well. Say, the dynamics of flat billiards is closely related to the behavior of the Teichm"uller flow on the moduli space of abelian and quadratic differentials on algebraic curves. The sum of Lyapunov exponents for the latter flow has an interpretation as a ratio of two intersection numbers (Kontsevich-Zorich), and the properties of the tau function on the moduli spaces of differentials may help to prove rationality of this ratio (Eskin-Kontsevich-Zorich, Chen, Korotkin-Zograf). A remarkable work of Mirzakhani gives an example of an opposite type. From the behavior of the hyperbolic geodesic flow she derived the Virasoro constraints for the Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces that, in particular, implied Witten's conjecture. We truly believe that the interactions between the experts in dynamics, integrable systems and algebraic geometry may appear quite fruitful.

We hope that the conference centered at these topics will be of high interest and will attract both leading senior specialists and talented young researchers, will stimulate interdisciplinary contacts and will generate new successful ideas and directions of research. In order to stress the anticipated interest in the workshop, we include below a partial list of people who we believe would be much interested in attending and would greatly contribute to the workshop success. We would also like to keep a few spots for postdocs, junior researchers and ''last minute" applicants.