The computation and enumeration of invariants of moduli spaces took a sudden turn with the conjecture of Witten that they could be combined into a formal series that solved the KdV hierarchy. This conjecture, subsequently proven by Kontsevich, was motivated by considerations of quantum gravity. It was followed by a series of developments in the same direction, notably in the computation of invariants for Hurwitz spaces and for Gromov-Witten invariants, for example in the work of Okounkov and Pandharipande, tying Gromov-Witten theory to the 2-Toda hierarchy. Kontsevichâs proof involved a detour through the theory of random matrices, and subsequently Eynard and Orantin proposed a vast generalization of the technique, with a wide variety of implications. The questions have physical motivations, and has advanced with the rapid mixture of calculation and heuristic reasoning which characterizes theoretical physics; mathematicians have in many but not all cases provided proof, and, it is hoped, some understanding.
Thus a first piece of the puzzle is the theory of moduli spaces. The moduli spaces of interest 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. 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 asymptotics. The paradigm of integrable systems continues to extend its influence: 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 gives yet another example.
A second part of the puzzle, brought to the fore most recently by Eynard and Orantin, lies in the theory of random matrices. The interaction with moduli goes back, and has a long history, with pioneers including not only Kontsevich but also the Saclay group of Itzykson and Zuber. 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; in each case, the generating object is a single plane curve. Elucidation of the universal role of the topological recursion relations (âloop equationsâ) 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.
A third theme comes with the theory of 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).
Dynamical systems seem also to play a role: for example, the dynamics of flat billiards is closely related to the behaviour 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.
The theory of integrable systems seems to lie at the heart of the subject, providing a thematic link. It is fair to say, though, that the way in which it happens is still ill-understood. Indeed, so far, it is more the tools, computational devices, and actual functions that appear, rather than flows and conserved quantities. It is hoped that recent physical input, again from the theory of quantum gravity, will help develop understanding.
Participants will include a sampling of theoretical physicists, mathematical physicists, geometers, and experts in random matrices. It is hoped that the interactions will shed some light on a long standing puzzle, at the core of some very beautiful mathematics.