Conference Programme

M. Mulase: Quantum curves for Hitchin fibrations

Quantum curves appearing in topological string theory seem to have a magical power of producing quantum invariants of various kinds. From a purely mathematical point of view, however, so far only a handful examples have been rigorously constructed, which are related to Hurwitz numbers and more generalized counting problems of coverings of a sphere.

In this talk I will present a mechanism of constructing the D-module of a spectral curve in the theory of Hitchin fibration, utilizing the Eynard-Orantin theory. This D-module is the quantum curve in the physics literature. From this point of view, the Eynard-Orantin integral recursion theory provides a method of constructing a generator of the D-module on an arbitrary base curve whose characteristic variety is the Hitchin spectral curve. The talk is based on the speaker's joint work with O. Dumitrescu.

Y. Lee: Functoriality of Gromov-Witten theory

I will explain two (new) techniques involved in Gromov--Witten theory (GWT) in the context of invariance of GWT under ordinary flops.

1. Quantum Leray-Hirsch: Lifting the quantum D-modules (in GWT) from the base to the bundle, for any toric bundle over an arbitrary base symplectic manifold.

2. Algebraic cobordism: Degenerating a non-split vector bundle to a split one in such a way that GWT of the former is reduced to the GWT of the latter, which can then be tackled via (fiber-wise) localization.

This is an on-going joint project with H.-W. Lin, C.-L. Wang and F. Qu.

D. Zvonkine: The A_2 Frobenius manifold, Witten's class, and relations in cohomology of Mbar_{g,n}.

Download presentation.

Denote by M_g the moduli space of smooth genus g compex curves. As g tends to infinitey, its cohomology ring stabilizes to Q[kappa_1, kappa_2, ...], where the kappa_m are the Mumford-Morita-Miller classes. For a fixed genus, however, there are polynomial relations between the classes kappa_m. The study of these relations was initiated by C. Faber in 1993 and is still not complete 20 years later.

One can consider a more difficult problem by putting marked points on the curve and passing to the Deligne-Mumford compactification of the moduli space. The number of both standard cohomology classes and relations between them increases significantly.

We construct a large family of relations between the strandard (so called "tautological") classes of the cohomology ring of Mbar_{g,n}. It includes all known relations and is conjecturally complete. The construction uses the Frobenius manifold A_2. More precisely, we study the behavior of the corresponding cohomological field theory (Witten's class) in the neighborhood of the discriminant.

Joint with A. Pixton et R. Pandharipande.

A. Takahashi: Orbifold projective lines and extended cuspidal Weyl groups

Download presentation.

We report on our recent study on a correspondence among orbifold projective lines, cusp singularities and cuspidal root systems. In particular, we discuss an isomorphism of Frobenius manifolds between the one from the Gromov-Witten theory for an orbifold projective line and the one associated to the invariant theory of an extended cuspidal Weyl group.

P. Rossi: Cohomological field theories with boundary.

Cohomological field theories were introduced by Kontsevich and Manin to encode and axiomatize the algebraic structure of Gromov-Witten invariants. They consist in a system of cohomology classes on the Deligne-Mumford moduli space of curves satisfying some properties and are known to produce (and in some measure be classified) by Frobenius manifolds. There are various natural bordifications of the Deligne-Mumford moduli space of curves obtained by different (real oriented) blowups of the nodal divisors and their intersections. After reviewing Kontsevich and Manin's definition, we introduce a version of cohomological field theories on these bordifications and analyze the ensuing algebraic structure.

O. Fabert: Mirror symmetry for open Calabi-Yau manifolds.

The classical mirror symmetry conjecture for closed Calabi-Yau manifolds can be formulated as an isomorphism of Frobenius manifolds, involving the Frobenius manifold structure on quantum cohomology on the symplectic side. Generalizing from closed to open symplectic manifolds, it is known that quantum cohomology needs to be replaced by symplectic cohomology, defined using Floer theory. In my talk I will show that the symplectic cohomology of an open symplectic manifold can be equipped with the structure of a cohomology F-manifold, a differential graded version of a Frobenius manifold. Since this indeed fits nicely with the expectations from mirror symmetry, I will formulate a classical mirror symmetry conjecture for open Calabi-Yau manifolds.

A. Buryak: Dubrovin-Zhang hierarchy for Hodge integralscontent.

To a semisimple Cohomological Field Theory one can associate the Dubrovin-Zhang hierarchy, that is a Hamiltonian hierarchy of PDEs. In the talk I will discuss the hierarchy associated to the CohFT formed by the Hodge classes. This hierarchy is a certain deformation of the KdV hierarchy.

Y. Zhang: On a certain generalization of Virasoro constraints for Frobenius manifolds

Download presentation.

We show that the Virasoro operators associated to an arbitray Frobenius manifold admit certain deformations which give additional constraints for the genus zero free energy of the Frobenius manifold. We discuss applications of such additional constraints and their analogues for higher genus free energies.

B. Bakalov: Kac-Wakimoto hierarchies and W-symmetries

We investigate the Kac-Wakimoto hierarchies using the language of vertex algebras. One application is the proof of Virasoro symmetries of these hierarchies. For the case of types sl(N) and so(2N), we realize the Kac-Wakimoto hierarchies as reductions of the N-component KP and DKP hierarchies, respectively. In these cases, we prove that the corresponding W-algebras act as symmetries.

F. Rodriguez Villegas: Distribution of Betti numbers

S. Shadrin: Local topological recursion and Givental theory

I'll explain a precise correspondence between two different theories: the Chekhov-Eynard-Orantin topological recursion, coming from the change of variables in a particular type of matrix models, and the Givental formula for the so-called ancestor potential that is a key ingredient in the theory of genus expansions of Frobenius manifolds.

There are many applications of this correspondence, and, if time permits, I'll try to explain how one can obtain some formulas for the usual Hurwitz numbers and Hurwitz numbers with completed cycles in terms of the intersection theory of the moduli space of curves (the ELSV formula, and Zvonkine's conjectural r-ELSV formula).

The talk will be based on joint works with P. Dunin-Barkowski, M. Kazarian, N. Orantin, L. Spitz, and D. Zvonkine.

H. Iritani: Fock Space of Givental Quantization

In this talk I will describe a joint work with Tom Coates (Imperial College London) where we construct a sheaf of Fock spaces for a variation of "semi-infinite" Hodge structures. This gives a framework to describe a global analytic behaviour of higher genus Gromov-Witten potentials.

S. Natanzon: Symmetric solutions of Toda hierarchy and Hurwitz numbers.

Download presentation.

We explicitly construct the series expansion for a certain class of solutions to the 2D Toda hierarchy in the zero dispersion limit, which we call symmetric solutions. We express the Taylor coefficients through some universal combinatorial constants and find recurrence relations for them. These results are used to obtain new formulas for the genus 0 double Hurwitz numbers. The talk is based on joint work with Anton Zabrodin

A. Zabrodin: Laplacian growth and Hurwitz numbers

Download presentation.

We report on the integrable structure of the 2D growth problems of Laplacian type with zero surface tension. The most familiar examples are the growth problems in the plane and in an infinite channel with periodic boundary conditions in the transverse direction. These problems can be embedded into the 2D Toda lattice hierarchy in the zero dispersion limit. We characterize the corresponding solutions of the hierarchy by the string equations and construct dispersionless tau-functions for these solutions. The Taylor coefficients of the tau-functions are shown to be given by double Hurwitz numbers counting connected ramified coverings of the 2D sphere of a certain ramification type.

K. McLaughlin: 1. Random matrices: a brief intro. 2. Asymptotics of Taylor polynomials and the normal matrix model.

Download presentation.

P. Miller: The semiclassical sine-Gordon equation and rational solutions of Painlevé-II

Download presentation.

We formulate and study a class of initial-value problems for the sine-Gordon equation in the semiclassical limit. The initial data parametrizes a curve in the phase portrait of the simple pendulum, and near points where the curve crosses the separatrix a double-scaling limit reveals a universal wave pattern constructed of superluminal kinks located in the space-time along the real graphs of all of the rational solutions of the inhomogeneous Painlev\'e-II equation. The kinks collide at the real poles, and there the solution is locally described in terms of certain double-kink exact solutions of sine-Gordon.

This study naturally leads to the question of the large-degree asymptotics of the rational solutions of Painlevé-II themselves. In the time remaining we will describe recent results in this direction, including a formula for the boundary of the pole-free region, strong asymptotics valid also near poles, a weak limit formula, and planar and linear densities of complex and real poles. This is joint work with Robert Buckingham (Cincinnati).

T. Claeys: Random matrices with equispaced external source

Download presentation.

I will talk about random matrix ensembles with a full rank external source, where the eigenvalues of the external source are equally spaced on an interval. Eigenvalue statistics of this model can be expressed in terms of multiple orthogonal polynomials with a growing number of orthogonality weights. I will set up a Riemann–Hilbert problem for these polynomials and explain how asymptotics for the Riemann-Hilbert problem can be obtained. The limiting mean eigenvalue distribution of the model will be described in terms of an equilibrium problem, and asymptotics for the multiple orthogonal polynomials will be discussed. This is based on joint work with Dong Wang.

E. Ferapontov: Dispersionless integrable systems in 3D and Einstein-Weyl geometry

Download presentation.

For several classes of second order dispersionless PDEs, we show that the symbols of their formal linearizations define conformal structures which must be Einstein-Weyl in 3D (or self-dual in 4D) if and only if the PDE is integrable by the method of hydrodynamic reductions. This demonstrates that the integrability of dispersionless PDEs can be seen from the geometry of their formal linearizations.

Based on E. V. Ferapontov and B. Kruglikov, Dispersionless integrable systems in 3D and Einstein-Weyl geometry, arXiv:1208.2728v1.

K.D. Shepelsky: A Riemann-Hilbert approach for the Degasperis-Procesi equation

Download presentation.

We present an inverse scattering transform approach to the Cauchy problem on the line for the Degasperis-Procesi equation $u_t-u_{txx}+3u_x+4uu_x=3u_xu_{xx}+uu_{xxx}$ in the form of an associated matrix (3x3) Riemann-Hilbert problem. This approach allows us to give a representation of the solution to the Cauchy problem, which can be efficiently used in studying its long-time behavior.

C. Klein: Computational Approach to Riemann

Download presentation.

We present a fully numerical approach to compact Riemann surfaces starting from plane algebraic curves. The code in Matlab computes for a given algebraic equation in two variables the branch points and singularities, the holomorphic differentials and a base of the homology. The monodromy group for the surface is determined via analytic continuation of the roots of the algebraic equation on a set of contours forming the generators of the fundamental group. The periods of the holomorphic differentials are computed with a spectral method along these contours. The Abel map is obtained in a similar way. The performance of the code is illustrated for many examples. As an application we study quasi-periodic solutions to certain integrable partial differential equations.

S. Abenda: Grassmannians and multi-soliton solutions to KP-II: an algebraic geometric approach

A (N,M)-soliton solution to KP-II is a real bounded regular solution u(x; y; t) which has M line soliton solutions in asymptotics in the $x,y$ plane whose directions are invariant w.r.t. to t. These solutions are defined as a torus orbit on the Grassmannian manifold Gr(N,M). They may be classified in terms of the matroid strata in the totally non-negative part of Gr(N,M) and to each such point there is associated a real and totally non-negative matrix A in reduced echelon form.

In this talk we address the classication problem of such multi-solitonic solutions from another point of view: we associate to a generic point in the totally non negative part of Gr(N,M) a compatible set of (N + 1) divisors sitting on a m-curve (perturbation of the rational curve associated to the multi-solitonic solution) and give an explicit representation of the matrix A in terms of such system of divisors. This research is in collaboration with Petr G. Grinevich (Landau institute of Physics)

H. Hedenmalm: Coulomb gas ensembles in 2D

Download presentation.

We will survey Coulomb gas ensembles in the plane. The setting involves an external potential $Q$ which is needed to keep the ensemble localized to a (spectral) droplet. We explain the semiclassical $\beta=2$ limit for a general inverse temperature $\beta$. In the determinantal case $\beta=2$ we go further and explore the fluctuation field in the bulk and near the boundary of the droplet. Further boundary analysis will require asymptotics of planar weighted orthogonal polynomials, which are not known. The bulk analysis is based on Bergman kernel asymptotics.

B. Eynard: Topological recursion and moduli spaces

Download presentation.

The topological recursion associates to a "spectral curve", an infinite sequence of n-forms, called its invariants. We show that for any spectral curve, the invariants compute some intersection numbers in some approrpiate moduli-space, related to the spectral curve by Laplace transform. For instance, the Laplace transform of the Lambert function is the Gamma function, and replacing the argument x^n in the Stirling series of the Log of the Gamma function by the Mumford's class kappa_n, yields the Hodge class in M_{g,n}, i.e. the invariants of the Lambert curve are the Hurwitz numbers. This is a general method which also recovers Marino-Vafa formula and much more.

G. Borot: Asymptotic expansion of matrix models in the multi-cut regime

Download presentation.

We consider the beta ensemble matrix models, which contain as a special case the NxN hermitian matrix model. The nature of the large N asymptotics of the partition function and the correlation function depends much on the topology of the support S of the large N density of eigenvalues. When S is connected, there is a 1/N expansion, and in the multi-cut case, the asymptotic features oscillatory behaviors at all scales N^{-k}. This phenomenon is famous in the study of integrable PDE's in the small dispersion limit. In the context of beta ensembles, we prove a conjecture of Eynard describing the all-order asymptotics in the multi-cut regime (and away from critical points), in terms of theta functions. As a corollary, we can deduce the asymptotic expansion of certain solutions of the Toda chain in the continuous limit, by a purely probabilistic method. This is based on a joint work with Alice Guionnet.

O. Lisovy: Painlevé functions and conformal blocks

Download presentation.

V. Schramchenko: Poncelet theorem and Painlevé VI

Download presentation.

In 1995 Hitchin constructed explicit algebraic solutions to the Painlevé VI (1/8,-1/8,1/8,3/8) equation starting with any Poncelet trajectory, that is a closed billiard trajectory inscribed in a conic and circumscribed about another conic. In this talk I will show that Hitchin's construction is nothing but the Okamoto transformation between Picard's solution and the general solution of the Painlevé VI (1/8,-1/8,1/8,3/8) equation. Moreover, this Okamoto transformation can be written in terms of an abelian differential of the third kind on the associated elliptic curve, which allows to write down solutions to the corresponding Schlesinger system in terms of this differential as well.

M. Mazzocco: Confluene of Painleve equations and q-Askey scheme

Download presentation.

In this talk we show that the Cherednik algebra of type $\check{C_1}C_1$ appears naturally as quantisation of the monodromy group associated to the sixth Painlev\'e equation. As a consequence we obtain an embedding of the Cherednik algebra of type $\check{C_1}C_1$ into $SL(2,\mathbb T_q)$, i.e. determinant one matrices with entries in the quantum torus. By following the confluences of the Painlev\'e equations, we produce the corresponding confluences of the Cherednik algebra and their embeddings in $Mat(2,\mathbb T_q)$. Finally, by following the confluences of the spherical sub-algebra of the Cherednik algebra in its basic representation (i.e. the representation on the space of symmetric Laurent polynomials) we obtain a relation between Painlev\'e equations and some members of the q-Askey scheme.

D. Masoero: String Equation and Scalar PDEs in the Semiclassical Regime

Download presentation.

I will consider a deformation of the method of characteristics based on the String Equation. I will prove its validity and I will use it to show a conjectural classification of the possible critical behaviours of scalar PDEs following Dubrovin Universality Conjecture. This will lead to a second order ODE with non-local terms which is a candidate for a novel Painleve equation.

M. Cafasso: Darboux transformations and random processes

Download presentation.

In this talk I will discuss some applications of classical Darboux transformations in the context of random processes such as, for instance, Dyson Brownian motions and spiked random matrices. The results I will expose have been obtained in collaboration with M. Bertola.

P. Van Moerbeke: Domino-tilings of Aztec diamonds, random surfaces and the Gaussian Unitary Ensemble (GUE)

Download presentation.

Eigenvalues of successive principal minors of an Hermitian matrix are well known to be interlacing. If an Hermitian matrix has independent Gaussian entries (Gaussian Unitary Ensemble, GUE), then the successive interlacing sets of (random) eigenvalues behave statistically like certain features in domino-tilings of large size Aztec diamond. They can best be explained in terms of associated random surfaces. I intend to present this and related models.

G. Falqui: Pole clashing in Gaudin Models and (stable) rational curves.

Download presentation.

We shall consider XXX homogeneous Gaudin systems with N sites, and describe a suitable limiting procedure for letting the poles of the Lax matrix clash. This yields new families of Liouville integrals (in the classical case) and of ``Gaudin algebras" (in the quantum case). Total collisions give rise to a generalization of the bending flows of Kapovich and Millson; other intermediate non-trivial cases can be obtained. We shall discuss - d'après Aguirre, Felder and Veselov -- how in this limit, somehow suggested by the Poisson geometry of these systems, Gaudin subalgebras are related with the moduli spaces of stable curves of genus zero with marked points.

A. Brini: Crepant resolutions and integrable systems

Download presentation.

Since Witten's influential conjecture on the relation between the KdV hierarchy and the moduli space of stable curves, classical integrable hierarchies have been long thought to underlie the Gromov--Witten theory of complex algebraic varieties (or orbifolds) with semi-simple quantum cohomology. When two given targets are obtained from each other by some geometrically meaningful operation, such as a resolution of singularities, a natural question that arises is how the respective integrable structures are related. I will discuss this in the context of type A surface resolutions and their relation to a class of reductions of the 2-Toda hierarchy; the main spinoff will be a proof of several conjectural correspondences which refine Yongbin Ruan's original Crepant Resolution Conjecture. This is based on joint works with Carlet-Romano-Rossi and Cavalieri-Ross.

V. Bazhanov: From Fuchsian differential equations to integrable QFT.

Download presentation.

I. Strachan: tt*-geometry on the big phase space

Download presentation.

The big phase space, the geometric setting for the study of quantum cohomology with gravitational descendents, is a complex manifold and consists of an infinite number of copies of the small phase space. The aim of this talk is to define a Hermitian geometry on the big phase space. Using the approach of Dijkgraaf and Witten, we lift various geometric structures of the small phase space to the big phase space. The main results state that various notions from tt*-geometry are preserved under such liftings.