The aim of this workshop is to bring together researchers working on integrable systems and interested in Dubrovin-Zhang hierarchies and related topics. The workshop will be centered around a minicourse by Zhe Wang.
All events will take place in the Seminar Room F3.20 on the 3rd floor of the NIKHEF building at Science Park 107.
| Wed 28 | Thu 29 | Fri 1 | |
|---|---|---|---|
| 10:00 | Wang | Wang | Wang |
| 12:00 | Lunch | Lunch | Lunch |
| 13:00 | Lorenzoni | Opanasenko | Blot |
| 14:00 | Break | Break | Break |
| 14:30 | Casati | van de Leur | Rossi |
| 18:30 | Social dinner (Restaurant Elixer) |
Z. Wang: Polynomiality theorem and beyond: a mini-course.
Abstract: In a series of talks, I would like to talk about the proof of the Dubrovin-Zhang polynomiality conjecture as well as new tools developed along the way of study. More precisely, given a calibrated semisimple Frobenius manifold, we prove that the topological deformation of its Principal Hierarchy possesses a bihamiltonian structure represented by differential polynomials.
In the first talk, I will review the overall picture of the Dubrovin-Zhang theory from an axiomatic view point. Then I will illustrate the idea of proving the polynomiality conjecture via the example of the one-dimensional Frobenius manifold given by the quantum cohomology of a point. Hopefully through this simplest case, the motivation of introducing new tools can be understood.
In the second talk, I will introduce the theory of super tau-cover and the theory of variational bihamiltonian cohomology, which are technical tools originally designed for solving the polynomiality conjecture. I hope to give as many details as possible.
In the third talk, I will give the proof of the polynomiality conjecture by combining the two tools introduced in the previous talk. I will also introduce other applications of these tools in the study of bihamiltonian integrable hierarchies. Finally I would like to list some open problems concerning the Dubrovin-Zhang theory.
X. Blot: The quantum Witten-Kontsevich series: new and old results.
Abstract: In this talk, I will first introduce the quantum Witten-Kontsevich series. This series, whose classical limit is the well-known Witten-Kontsevich series, is the simplest example of a logarithm of a quantum tau function. The notion of quantum tau functions was introduced by Buryak, Dubrovin, Guéré and Rossi in the context of the quantum DR hierarchies. The purpose of the talk is to give a new interpretation of the coefficients of the quantum Witten-Kontsevich series in terms of Gromov-Witten invariants of the projective line (joint with A. Buryak). If times allows, I will explain older results on the quantum Witten-Kontsevich series: its connection with Hurwitz numbers and a conjectural link with the Chiodo class (joint with D. Lewanski).
M. Casati: Two-dimensional Hamiltonian structures and BiHamiltonian pairs.
Abstract: Since the early work of Dubrovin and Novikov (1984), it has been known that a pair of Hamiltonian structures (in particular, but not necessarily, of hydrodynamic type) may define - subject to further constraints - both a (1+1 dimensional) biHamiltonian pencil and a Hamiltonian structure for 2+1 dimensional systems. In this talk I will draw a comparison between the corresponding biHamiltonian and Poisson cohomologies for brackets of hydrdodynamic type (joint work with G. Carlet and S. Shadrin) and will present some recent further developments in the non-homogeneous case (joint work with Hu X.)
P. Lorenzoni: Non semisimple structures in the theory of Dubrovin-Frobenius manifolds and integrable PDEs.
Abstract: In this talk I will give a survey of the results I have obtained in the last years in the study of non semisimple structures. This includes the classification of regular Dubrovin-Novikov Hamiltonian structures in 2+1 dimensions (joint work with E. Ferapontov and A. Savoldi), the study of infinitesimal deformations of bi-Hamiltonian structures of hydrodynamic type related to Balinski-Novikov algebras (joint work with A. Della Vedova and A. Savoldi), the classification of 3-dimensional regular bi-flat structures in terms of Painlevé transcendents (joint work with A. Arsie), the study of regular non-semisimple Dubrovin-Frobenius manifolds in dimensions 2,3,4 and the construction of regular Lauricella bi-flat F-manifolds from Frölicher–Nijenhuis bicomplexes (joint works with S. Perletti).
S. Opanasenko: Bi-Hamiltonian geometry of WDVV equations: general results.
Abstract: It is known that a system of WDVV equations can be written as a pair of commuting quasilinear systems (first-order WDVV systems). In recent years, particular examples of such systems were shown to possess first- and third-order Hamiltonian operators. Together with R. Vitolo, we present that this is the case for any first-order WDVV system, with the former operator being in general Ferapontov-type nonlocal one.
P. Rossi: Integrable hierarchies and the moduli space meromorphic differentials.
Abstract: In a recent series of papers and in a work in progress we are exploring the relation between intersection theory on the moduli space of meromorphic differentials on stable curves and integrable systems. There are several aspects to this story: the fundamental class of the moduli space of residueless meromorphic differentials can be pushed forward to the moduli space of curves to produce a partial CohFT whose corresponding integrable hierarchy contains KP as a reduction (joint work with A. Buryak and D. Zvonkine). Alternatively one can use the space of meromorphic differentials without residue conditions (and at least one simple pole) as an alternative to the DR cycle in an analogue to the DR hierarchy construction (joint work in progress with X. Blot). This idea can be generalized to include residueless poles and the effect on the hierarchy is a deformation into the reals of non-autonomous integrable systems. This is a recent idea I am exploring that is related to certain tautological relations discovered by A. Sauvaget whose role in integrability was not explored before.
J. van de Leur: CKP and reductions associated to conjugacy classes of the Weyl group of $C_n$.
Abstract: The KP hierarchy and its multi-component versions can be defined in terms of a Clifford algebra, a spin module and an equation in terms of fermions, the generators of the Clifford algebra. The vertex operator construction of the fermionic fields then leads to the KP and multi-KP hierarchy. Reductions of the $s$-component KP hierarchy, associated to partitions $\lambda=(\lambda_1,\lambda_2, \ldots,\lambda_s)$ of $n$, that parametrize the conjugacy classes of the Weyl group of $sl_n$, then give multi-component versions of the KdV hierarchy. In this talk I will discuss the CKP hierarchy. The Clifford algebra, spin module and fermions get replaced by a Weyl algebra, a Weyl module and symplectic bosons, respectively. The various conjugacy classes of the Weyl group of $sp_{2n}$ lead to two versions of symplectic-bosonic fields and to two vertex operator constructions for these fields. A combination of the two then gives the to $sp_{2n}$ reduced hierarchy. This is based on joint work with Victor Kac: Multicomponent KP type hierarchies and their reductions, associated to conjugacy classes of Weyl groups of classical Lie algebras
The workshop will take place in Room F3.20 at the Korteweg-de Vries Institute for Mathematics, located on the third floor of the NIKHEF building at Science Park 107 in Amsterdam, map.
The social dinner will take place at the Restaurant Elixer on the 28th February at 18:30.
To register send an email to guido.carlet@u-bourgogne.fr or s.shadrin@uva.nl
There will be a maximum of 35 participants.
Korteweg - de Vries Institute for Mathematics and Netherlands Organisation for Scientific Research.
G. Carlet
S. Shadrin