Double ramification cycles and integrable systems

Double ramification cycles and integrable systems

This workshop, sponsored by AIM and the NSF, will be devoted to the study of the double ramification cycle in the moduli space of stable curves and its relation with the theory of integrable systems of PDEs, with a special stress on the double ramification hierarchy, a construction associating to any cohomological field theory an integrable system of PDEs and its quantization. The goal is bringing together experts in the geometry of moduli spaces of curves, both algebraic and symplectic, and exponents of the integrable systems community to approach several specific open problems at the boundary between these two disciplines.

The main topics for the workshop are:

• DR/DZ equivalence conjecture
  The Dubrovin-Zhang hierarchy is another construction of an integrable system of PDEs associated to a given semi-simple cohomological field theory. Guided by the evidence contained in the first computed examples, A. Buryak proposed the conjecture that, for semi-simple CohFTs, the Dubrovin-Zhang hierarchy is equivalent, through some coordinate transformation of the phase space, with the DR hierarchy. In subsequent publications a stronger and more precise conjecture was formulated, which uniquely identifies the coordinate transformation. We feel that the moment is ripe for a full proof and that bringing together the two communities would provide the needed expertise and technical tools to overcome the technical difficulties.
• Generalizations of CohFTs and Givental-Teleman classification
  The DR hierarchy works with more general objects than just cohomological field theories. In fact the axioms of CohFT can be relaxed to partial CohFTs to still produce integrable Hamiltonian hierarchies and even more, to F-CohFTs, to obtain still integrable, albeit not Hamiltonian, systems. While for actual CohFTs, powerful classification and reconstruction theorems exist (by Givental and Teleman, in particular), their generalizations have never been studied in such depth. Partial results have recently been obtained in this direction for F-CohFTs and with application to the existence of dispersive deformation of integrable systems of conservation laws. We propose to develop this theory and compute several explicit examples.
• DR cycles and admissible covers DR cycles
  The DR locus is an extension to the moduli space of stable curves of the locus in the open moduli space parameterizing smooth curves with a map to the projective line of prescribed ramification. An alternative extension, the locus of "admissible covers", is given by simply taking the closure of the locus of such smooth curves. A direct relationship between these two compactifications has been explicitly studied in only a small number of cases, but recent work of Schmitt and van Zelm has provided new tools to investigate this geometrically compelling question. A natural goal would be to produce graph formulas for admissible covers cycles analogous to Pixton's formula for the DR cycle.

This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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