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Moduli spaces of curves and integrable hierarchies

Periodic Reporting for period 1 - MSCIH (Moduli spaces of curves and integrable hierarchies)

Reporting period: 2018-03-19 to 2020-03-18

Objective O1. Proving the DR/DZ equivalence conjecture, saying that the hierarchy of topological type is Miura equivalent to the double ramification hierarchy, is the main step needed to prove the Dubrovin-Zhang conjecture about a polynomial bihamiltonian structure for the hierarchies of topological type. In my paper I made a crucial step for the proof of the DR/DZ equivalence conjecture. I reduced this conjecture to a system of relations in the cohomology of the moduli spaces of stable algebraic curves and proved these relations up to genus 2. This proves the DR/DZ conjecture at the approximation up to genus 2. Another important result was obtained in my other paper, where I proved the DR/DZ equivalence conjecture for cohomological field theories of rank one at the approximation up to genus 5.

Objective O2. My two papers are devoted to the Frobenius manifolds associated to the curve singularities of types A and D. More precisely, I constructed solutions of the associated open WDVV equations that should correspond to open FJRW invariants of the curve singularities of types A and D. I proved that the coefficients of the solutions of the open WDVV equations coincide with the transition functions between two natural coordinate systems on the Frobenius manifolds associated to the curve singularities of types A and D. In other words, the solutions of the open WDVV equations describe the mirror map for these Frobenius manifolds.

Objective O3. In my paper I constructed a tau-structure for the double ramification hierarchy, and in my other paper I constructed its generalization for the quantum double ramification hierarchy. As a result, this gives a construction of a quantum tau-function for a large class of integrable systems, which are important in mathematical physics.

Objective O4. A construction from my paper defines a quantum tau-function for the KdV hierarchy and, more generally, for higher Gelfand-Dickey reductions of the KP hierarchy. From another perspective, in my other paper I generalized the open Virasoro equations for the open extension of the Gelfand-Dickey hierarchy, finding open Virasoro equations for an arbitrary homogeneous solution of the open WDVV equations.
"During the fellowship I made a key step towards a proof of the DR/DZ equivalence conjecture, which allowed me to prove this conjecture in a very large class of cases. I obtained a construction of a quantum tau-function for the double ramification hierarchies. Such a general construction of tau-functions in the world of quantum integrable systems never appeared in the literature before. I made a substantial progress in the study of the open WDVV equations, finding open Virasoro equations and relating the solutions of the open WDVV equations for the singularities of types A and D to the mirror map. The results of the fellowship are published in 6 papers in leading journals: Geometry and Topology, Communications in Mathematical Physics, International Mathematics Research Notices, Moscow Mathematical Journal, Arnold Mathematical Journal.

Transfer of knowledge was achieved through various activities: seminar talks at the University of Leeds, other universities in the UK (King's College London, University of Glasgow, University of Sheffield, Loughborough University), and worldwide (Tsinghua University in Beijing, National Research University Higher School of Economics in Moscow, International School for Advanced Studies in Trieste); research visits (University of Milano-Bicocca, University of Padua); conference talks; invitations of researchers to visit the University of Leeds; organization of the conference ""Curve counting theories and related algebraic structures'' (University of Leeds, 9-11 September 2019); supervision of a PhD student (Oscar Brauer, starting date 1st September 2019).

I significantly expanded my network of collaborators inviting the following researchers to visit the University of Leeds during the fellowship: Alexey Basalaev (National Research University Higher School of Economics in Moscow), Mattia Cafasso (University of Angers), Tamara Grava (University of Bristol), Sabir Gusein-Zade (Moscow State University), Jules Lamers (University of Melbourne), Andrey Mironov (Novosibirsk State University), Paolo Rossi (University of Padua), Vladimir Rubtsov (University of Angers), Ian Strachan (University of Glasgow), Di Yang (Max Planck Institute for Mathematics in Bonn).

Conference talks:
1. 3rd IBS-CGP and BICMR Joint Symplectic Geometry Workshop. 24-26 September 2019, Pohang, South Korea.
2. ''Solitons, Collapses and Turbulence: Achievements, Developments and Perspectives'', in honour of Vladimir Zakharov's 80th birthday. 5-9 August 2019, Yaroslavl, Russian Federation.
3. Dynamics in Siberia. 25 February - 2 March, 2019, Novosibirsk, Russian Federation.
4. Integrable Day 2018. 30 November 2018, Loughborough, United Kingdom.
5. Workshop on Geometry and Integrable Systems. 19-23 November 2018, Chengdu, China.
6. Flat Surfaces and Algebraic Curves. 16-22 September 2018, Oberwolfach, Germany.
7. Russian-Chinese Conference on Integrable Systems and Geometry. 18-26 August 2018, St. Petersburg, Russian Federation.
8. Differential Algebra and Related Topics IX. 30 July - 2 August 2018, Leeds, United Kingdom.
9. 6th Workshop on Combinatorics of Moduli Spaces, Cluster Algebras and Topological Recursion. 4-9 June 2018, Moscow, Russian Federation.
10. Yorkshire and Durham Geometry Days. 2 May 2018, Leeds, United Kingdom.

Lecture courses:
Mini-course ''Moduli spaces of algebraic curves with marked points'', Novosibirsk State University, 4-7 February 2020, Russian Federation.

Outreach:
Lecture ''Counting of partitions and generating functions'' at the high school ""Specialized Educational Scientific Center of the Novosibirsk State University"", 3 February 2020."
"During the fellowship I made a key step towards a proof of the DR/DZ equivalence conjecture, which allowed me to prove this conjecture in a very large class of cases. I obtained a construction of a quantum tau-function for the double ramification hierarchies. Such a general construction of tau-functions in the world of quantum integrable systems never appeared in the literature before. I made a substantial progress in the study of the open WDVV equations, finding open Virasoro equations and relating the solutions of the open WDVV equations for the singularities of types A and D to the mirror map.

My standing in the community as a leading researcher was evidenced by a Whitehead Prize of the London Mathematical Society in 2019.

My works are having a significant impact on the academic community. My DR/DZ equivalence conjecture was one of the main topics of the workshop ""Double ramification cycles and integrable systems"", which was held at the American Institute of Mathematics, San Jose, California, USA, 7-11 October 2019. The influence of my research is confirmed in the overview of the External Satellite Conference of the 8th European Congress of Mathematics ""Integrable systems in geometry and mathematical physics. In memoriam of Boris Dubrovin (1950-2019)"", which was planned to be held in Trieste, Italy, 13–17 July 2020 (rescheduled because of COVID-19 pandemic)."
Photo of Alexandr Buryak