INdAM-COFUND INdAM Outgoing Fellowships in Mathematics and/or Applications cofunded by Marie Curie actions. MPoisCoho

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1 INdAM-COFUND-2012 INdAM Outgoing Fellowships in Mathematics and/or Applications cofunded by Marie Curie actions MPoisCoho Poisson cohomology of multidimensional Hamiltonian structures Report 2017 Period covered from 01/10/2016 to 30/09/2017 Start date of the project 01/10/2016 Date of the report 11/10/2017 Fellow dr. Matteo Casati Personal supervisor in the Outgoing phase prof. Evgeny Ferapontov Host organisation for the Outgoing phase Loughborough University Page 1 of 6

2 Table of contents 1 Work progress and achievement during the period Progress towards the objectives Progress of the researcher training activities Deviation from the proposed workplan Summary of the activities Publications Short visits Visitors Seminars and conferences Justification of the costs Updated timeline 6 1 Work progress and achievement during the period 1.1 Progress towards the objectives Objective #1 The computer software developed to address the computation has been optimised and made faster and more reliable. The fellow has computed the third degree component of H 2 for the n = d = 2 Poisson brackets of hydrodynamic type and proved that H 3 is not trivial as well, drawing partial conclusions in the direction of finding the normal form for n = d = 2 Hamiltonian operators. Objective #2 The theorem proved in the scalar, multidimensional case 1 has been extended to a class of n component operators, in particular including an explicit description of all the cohomology classes for one of the three normal forms of the two-component Poisson bracket of hydrodynamic type. Objective #3 As for the project timeline, the objective has been left for the second year. 1.2 Progress of the researcher training activities The crucial training objective is the achievement of all the skills that are suitable for the pursuing of an independent research career. The fellow has started working within the renowned research groups in the field of Integrable Systems at Loughborough University and establishing new collaborations. In particular he is working on a side project on nonlocal Hamiltonian structures with prof. Ferapontov and prof. Vitolo from University of Salento. 1 Guido Carlet, Matteo Casati, and Sergey Shadrin. Poisson cohomology of scalar multidimensional Dubrovin- Novikov brackets. In: J. Geom. Phys. 114 (2017), pp Page 2 of 6

3 The fellow has qualified for teaching modules at Loughborough University. In the spring term he shared the teaching duties of the module of Mathematical Methods 2, one of the core teachings for first year undergraduate students in Mathematics. This summed up to 20 hours of frontal lectures and to preparing and marking the exams for 200 students. Moreover, he was involved in the activities (guided tours) for the University Open Day, July Deviation from the proposed workplan During the past year, the focus of the proposed researcher shifted from the explicit computation of cohomology groups for the multidimensional Hamiltonian structure to the more comprehensive problem of studying the theory of deformations of the named structures. The two problems are related but not equivalent as opposite as the case of one-dimensional structures because of the non triviality of the second cohomology group. It should be remarked that studying the theory of the deformations allowed us to obtain the complete classification and the normal form of two-dimensional Hamiltonian structures in the scalar case 2. In fact, the establishment of normal forms for multidimensional Hamiltonian structures is the leading aim of the project, since we acknowledged in the proposal that studying their Poisson cohomology is a necessary prerequisite. The opening of this deeper line of research has exchanged the priority between the objectives of the project. We regard now as a priority the computation of the full Poisson cohomology with homological algebraic methods, and the explicit computation of some cohomology groups for different forms of hydrdoynamic type brackets as a supporting task. 2 Summary of the activities 2.1 Publications 1. G. Carlet, M. Casati, S. Shadrin. Poisson cohomology of scalar multidimensional Dubrovin Novikov brackets, J. Geom. Phys. 114 (2017), pp M. Casati, D. Valeri, MasterPVA and WAlg: Mathematica packages for Poisson vertex algebras and classical affine W-algebras, submitted. 3. G. Carlet, M. Casati, S. Shadrin. Normal forms of dispersive scalar Poisson brackets with two independent variables. arxiv/ , submitted. 4. M. Casati, Higher order dispersive deformations of multidimensional Poisson brackets of hydrodynamic type, in preparation. Paper [1] was a preprint when the application was submitted. In the paper we find the dimension of the Poisson cohomology for scalar multi-dimensional Hamiltonian operators. In the twodimensional case, the formula is made explicit. In [2] we present the computer 2 Guido Carlet, Matteo Casati, and Sergey Shadrin. Normal forms of dispersive scalar Poisson brackets with two independent variables. arxiv/ Page 3 of 6

4 algebra software developed for performing most of the computations we deal with within the framework of Poisson vertex algebra. An early version of the software was used to obtain the results of some previously published papers 3 ; in the paper we discuss also the package WAlg, useful to study classical W -algebras and to derive the PVA (or, equivalently, Hamiltonian) structure after the generalised Drinfeld-Sokolov reduction of an affine simple Lie algebra. In [3] we provide the normal forms of two-dimensional scalar Hamiltonian structures, building onto the results of [1] providing the explicit form for the elements of the cohomology classes and proving that only the constant ones can be extended to define Hamiltonian operators. In [4] we report the explicit computations performed for the n = d = 2 Hamiltonian operators of hydrodynamic type proving the nontriviality of the second cohomology group. Moreover, we use the results of [1] to compute the dimension of the full cohomology for one class of such Hamiltonian structures and demonstrate that the thrid cohomology group is not only nontrivial, but effectively obstructs the extension of most of the elements of H 2. Part of the explicit computations of the paper were already discussed in the PhD thesis of the authors but have never been published. 2.2 Short visits 1. Korteweg-De Vries Institut, Universiteit Van Amsterdam. December 11th 15th, Hosts: Sergey Shadrin and Guido Carlet. 2. Dipartimento di Matematica e Applicazioni, University of Milan Bicocca. January 24th 26th, Host: Paolo Lorenzoni. 2.3 Visitors 1. Guido Carlet. May 22nd-26th. 2.4 Seminars and conferences Seminars 1. University of Kent, November 4th Multidimensional Hamiltonian operators and Poisson vertex algebras 2. Loughborough University, November 9th Poisson Vertex Algebras. 3. University of Leeds, February Algebraic methods for the theory of deformations of multidimensional Hamiltonian operators 4. University of Glasgow, March Dispersionless limit for the BiHamiltonian structure of classical affine W-algebras: scouting for algebraic Frobenius manifolds 3 Matteo Casati. On deformations of multidimensional Poisson brackets of hydrodynamic type. In: Comm. Math. Phys (2015), pp ; Matteo Casati. Dispersive deformations of the Hamiltonian structure of Euler s equations. In: Theor. Math. Phys. 188 (2016). Page 4 of 6

5 Workshop, conferences and schools 1. Integrable Day, November 25th, Workshop at Loughborough University. 2. Schrödinger Equation: asymptotics, integrability and beyond. Workshop at University of Bristol, March 28th, National meeting of INdAM-GNFM, Montecatini (PT, Italy), May 4th 6th, Communication: Scalar multidimensional Hamiltonian operators of hydrodynamic type. 4. INdAM Day, Messina, June 7th, Schrödinger Equation: asymptotics, integrability and beyond. Workshop at Loughborough University, May 16th Physics and Mathematics of Nonlinear phenomena years of IST. International conference at Gallipoli (LE, Italy). June 17th 24th, Speaker: Normal forms of dispersive scalar Poisson brackets for two-dimensional systems. 7. Discrete models in Geometry and Physics, Summer School. TU Berlin, September 11th 18th, Modern geometry and physics, Workshop. Loughborough University, 18th 21st September, Justification of the costs The fellow has ascribed on RCC1 and RCC2 the latter as for an agreement with the Host Institution, that allotted up to 750 in the 2017 budget the following expenses: Trip to Canterbury, visit and seminar at University of Kent. Payment of the trip, 175 (195 C) on RCC1 Visit of dr. Carlet to Loughborough. 450 on RCC2 Conference PMNP 2017 in Gallipoli. Trip, accommodation, conference fee C on RCC1 Page 5 of 6

6 3 Updated timeline As already discussed, it is now clearer that for the ultimate aim of classifying the multidimensional Hamiltonian operator the Poisson cohomology plays a role only with its second and third cohomology groups. The explicit computation of the dimensions for higher cohomology groups has a lesser relevance than anticipated. On the other hand, rather then enriching the zoology of Hamiltonian operators for which we can compute the cohomology groups, we deem that it is more important to deepen the understanding of the Poisson brackets we have already studied, to extract general results out of our explicit computation. Objective #1 will still be pursued alongside the Objective #2 and #3 that will get the priority. Time (months) Outgoing phase Return phase Tasks Accommodation, administrative tasks, involvement in LU research group Optimization and parallelization of the code Explicit computations (objective #1) Computation of the Poisson Cohomology of constant structures (objective #2) Study of homological algebra Relocation to Italy, involvement in SISSA research group Computation of the Poisson Cohomology of linear structures (objective #3) Writing scientific publications Dissemination Submit final report Page 6 of 6

CURRICULUM VITAE. Di Yang. Date of Birth: Feb. 8, 1986 Nationality: P. R. China

CURRICULUM VITAE. Di Yang. Date of Birth: Feb. 8, 1986 Nationality: P. R. China CURRICULUM VITAE Di Yang PERSONAL INFORMATION Name: Di Family Name: Yang Date of Birth: Feb. 8, 1986 Nationality: P. R. China Address: MPIM, Vivatsgasse 7, Bonn 53111, Germany Phone Number: +49-15163351572