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Laboratory for Geometrical Methods in Mathematical Physics

Contract number
Time span of the project

As of 30.01.2020

General information

Name of the project: Geometric method of mathematical physics

Goals and objectives

Research directions: Development of geometrical of mathematical physics

Project objective: Research of geometric structure of mathematical physics

The practical value of the study

  • We have computed correlation numbers of Louisville minimal gravitation from Douglas structural equation
  • Our team has researched Baker-Ahiezer discrete modules over a ring of difference operators
  • Invariants have been calculated for symplectic field theory defining quantum integrable system
  • An axiomatic approach has been developed to symplectic field theory
  • Commuting operators have been produced by quantization of hamiltonians of Hopf hierarchy. Potential of a disc in symplectic field theory has been calculated
  • Our neat has studied integrable hierarchies of topological type in zero power approximation for quantum cohomologies of torous Fano manifolds.
  • We have described the structure of Poisson-Lee group of dressing transforms of the Darboux-Egorov system.
  • We have studied nonlinerar cinematic equation describing propagation of concentration waves in dispersed bubble fluids.
  • We have proposed differential conservation laws approximating the initial integro-differential equation. On the basis of these laws we have completed quantitative computations of wave propagation demonstrating possibility of kinetic tripping of distribution function.
  • Mathematical provisions have been developed for a nonlinear acoustic tomograph (new generation). First such tomographs have been manufactured.
  • We have developed a theory of isomonodromic deformation of nonlinear differential operators of infinite order.
  • General type weak singularities have been classified to solve quazilinear Hamiltonian partial differential equation

Implemented results of research: We have developed software for computation and visualization of modularized theta-functions. The software allows to work with Whitham asymptotics of higher orders of evolutionary partial derivative equation

Education and career development:

  • We have developed a lecture course for masters «Geomethric methods of mathematical physics», the course has been read at the Faculty of Physics and Mathermatics of the Moscow State University
  • 2 doctoral dissertations and 7 candidate dissertations have been defendeed
  • 6 national scientific summer schools for geomethrical methods for mathematical phyisics for undergraduates of final years and postgraduates

Organizational and structural changes:

A computer classroom is operating that is equipped for visualization of computational processes of mathematical physics

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B.A. Dubrovin, Di Yang, Don Zagier
Classical Hurwitz numbers and related combinatorics, Moscow Mathematical Journal 17, 601—633 (2017)
Dubrovin B., Di Yang,
Jn cubic Hodge integrals and random matrices. Communications in Number Theory and Physics 11, 311-336 (2017)
Dubrovin B., Di Yang
Generating series for GUE correlators. Letters in Mathematical Physics. 107 1971-2012 (2017)
Dubrovin B., Si-Qi Lu, Di Yang, Youjin Zhang
Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutional PDE Advances in Mathematics. 293, 382-435 (2016)
Dubrovin B.
Symplectic field theory of a disk, quantum integrable systems, and Schur polynomials Annales Henri Poincaré, 17:7, 1595-1613 (2016)
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