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International Laboratory for Mirror Symmetry and Automorphic Forms

Invited researcher Lyudmil Vasilev Katzarkov
Contract number
14.641.31.0001
Time span of the project
2017-2021
Head of the laboratory

As of 20.05.2020

38
Number of staff members
80
scientific publications
General information

Many new category structures have been discovered by physicists over the last several years. It is obvious that the language of higher categories suits perfectly for describing the cornerstone concepts in the modern theoretical physics. All of this has lead to the separation of many domains of mathematics into categories. The starting point of the Laboratory's research is homological mirror symmetry. Homological mirror symmetry currently serves as the basis for a wide range of absolutely new studies in mathematics that is promoted and constantly updated by many researchers. One of the main flaws of the rich but very technically complex method of homological mirror symmetry is a lack of applications. Focusing namely on applications, the Laboratory researchers are planning to develop the missing link — geometrisation of the category theory based on the interaction with mathematical physics.

Name of the project: Mirror symmetry and automorphic forms

Strategy for Scientific and Technological Development Priority Level: а


Goals and objectives

Research directions: Mathematics, mirror symmetry and automorphic forms

Project objective: Uniting efforts of specialists from various domains such as geometry, topology, automorphic form theory and Lie algebras, number theory, mathematical physics for solving the main theoretical problems of mathematics and physics related to mirror symmetry. Study of category, geometry and automorphy, geometric and arithmetic aspects of homological mirror symmetry.

Finding principally new applications of category and Kähler geometry to problems of geometry, to geometric problems of rationality, in automorphic form theory and Lorentzian Kac-Moody algebras.

Geometrization of category theory, and its starting point is the theory of homological mirror symmetry formulated by Maxim Kontsevich.


The practical value of the study

  • We have obtained results in the fields in categorical, Kähler and algebraic geometry, as well as automorphic form theory and Lee algebra in number theory and theoretical physics.
  • We have proven the conjecture on homological mirror symmetry for the 3D projective space, special cases of the Katzarkov-Kontsevich-Pantev conjectures on filtration of Landau–Ginzburg models, the first-order theta-block conjecture of Gritsenko-Poor-Yuen, proven the Kontsevich conjecture on homotopy finiteness of derived categories of coherent sheaves on finite type schemes.

  • A new approach to proving the conjecture on the conjecture on homological mirror symmetry for general type manifolds has been initiated.
  • The laboratory has proposed a new method to solve the classical problem of the irrationality of the general 4D cubic plane curve that was formulated in the 1920s.

Implemented results of research:

  • Our research bears important multidisciplinary value within theoretical mathematical (Kähler, algebraic and differential geometry, infinite dimensional Lie algebra, automorphic form theory, numbers theory) and modern theoretical physics (quantum field theory, quantum gravitation, string theory).

Education and career development:

  • Three doctor dissertations and one candidate dissertations have been defended, as well as 9 master degree thesis.
  • 184 researchers have received additional training completion certificates at events organised by the Laboratory: the «Summer Mathematics School on Fontanka. Geometry 2017», a series of master classes by Maxim Kontsevich, the School of Advanced Research for Young Mathematicians «Hodge theory: the past and the future», the International Siberian Summer Schools «Modern Geometry» in Novosibirsk, the International Winter School in Dubna «Statistical sums and automorphic forms».

  • We have conducted courses by the head of the Laboratory Ludmil Katzarkov «Categories theory», «Weinstein manifolds and symplectic cohomologies - 1, 2», «Birational invariants from the symplectic topology».

  • The Laboratory has organized the seminar «Jakobi forms 30 years later» for final-year students by the head of the laboratory Prof. Gritsenko at the Coursera international Internet platform (1240 active listeners all over the world as of the end of 2017).
  • Three weekly scientific seminars have been organized: «Automorphic forms and their applications», «Categories theory and its applications» and «Seminar in arithmetic geometry».
  • Employees of the Laboratory have compiled and delivered 22 education courses.

Other results:

The Laboratory researcher A. I. Efremov receive a Russian Academy of Sciences Gold Medal with an award for young scientists in 2017 for the series of works entitled «Derived categories and cyclic homologues» and in 2020 he was awarded a European Mathematical Society prize.

In 2019 the deputy head of the Laboratory V. V. Przhiyalkovsky received a prize from the Government of Moscow and a grant from the president of the Russian Federation for young Russian scientists.

Collaborations:

  • V. A. Steklov Mathematical Institute of the Russian Academy of Sciences (Russia): joint international conference «Mirror symmetry and its applications», International Conference «Algebra, algebraic geometry and number theory» commemorating the academician Igor Shafarovich, International conference «Birational geometry and Fano manifolds».
  • Saint Petersburg Branch of the V. A. Steklov Mathematical Institute of the Russian Academy of Sciences (Russia): joint Summer scientific school on Fontanka «Geometry 2017», International conference «Automorphic forms and algebraic geometry», International school-conference «Modular Forms», International conference «Algebraic geometry and applications» (events were conducted with support from the Simons Foundation )
  • Max Planck Institute for Gravitational Physics (Germany): International winter school in Dubna «Partition functions and automorphic »
  • Max Planck Institute for Mathematics (Germany), International Centre for Theoretical Physics (Italy): working meeting of the Four Cities Alliance: Mirror symmetry and Arithmetic of Differential Equations.
  • Institute of Advanced Scientific Studies (France): School of Advanced Research for Young Scientists «Hodge theory — old things and new things», International conference «Meeting of gloal leaders in homological mirror symmetry».
  • Institute for the Physics and Mathematics of the Universe (Japan) - conference «Noncommutative deformation and modular spaces»
  • Moscow State University (Russia): joint international school-conference «Lie algebras, algebraic groups and theory of invariants»
  • CNRS Poncelet Laboratory in Moscow (Russia): weekly joint scientific seminar «Automorphic form and their applications», International school «Algebra and Number Theory» in Voronovo.
  • «Sirius» mathematical centre, Sochi (Russia): the International conference «Integrable Systems and Automorphic Forms» in Sochi.

  • Novosibirsk State University, S. L. Sobolev Institute of Mathematics of the Russian Academy of Sciences: annual International Siberian Summer School «Modern geometry», International conference «Geometry days in Novosibirsk — 2019.

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Haiden F., Katzarkov L., Kontsevich M.
Flat Surfaces and Stability Structures. Publications math´ematiques de l’IHES 126(1): 247–318 (2017)
A. Blanc, L. Katzarkov, P. Pandit
Generators in formal deformations of categories, Compositio Mathematica, vol. 154:10 (2018), 2055–2089
A. Petrov, D. Vaintrob, V. Vologodsky
The Gauss-Manin connection on the periodic cyclic homology, Selecta Mathematica, New Series, vol. 24 (2018), 531–561.
A. Beilinson, G. Kings, A. Levin
Topological polylogarithms and p-adic interpolation of L-values of totally real fields, Math. Ann., vol. 371 (2018), 1449–1495
V. Lunts, V. Przyjalkowski
Landau–Ginzburg Hodge numbers for mirrors of del Pezzo surfaces, Adv. Math., vol. 329 (2018), 189–216
А. И. ЕФИМОВ
О гомотопической конечности DG-категорий, УМН, том 74 (2019), выпуск 3(447), страницы 63–94
AKHMEDOV A., KATZARKOV L.
Symplectic surgeries along certain singularities and new Lefschetz fibrations, Advances in Mathematics vol.
GRITSENKO V., WANG H.
Theta block conjecture for paramodular forms of weight 2, Proc. of American Math.
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