International Laboratory for Mirror Symmetry and Automorphic Forms

Scientific results:

• Categorical, algebraic and Kähler geometry
  

Our researchers have created the foundations of the theory of spectra (or spectral networks), which is a new categorical generalization of the classical theory of vanishing cycles. We have built a new Hodge type theory — the Hodge theory of extensions which binds the classical theory of partial differential equations with the theory of categories.

A new approach has been proposed to the mixed non-Abelian Hodge theory from the categorical point of view.

We are continuing to work successfully on solving the problem of the irrationality of the general four-dimensional cubic. The latter problem has a hundred-year history. To solve it, we are building a radically new theory of spectra that will provide an interpretation off the Landau-Ginzburg theory of models as generalized singularity theory. We are planning to prove that the Gromov-Witten spectrum, i. e. the categorical spectrum, is a birational invariant. Results in this domain will have exceptionally important applications in geometry, mathematical physics and theoretical physics.

Our researchers have proven the Kontsevich conjecture that the derived category of coherent sheaves on a separable finite-type scheme over a field of characteristic 0 is homotopically finite.

We have proven the conjecture on mirror symmetry for the three-dimensional projection space.

The Laboratory has proven the Katzarkov–Kontsevich–Pantev conjecture that binds the Hodge numbers of a Fano manifold and the Hodge numbers of its Landau–Ginzburg model in dimension 2 or 3. We have established the connection of this conjecture with the P = W conjecture that is in the foundation of the modern interpretation of the geometric Langlands program.

Our researchers have initiated a new approach to proving the conjecture on homological mirror symmetry for manifolds of general type.

We have provided a proof of the existence of toric Landau-Ginzburg models for the three-dimensional case and for the case of complete intersections. We have provided a description of derived categories of fibrations on a hypersurface of an arbitrary order, groups of homologies of tropical manifolds have been built.

• Automorphic forms and applications
  

  We have build a theory of modular theta-blocks and reflective automorphic forms and found important applications to the theory of moduli of polarized hyper-Kähler manifolds and  polarized Kummer surfaces.

  We have provided a complete classification of Lorentzian Kac–Moody algebras, with a Weyl group generated by all the 2-reflections.

  We have proven the conjecture on arithmetic mirror symmetry for a wide class of K3 surfaces.

  Our researchers have provided a solution to the Yoshikawa problem (2000) on the explicit description of Lorentzian Kac-Moody algebras built on the analytical torsion of del Pezzo surfaces.

  In the terminology of automorphic forms we have provided an answer to the long-standing Frenkel-Feingold problem (1982) concerning the possible relation between the Kac-Weyl denominator of affine and hyperbolic Kac–Moody algebras.

  We have obtained new automorphic confirmations of a two-dimensional analog of the famous Taniyama–Shimura conjecture (the Brumer–Kramer  conjecture on the modularity оf Abelian surfaces).

  Our researchers have proven the Rodriguez Villegas conjecture (2003) on the existence  of supercongruences for 14 rigid hypergeometric Calabi-Yau threefolds over Q.

  We have found an expression of values of the Dedekind zeta-functions as periods of real-valued automorphic forms from real-valued Heegner cycles and the rationality and the congruence properties of these zeta-values that follow from this.

  We have proven new lower bounds on large gaps between integers that are sums of two squares.

Implemented results of research:

The research has high multidisciplinary importance within theoretical mathematics (Kähler, algebraic and differential geometry, the theory of infinite-dimensional Lie algebras, the theory of automorphic forms, numbers theory) and modern theoretical physics (quantum field theory, quantum gravitation, string theory). 

Education and career development:

During six years, employees of the Laboratory have defended 4 Doctor of Sciences and 5 Candidate of Sciences dissertations, delivered more than 60 original lecture courses. 135 young researchers, specialists and lecturers have completed occupational retraining (additional training) at the Laboratory in the domain of the scientific research project. Every year the Laboratory stages summer student schools in geometry in Novosibirsk, student schools were also organized in Saint Petersburg (2017) and on the grounds of the Training Center of the Higher School of Economics in Voronovo (2019). 

Other results:

• The research fellow of the laboratory Alexander Yefimov received Gold Medal of the Russian Academy of Sciences with a prize for young scientists in 2017. He also received an award from the European Mathematical Society in 2020.
• The deputy chair of the Laboratory Viktor Przhilkovskiy received an award from the Government of Moscow and a grant of the President of the Russian Federation for young scientists in 2019.
• The employees of the Laboratory Valeriy Gritsenko and Andrey Levin have supervised the ANR-RSF French-Russian grant «Symmetries and moduli spaces in  algebraic geometry and physics» for the years 2021-2023. The uniqueness of this success is that the employees of the International Laboratory for Mirror Symmetry and Automorphic Forms lead both parts of the grant: Valeriy Gritsenko heads the French team, Andrey Levin leads the Russian part. The grant encompasses research in the domain of mathematical physics. The budget of the project for 2021-2023: Russian Science Foundation – 6 000 000 rubles per year, ANR – 700 000 Euro per  year.

Collaborations:

• Novosibirsk State University, Mathematical Center in Akademgorodok, Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences (Russia): joint international conferences, annual joint summer student school «Current advancements in geometry», joint research.
  
• Institut des Hautes Études Scientifiques (France): joint scientifc research.
  
• Steklov Institute of Mathematics of the Russian Academy of Sciences (Russia): joint international conferences, scientific seminars, research projects.
  
• The Institute of the Mathematical Sciences of the Americas at the University of Miami (USA): joint scientific seminars, workshops, conferences.
  
• International Center for Mathematical Sciences – Sofia – Bulgarian Academy of Sciences (Bulgaria): joint international scientific conferences in Bulgaria and Russia, academic exchange.