Attraction of the leading scientists to Russian institutions of higher learning, research organizations of the governmental academies of sciences, and governmental research centers of the Russian Federation

Mirror Symmetry Laboratory

About Laboratory

Grant Agreement No.: 14.641.31.0001

Project name: Mirror Symmetry and automorphic forms

Name of the institution of higher learning: Higher School of Economics - National Research University

Fields of scientific research: Mathematics

Project goal:

The main goal of the project is geometrization of categories. The starting point of this project is Homological Mirror Symmetry. This is one of the most fundamental conjectures of modern mathematics. It was put forward by Kontsevich and has originated in physics as a duality between superconformal quantum field theories. The project is aimed to developing of a new cutting edge direction in mathematics — Categorical Kähler Geometry. This is a totally new, bold idea under implementation Katzarkov, Kontsevich, Simpson.

Mirror symmetry, as a foundational attribute of models in elementary particles theory, was discovered by physicists in 90-th. Categorical Kähler geometry is deeply related to such models. Partition  functions in physical theories and generating functions for geometric invariants of manifolds are special functions such as Borcherds products or elliptic hypergeometric integrals determining Lorentzian Kac–Moody algebras, Seiberg duality and mathematical invariants of manifolds standing behind these theories.

The concept of Mirror symmetry creates new mathematical disciplines and new way of thinking in theoretical physics. The next steps in the theory of fundamental interactions in the Nature are closely related to the development of these disciplines and new ideas in mathematics and theoretical physics. We propose to open a new multidisciplinary laboratory in order to focus the research of specialists in different mathematical domains like geometry, topology, number theory and automorphic forms, Lie algebras and mathematical physics on the major theoretical problems in mathematics and physics related to Mirror symmetry. Besides scientific achievements this proposal will have a broad impact on Russian Mathematics and on Russian science in keeping their leading position in the world.

The Homological Mirror Symmetry conjecture stated by Kontsevich relates objects in two different mathematical worlds. One is the world of complex geometry, which is fairly robust and rigid, the other is the world of symplectic topology, which is a kind of flabby and has a lot of wiggle space in it, and is hard to pin down. The conjecture is that every symplectic manifold has a mirror in the complex geometry world and that an invariant of the symplectic manifold, known as the Fukaya category, is the same as an invariant called the derived category of its mirror space. Homological Mirror Symmetry conjecture is one of the most fundamental conjectures of modern mathematics bringing as well new methods in theoretical physics.

In this project we plan to capitalize on geometric consequences of Homological Mirror Symmetry and develop the following theories:

1.            Categorical Kähler Geometry;

2.            Theory of Perverse Sheaves of Categories and its parallel with Nonabelian Hodge theory and integrable systems;

3.            Categorical Lefschetz theory;

4.            Theory of categorical multiplier ideal sheaves.

On this basis we plan to find

1.            a connection between automorphic forms and categorical Noether–Lefschetz loci;

2.            a connection between automorphic forms and partition functions from the new prospective of categorical Kohler Geometry

and to construct

1.            classification of Lorentzian Kac–Moody algebras, corresponding automorphic forms, string partition functions and mirror symmetry;

2.            classification of Seiberg dualities and related elliptic hypergeometric function identities.

Leading scientist

katzarkov l00 

Katzarkov Ludmil Vasilev

 

Date of Birth: 19.12.1961

Citzenship: United States, Bulgaria

Academic degree and title: Ph.D.

Field of scientific interests: Algebraic Geometry, Homological Mirror Symmetry.

Job Title: University of Miami, Coral Gables, FL

Academic recognition:

Katzarkov have got several distinguished prizes such as Sloan Fellowship, NSF Career Award, Clay Fellowship, Simons Fellowship. Since 2008 he is a main organizer of big annual conference in Miami and a series of conferences and congresses. Katzarkov is one of organizers of the project Simons Collaboration (2015-2020), Miami-Harvard-UCBerkeley-Columbia-UPenn (10 millions USD). Under his supervision at least 11 students defended PhD; now they are either have permanent positions or are postdocs. He also hosted 26 postdoscs.

In the last 10 years Katzarkov has written 39 papers, many of which were published in highest level journals. Katzarkov has collaborated with three Fields medal winners: Borcherds, Donaldson, Kontsevich. Katzarkov covers several areas with his research – from classical algebraic geometry to most modern categorical methods, and symplectic geometry, dynamical and integrable systems. This has allowed him to bring new unexpected approaches to classical old problems – e.g. the solution of Shafarevich’s conjecture for linear groups, connection of Moishezon program with theory of perverse sheaf of categories, dynamical systems for categories.

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