### M Center

During past 25 years there has been intensive interaction between string theory and geometry which has led to a creation of entirely new mathematical areas. String theory has suggested that "conventional" geometry emerges from the quantum theory at certain limits. Then various string dualities give equivalent but mathematically very different descriptions of the same physical quantities. A beautiful and deep example illustrating all these ideas is mirror symmetry.

The M-Center supports and promotes research in mathematical questions arising from string theory, in which mirror symmetry and tropical geometry play a central role.

**Research**

Homological Mirror Symmetry program laid out by Maxim Kontsevich at the beginning of 90's is still the main topic of the worldwide research in Mirror Symmetry. This includes a variety of algebraic,geometric and categorical questions as well as new problems, some of themstrongly motivated by quantum physics, e.g. string theory, gauge theories.Tropical geometry plays an important role in formulation and solutions of those problems.

The M-Center research team is on the frontier of all new developments in the area.

Some of the current research directions pursued at the M-Center include:

- study of the structure of Fukaya categories of non-compact symplectic manifolds
- mirror symmetry for Fano varieties and manifolds of general type
- non-archimedean integrable systems and mirror symmetry for them
- wall-crossing formulas
- tropical enumerative geometry of real and complex curves
- non-commutative Hodge theory
- derived algebraic geometry
- perverse sheaves of triangulated categories (schobers)
- moduli spaces of instantons arising in gauge theories
- tropical homology and its relation to classical homology and the Hodge conjecture
- Welschinger invariants, open Gromov-Witten theory.