Title:

Quantitative approach to Mirror Symmerty: weak Landau--Ginzburg

models, their properties and invariants.

Abstract:

Given a smooth Fano variety, Mirror Symmetry predicts the existence

of a so called Landau--Ginzburg model --- a pencil, whose symplectic

geometry reflects the algebraic geometry of the Fano variety, and

viceversa. Mirror symmetry conjecture of Hodge structure variations

that translates this relation to a quantitative level.

This conjecture enables one to construct explicitly mirror

Landau--Ginzburg models for a large class of varieties.

Under some conditions on these models we assume that they

(or their minimal compactifications) are dual models for Homological

Mirror Symmetry conjecture. Studying them from this point of view

one can predict some (numerical) invariants of initial Fano variety that can be

extracted from Landau--Ginzburg model. In the talk we discuss

how to calculate some of such invariants, in particular

Gromov--Witten invariants, Hodge numbers, characteristic numbers,

the birational type of Fano variety. We also discuss connections

between different weak Landau--Ginzburg models for given Fano variety

and their relation with toric degenerations.