Speaker: Victor Przyjalkowski


Quantitative approach to Mirror Symmerty: weak Landau--Ginzburg
models, their properties and invariants.


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.