On October 7-9, Professor Yan Soibelman from Kansas State will give a
series of 6 lectures "Commutative and non-commutative geometry of mirror
symmetry".
He will explain the approach to Mirror Symmetry suggested in a joint
project with Maxim Kontsevich. The aims is to explain the phenomenon of
Mirror Symmetry in terms of homological algebra and non-commutative
geometry. Discovered by physicists as a duality on a certain class of
string theories, Mirror Symmetry turned out to be related to many deep
questions of algebraic and symplectic geometry, algebra, number theory
and differential equations. Non-commutative geometry provides an
appropriate framework for study of what is called "D-branes" in the
String Theory.
It combines physical idea of degenerating Conformal Field Theories with
mathematical idea of Gromov-Hausdorff collapse of Calabi-Yau manifolds,
as well as with unexpected relation to rigid analytic geometry. We
suggest to view a given Conformal Field Theory as a kind of
non-commutative space. Such non-commutative spaces can degenerate "at
infinity". Mirror symmetry can be explained in terms of the residual
commutative geometry. On the algebraic side we will meet homotopy
categories associated with compact symplectic manifolds. I am going to
explain non-commutative formal geometry of those homotopy categories.
There is another kind of non-commutative geometry of Mirror Symmetry. It
is geometry of deformed Calabi-Yau manifolds (a kind of deformation
quantization). I plan to discuss the way to construct such spaces
starting with real manifolds equipped with an integral affine structure.
This part of my lectures is also related to the so-called "tropical
geometry".
The lectures will be accessible to graduate students. We shall have two
lectures on Friday, October 7 from 3 to 5 pm, Saturday October 8,
10-12:30 am, and Sunday October 9, 10-12:30 am.
Contact person: Mark Sapir