250091 VO Fano versus Calabi-Yau (2012W)

We will start with elementary inroduction to the theory of Fano and Calabu Yau manifolds.

The study of Geometry in the 20th Century was devoted, in large part and with astounding success, to the classification and parametrization of geometrical objects. However, these objects, of various kinds, were uniformly viewed somehow as ``sets of points''. Along the way, the relationship with categorical structures grew steadily, With PI Kontsevich's introduction of HMS - Homological Mirror Symmetry, a subtle change was introduced, in that ``Geometry'' began to be seen {\em within} a categorical structure. And the concurrent development of the theory of higher stacks meant that geometric structures were no longer viewed just as ``sets of points'' but rather as objects enclosing a higher structure.

Following this idea we inroduce classical TQFT approach by Athyah and Segal dressed by recent work of Lurie. The last part of the course will be dedicated to constructing categorical invariants. No prerequisits will be needed.