27/02/2014 Misha Shkolnikov: Yukawa couplings, large complex structure and mirror symmetry for the quintic, part 1
1/03/2014 Misha Shkolnikov: What is Maslov index? Ivan Bazhov: What is a Batyrev construction?
3/03/2014 Misha Shkolnikov: Yukawa couplings, large complex structure and mirror symmetry for the quintic, part 2
8/03/2014 Sergei Lanzat: What is a Special Lagrangian fibration?
22/03/2014 Nikita Kalinin: What is a Mixed Hodge Structure?
27/03/2014 Sergey Galkin: Rationality problems, with special regard to cubic hyper surfaces.
Abstract:The first (out of two) talks will be introductory. I will tell about the problems in birational geometry and tools for solving them. I will introduce various flavours of rationality: rational, unirational, stably rational, and rationally connected varieties, and describe the subtle differences between them. Also I'll tell known theorems, examples, conjectures and counterexamples, regarding rationality of cubic hypersurfaces and other related varieties.
29/03/2014 Sergey Galkin: Grothendieck's constructor and irrationality of cubic fourfolds.
Abstract: This is second part of the story. I will introduce Grothendieck ring of varieties (ring of generalized Euler characteristic or so-called “motivic measures”) and show how it could be used for solving rationality problems, using theorem of Larsen and Lunts. I will demonstrate a beautiful identity in this ring, that relates symmetric square of cubic hypersurface with its Fano variety of lines. We will use this identity to show, that if class of an affine line is not a zero divisor in the Grothendieck ring of complex varieties, then variety of lines on a rational cubic fourfold is birational to symmetric square of a K3 surface, in particular, generic cubic fourfold is irrational. This is a joint work with Evgeny Shinder.
5/04/2014 Sergei Lanzat: What is a Picard-Fuchs equation? Mirror Symmetry of the quintic 3-fold.