Abstracts - Spring 2011 

Speaker: Jun Li

Title: Finding rational curves in a K3 surface

Abstract: Finding rational curves in a polarized K3 surface is a classical and a multi-facet problem. After briefly reviewing the classical and the moduli approach to produce rational curves in such a K3, I will concentrate on the method of reduction to finite field that produces infinitely many irreducible rational curves in a K3 of odd Picard number. 

Speaker: Raphael Rouquier

Title: Dimensions in commutative and non-commutative geometry

Abstract: We will discuss various finiteness questions and associated dimensions: dimensions of triangulated categories, Kimura finiteness and Carlsson's conjecture.

Speaker: Bernd Siebert

Title: Logarithmic Gromov-Witten invariants

Abstract:  So far Gromov-Witten invariants relative to a divisor $D$ could only be defined under the assumption that $D$ is smooth. In the talk I will survey recent joint work with Mark Gross (http://arxiv.org/pdf/1102.4322v1) establishing Gromov-Witten theory in log smooth situations. In particular, this includes the cases that $D$ has (simple) normal crossings, or of fibers of semi-stable families. There is an interesting appearance of tropical geometry in the story.

Speaker: Andrei Suslin

Title: The group K_2 for division algebras

Abstract:  Let D be a division algebra over F.  Denote by \bar(K)_2(D) the quotient of K_2(D) modulo the subgroup generated by elements coming from splitting fields of D.  For algebras of squarefree degree it was proved by Merkurjev and Suslin in 1981 that \bar(K)_2(D) = 0.  The question whether \bar(K)_2(D) vanishes for other algebras remains open.  We study the case of a biquaternion algebra and relate in this case \bar(K)_2(D) with Galois cohomology of F.