Lecture series title: Derived categories in algebraic geometry

1) Birational geometry and derived categories 

The derived category of coherent sheaves was originally conceived as a technical tool for studying sheaf cohomology in algebraic geometry, but it has since emerged as a powerful invariant of algebraic varieties in its own right. I will survey some of the known and conjectural relations between the structure of this category and birational geometry, especially in the cases of Calabi-Yau and Fano varieties. This perspective naturally leads to a "noncommutative" enlargement of classical algebraic geometry, in which spaces are reimagined as categories (as opposed to sets) with extra structure. The talk is meant to be a general introduction to these ideas; no previous knowledge of derived categories or birational geometry will be assumed.

2) Hyperkähler varieties and noncommutative Calabi-Yau surfaces 

Hyperkähler varieties form one of the three building blocks for projective varieties with trivial canonical bundle. Their classification is widely open, but there is growing evidence that the problem is closely related to the classification of noncommutative Calabi-Yau surfaces. In particular, given such a surface, one can hope to construct hyperkähler varieties as moduli spaces of Bridgeland stable objects. I will discuss this circle of ideas, including recent work with Arend Bayer, Laura Pertusi, and Xiaolei Zhao completing the story for hyperkähler varieties of Kummer type. 

3) The period-index conjecture and the noncommutative Hodge conjecture 

The Brauer group is a fundamental invariant classifying the central division algebras over a field, with numerous applications in geometry and arithmetic. In the case of the function field of a variety, the period-index conjecture proposes a precise bound on the most basic invariant of a division algebra, its dimension, in terms of its order in the Brauer group. I will explain recent progress on this conjecture, including a proof for unramified division algebras over the function field of an abelian threefold, based on joint work with James Hotchkiss. This depends on a reinterpretation of the conjecture in terms of a noncommutative version of the integral Hodge conjecture, which for noncommutative Calabi-Yau threefolds can be approached using enumerative geometry.

All lectures will take place from 4:00 – 5:00pm in SCGP room 102

Live streaming: scgp.stonybrook.edu/live

Inaugural Stony Brook Lectures in Algebraic Geometry

Speaker: Professor Jacob Tsimerman (University of Toronto)
“Transcendence in Algebraic Geometry”

Tuesday, November 7. Title: Transcendence of period integrals over function fields. ABSTRACT

Wednesday, November 8. Title: Definable o-minimal structures and applications to Periods. ABSTRACT

Thursday, November 9. Title: Unlikely Intersection problems and Functional Transcendence. ABSTRACT

All lectures will take place from 4:00 – 5:00pm in SCGP room 102

Live streaming: scgp.stonybrook.edu/live