Summer Graduate School

Schur quartic x 4−xy3 = z 4−zu3 and several of the 64 lines that it contains

Derived algebraic geometry is an ‘update’ of algebraic geometry using ‘derived’ (roughly speaking, homological) techniques. This requires recasting the very foundations of the field: rings have to be replaced by differential graded algebras (or other forms of derived rings), categories by higher categories, and so on. The result is a powerful set of new tools, useful both within algebraic geometry and in related areas. The school serves as an introduction to these techniques, focusing on their applications.

The school is built around two related courses on geometric (‘derived spaces’) and categorical (‘derived categories’) aspects of the theory. Our goal is to explain the key ideas and concepts, while trying to keep technicalities to a minimum.

School Structure

Two lectures will be held in the mornings. The afternoons will consist of problem sessions in which students will be given exercises directly related to the course subject. There will also be discussion sessions where interaction will be more free form and would cover higher-level topics, making the material exciting for advanced students while also provided less advanced students with a broader view of the field.

Suggested Prerequisites

• The main prerequisite is familiarity with derived categories in algebraic geometry.
• Traditional references on these topics would be Hartshorne (Algebraic Geometry), particularly the first 3 chapters, and Weibel (An Introduction to Homological Algebra, errata) particularly Chapters 1–5 and 10. (Chapter 10 of Weibel’s book introduces to derived categories; it is essentially independent from Chapters 6–9.)
• Familiar with basic objects of geometric representation theory, such as constructible sheaves and representations of algebraic groups.
• Some familiarity with infinity-categories would be helpful.  For example, Rezk (Introduction to Quasicategories), Groth (A Short Course on Infinity-Categories), Riehl and Verity (Elements of Infinity-Category Theory, alternate), Cisinski (Higher Categories and Homotopical Algebra, alternate, errata), or Land (Introduction to Infinity-Categories).
• The functor of points approach to algebraic geometry as in chapter VI "Schemes and Functors" of Eisenbud and Harris (The Geometry of Schemes).

Suggested Readings

• D. Gaitsgory. Generalities on DG categories The notes (only 17 pages!) summarize properties of dg categories for future applications
• V. Drinfeld. DG quotients of DG categories This (essentially self-contained) paper focuses on quotients and localization (which, in particular, is necessary to define derived categories). Section 1 is recommended reading in preparation for the school; the rest of the paper may be best left until after the school.

For eligibility and how to apply, see the Summer Graduate Schools homepage