Summer Graduate School

<p>Homotopy measures how spheres can be tangled in spaces; the logo shows a sphere tangled in a grove of California redwoods<br />The background painting is &ldquo;Giant Redwood Trees of California&rdquo; by Albert Bierstadt in 1874</p>

The goal of this school will be to introduce students to several powerful cohomological tools that were brought to commutative algebra by Avramov in the 80s and 90s: Lie algebra methods from homotopy theory, and support theoretic methods from the representation theory of finite groups. These tools have have seen a huge array of applications that continue to grow, with several major developments in recent years, opening new connections to algebraic topology, noncommutative algebraic geometry, and representation theory.

Students will be introduced to cohomological support varieties and the homotopy Lie algebra, and will get a first glance at how to use these tools to understand the new developments mentioned above. To build towards this, students will become familiarized with differential graded algebras and modules, minimal models, and derived category techniques. The school will also include an introduction to the computer algebra system Macaulay2, with the goal of making seemingly abstract calculations concrete.

School Structure

There will be two lectures each day, each followed by an accompanying problem sessions. In addition to the problem sessions, there will be two Macaulay2 sessions (one on the very first day, to get set up), and the problem sessions will include Macaulay2 questions.

Prerequisites

• Basic commutative algebra: prime ideals, localization, dimension theory, graded rings, regular sequences and depth. Knowledge of the classical singularities would be helpful but not strictly necessary: Cohen-Macaulay, Gorenstein, complete intersection and regular rings. Suitable references for this material are chapters 1-2 and 5-7 from Matsumura’s book Commutative Ring Theory, as well as chapters 1-3 from Bruns and Herzog’s text Cohen-Macaulay rings.

• Basic homological algebra: complexes and homology/homotopy, projective resolutions, derived functors like Ext and Tor. Knowledge of the Koszul complex would be helpful. This material can be found in Rotman’s An introduction to homological algebra (second edition), chapter 2 and sections 6.1, 6.2, 7.1, and 7.2, or Weibel’s An introduction to homological algebra chapters 1, 2, and 3.

Application Procedure

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