Workshop on Homotopy theory and Derived Algebraic Geometry

Northwestern University, Evanston IL

Investigators

Abstract

In May 2007 there was a workshop at the Fields Institute on stacks in geometry and topology; with this workshop, we saw a snapshot of the emerging field of derived algebraic geometry. In a remarkable series of talks, many by mathematicians with relatively recent PhDs, we saw the implementation and application of derived schemes, derived stacks, higher categories, and the attendant homotopy theory across a broad spectrum of geometric and topological subjects. This new workshop is a follow-up to the 2007 conference: the main point is to revisit the field three years later, to assess what has happened and to to see where we are going. In particular, the field of derived algebraic geometry and its interplay with higher category theory field has grown rapidly since 2007 and is now central to several developing areas of algebraic topology. It is an ideal moment to explore this interplay. Researchers who have agreed to participate include Mark Behrens (MIT), D-C. Cisinski (Paris 13), Ralph Cohen (Stanford), Andre Henriques (Utrecht), Gerd Laures, (Bochum), Tyler Lawson (Minnesota), Mike Mandell (Indiana), Niko Naumann (Regensburg), and Charles Rezk (UIUC). Algebraic geometry is a classical field of mathematics, arising from the study of solutions of systems of polynomial equations in many variables. The focus on polynomials make the geometric objects studied very rigid, in contrast to topology, which is the study of phenomena which remain unchanged under any continuous deformation. Derived algebraic geometry seeks to import techniques from algebraic topology into algebraic geometry in order to capture and calculate some of the finer structure apparently hidden by the inherent rigidity. There have been remarkable recent successes. This grant will be used to fund the attendance of US research mathematicians near the beginning of their careers, in this way promoting the spread of these ideas among the broader research community.