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.
In May 2007, we organized a workshop at the Fields Institute for Research in Mathematical Sciences in Toronto on stacks in geometry and and topology. Despite the vague name, stacks are a fundamental tool used for classifying objects with continuously varying geometric properties. The workshop was the centerpiece of a two-month program in Spring 2007 intended to explore interactions between algebraic geometry, algebraic topology, and homotopy theory. What we did not realize at the time was that this conference and the entire program caught the wave of the new and 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, topological, and even number theoretic subjects. This 2010 workshop partially funded by this grant was intended as a follow-up to the 2007 conference: we envisioned revisiting the field three years later to assess what had happened and to see where we are going. In the end, however, the conference became much more than that, for we found that derived algebraic geometry and the associated homotopy theory has applications across a broad selection of fields, including algebraic topology,algebraic geometry, homotopy theory, category theory, and the more topological side of mathematical physics. This led to an eclectic and wide-ranging series of talks, some from established research mathematicians, but many from some of the emerging stars in the field. It also led to a large, vibrant and young audience: we filled the main lecture hall at the Fields Institute for a week with research mathematicians and graduate students from all over the world. The funds from this grant were spent entirely to fund travel by graduate students, postdocs, and non-tenured research mathematicians to the conference; in short, we enabled the participation of twenty-five researchers who might otherwise not have been able to attend this important conference.