Localization techniques in equivariant cohomology

March 15 to March 19, 2010

at the

American Institute of Mathematics, Palo Alto, California

organized by

William Fulton, Rebecca Goldin, and Julianna Tymoczko

This workshop, sponsored by AIM and the NSF, will be devoted to localization techniques in equivariant cohomology. Localization techniques in equivariant cohomology are a powerful tool in computational algebraic topology in the context of a topological space with the action of a Lie group. The development of these techniques has led to an explosion of research, including groundbreaking work in many different fields, especially algebraic combinatorics, algebraic geometry, symplectic geometry, and algebraic topology.

The goal of this conference is to relate the geometric and combinatorial aspects of what has been dubbed ''GKM'' theory by bringing researchers from combinatorics and geometry/topology together, so that workers in each of these fields can benefit from the insights and experience of those in the other. The conference will address open questions such as the following:

  1. Can the combinatorial techniques of GKM theory be extended to a wider class of spaces, even if the image of the inclusion map cannot be described? These families could include singular spaces, infinite dimensional spaces, stacky spaces (including Deligne-Mumford stacks), and spaces with specific kinds of group actions or geometric structures.
  2. For what larger class of manifolds than $G/P$ does the Kostant program extend? Resolving this would entail finding geometric reasons behind the combinatorial rules in $G/P$. What specific kinds of group actions or geometric structure would be required? Answering this question would allow us to apply these powerful combinatorial techniques to reveal geometric and algebraic structure (such as intersections and cohomology rings of all types) in a much broader context.
  3. Can the techniques of GKM theory be used to prove combinatorially positive results in Schubert calculus or its generalization in an appropriate geometric setting? A combinatorially positive result interprets structure constants in the cohomology ring as the cardinality of a set; an answer to this question might generalize combinatorial objects such as puzzles, cartons, or tableaux to an algorithm that counts paths within graphs, and give closed positive formulas for structure constants in an equivariant cohomology ring.
This conference will gather researchers from each of the main approaches to equivariant cohomology: algebraic geometry, symplectic geometry, algebraic topology, and algebraic combinatorics.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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