Localization techniques in equivariant cohomology

March 15 to March 19, 2010

at the

American Institute of Mathematics, San Jose, California

organized by

William Fulton, Rebecca Goldin, and Julianna Tymoczko

Original Announcement

This workshop 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.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Equivariant Schubert calculus and jeu de taquin
by  Hugh Thomas and Alexander Yong
Equivariant cohomology, syzygies and orbit structure
by  Christopher Allday, Matthias Franz, and Volker Puppe,  Trans. Amer. Math. Soc. 366 (2014), no. 12, 6567-6589  MR3267019
What is equivariant cohomology?
by  Loring Tu,  Notices Amer. Math. Soc. 58 (2011), no. 3, 423-426  MR2789121
Singularities of generalized Richardson varieties
by  Sara Billey and Izzet Coskun
Patch ideals and Peterson varieties
by  Erik Insko and Alexander Yong
Singularities of generalized Richardson varieties
by  Sara Billey and Izzet Coskun