Stable Cohomology: Foundations and Applications

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This award supports participation in a 5-day workshop to be held May 28-June 1, 2018 in Snowbird, Utah. The workshop will bring together world experts and early career researchers to explore some fundamental questions in algebra and representation theory, subjects that are important in many scientific and mathematical settings. The workshop will lead to a better understanding of stable cohomology, its foundations, and its applications. The workshop will thus have substantial impact on research in the many fields to which stable cohomology can be applied. At the same time, the workshop is an opportunity for early career mathematicians to learn from experts in a collaborative environment. Further details are on the conference website: http://www.math.ttu.edu/~lchriste/snowbird2018.htmlStable cohomology goes back to Tate, who in 1952 announced a new cohomology theory for finite groups, with a view towards applications in number theory. It deviates only slightly from classical (co)homology of finite groups, but the difference is decisive: it essentially combines group homology and group cohomology into one theory. Tate cohomology has since evolved along several different tracks, and there are applications in several branches of mathematics, including local algebra, singularity theory, representation theory, and homological and categorical algebra. Evolutions include generalizations to other groups and rings, as well as a Hochschild analogue for some associative algebras. Today one has several theoretical frameworks, depending on context, for treating stable cohomology. They expose different facets of the theory, and some properties are as evident in one framework as they are obscure in another. The purpose of the workshop is to bring together experts on homological algebra and mathematicians who apply stable cohomology in their research to address such broad questions as: Is there a unifying theoretical framework for stable cohomology in which all its properties emerge naturally? What kind of information, exactly, is encoded in stable cohomology?This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.