This workshop brings together experts in algebraic topology, commutative algebra, and the modular representation theory of finite groups. There are many points of contact between these areas of mathematics. The focus of the workshop will be on duality phenomena and also on classification problems in various tensor-triangulated categories that arise naturally in these fields.
The derived category of a commutative ring—an interesting object in its own right—often serves as a (simpler) model for the stable category of a finite group, and also the stable homotopy category in topology. There are many similarities between these categories, for example the possession of a triangulated structure with a compatible symmetric product, but also notable differences; in particular, the stable homotopy category is significantly more complicated. Nevertheless, ideas and notions that arose in commutative algebra and algebraic geometry have been transplanted fruitfully to modular representation theory and stable homotopy category. Notable among these are Grothendieck duality theory, and the notion of Gorenstein rings and schemes, that, in the context of algebraic topology, yield far-reaching generalizations of classical Poincaré duality for manifolds. On the other hand, developments arising in algebraic topology, abstract homotopy theory, and modular representation theory, have provided powerful new tools and perspectives in algebra and geometry. The classification of thick and localising subcategories of various triangulated categories, and the computation of their spectra in the sense of Balmer, is a striking example. Another, and closely related, is the theory of cohomological support varieties that was first developed in modular representation theory.
The workshop will comprise 8 different lecture series, all of which connect to the ideas described above. These will be complemented by background lectures and discussion sessions. There will be around 5 hours of lectures everyday over the course of two weeks. The rough background required is a first course in algebraic topology, in commutative algebra, and a familiarity with homological algebra.
ICTS is committed to building an environment that is inclusive, non-discriminatory and welcoming of diverse individuals. We especially encourage the participation of women and other under-represented groups.
Eligibility Criteria: All faculty, postdocs, Ph.D students, advanced Master’s students are welcome to apply.