Monday, September 16, 2019
09/16/2019 - 4:15pm
We will report on a series of works in the last 2 decades aroound the following question: “For a given simple Lie group G, how many lattices (i.e., discrete subgroups of finite covolume) does it have of covolume at most x?” Equivalently: “How many manifolds (of volume at most x) are covered by the associated symmetric space.?”As many of these lattices are arithmetic, these questions often lead to deep number theoretic problems: counting primes etc. A recent joint work (joint with M. Belolipetsky) gives a sharp estimate for the number of non unoform lattices in high rank simple groups.
Equivariant Localization in Factorization Homology and Vertex Algebras from Supersymmetric Gauge Theory
09/16/2019 - 4:30pm
I will introduce factorization algebras and their factorization homology, following Beilinson-Drinfeld and Francis-Gaitsgory, and explain how these generalize the notions of vertex algebras and their conformal blocks. I will give the definition of an equivariant factorization algebra on a variety with group action, and prove an analogue of the equivariant localization theorem for factorization homology. Finally, I will explain a new family of constructions of factorization algebras on curves and manifest relations between them (in particular yielding concrete results about vertex algebras) which are motivated by higher dimensional physics via the preceding localization principle.