For a smooth surface S, the Hilbert scheme of points on S gives a smooth compactification of the configuration space of n distinct points on S. Its cohomology is by now well understood and exhibits deep connections with representation theory. Understanding its quantum cohomology, a deformation of ordinary cohomology involving curve counting invariants, has since received a lot of…

In the study of dualizability of tensor categories, Brochier-Jordan-Snyder observe that the tensor product of two (compact-)rigid monoidal categories may fail to be (compact-)rigid, and hence the collection of such categories fails to form a well-behaved higher category, in which fully extended topological field theories could take values. To remedy this, one weakens the definition of a tensor category and instead requires only rigidity of (compact) projective objects, known as cp-rigidity.

In this talk, I will present some initial structural results for module categories over finite cp-rigid categories, which we term pre-tensor categories, following Décoppet. This includes variants of the standard reconstruction results via algebra and bimodule objects, a description of exact module categories, and, perhaps most importantly, a characterization of pre-tensor categories which are Morita equivalent to (rigid, finite) tensor categories. Higher-categorical applications include a characterization of fully dualizable pre-tensor categories, and of Witt equivalence as equivalence internal to the Morita 4-category of braided tensor categories.

This talk is based on work in progress joint with Thibault Décoppet.

Abstract: In this talk we will describe an interpretation of a certain moduli space of vector bundles of rank 3 on a curve of genus 2 as an orbital degeneracy locus on the Grassmannian G(3,9). This allows to extract interesting geometric data about the moduli space and has also some consequences on models of K3 surfaces of genus 19. This is a…

Abstract: TBA

A representable natural transformation u : U* —> U in the category Psh(C) of presheaves on a small category C is a “natural model" of dependent type theory. The type-forming operations may be described as algebraic structure on u, representing corresponding operations on the type-families classified by u. For example, the dependent product or “Pi-type” is an algebra structure for the polynomial endofunctor

P_u : Psh(C) —> Psh(C) .

Similar operations on u represent the other type-formers of unit type, dependent sums, and identity types. The latter are given by a newly discovered “path-type” structure, which relates such models to cubical (Quillen) model categories.

Abstract: TBA