We discuss how one can “categorify” the Temperley-Lieb, Hecke algebras, and their modules. We then discuss what is knowable (and unknowable!) about the representation theoretic structure of these categorifications.

Abstract: Let X_0 be a rational surface with a cyclic quotient singularity (1,a)/r. Kawamata constructed a remarkable vector bundle F_0 on X_0 such that the finite-dimensional algebra End(F_0) "absorbs" the singularity of X_0 in a categorical sense. If we deform over an irreducible component of the versal deformation space of X_0 (as described by Kollár and…

This talk concerns monads and a variant of them called relative monads. I'll begin by recalling Moggi’s metalanguage for monads, and explain how it extends to relative monads. A central issue in doing so is the study of “strength” for relative monads. I will illustrate the framework with examples drawn from algebraic theories and “graded” monads (no prior familiarity assumed).

The magnitude of a metric space is a real-valued invariant that counts the "effective number of points" in the space. In this talk, I will describe a relation between the magnitude of a (nice enough) metric space and an invariant from Euclidean geometry: to each space X, one can associate a simplex S such that the magnitude of X is a simple transformation of the circumradius of S. Furthermore, this relation extends to all subspaces X' of X and corresponding faces S' of S, and is uniquely defined by this property. As one application, I will discuss how this geometric perspective identifies a new regime in the asymptotic behaviour of metric spaces when rescaling the metric. Finally, if time permits, I will say a few words on connections to graph theory and potential theory; some keywords which may or may not be covered are: Laplacians, discrete curvature, capacity, second moment.

Abstract: TBA