October 27: Michael R. Montgomery. Spin model subfactors

Complex Hadamard matrices generate a class of irreducible hyperfinite subfactors with integer Jones index coming from spin model commuting squares. I will prove a theorem that establishes a criterion implying that these subfactors have infinite depth. I then show that Paley type II and Petrescu's continuous family of Hadamard matrices yield infinity$

November 3: Anup Poudel. Diagrammatic methods for ribbon tensor categories

It is known that given a tensor category, one can associate string diagrams to represent its morphism space. When the category has other nice structures (braided and spherical), one can view the morphism$

November 17: Kathlyn Dykes. MV polytopes of highest vertex w

Mirkovic and Vilonen defined a useful basis for the irreducible representations of an algebraic group G, which is indexed by certain combinatorial objects called MV polytopes. From the works of Kamnitzer and Goncharov-Shen, MV polytopes are in correspondence with the non-negative tropical points of the unipotent group of G. We consider a certain subclass of MV polytopes with highest vertex w and show that they are in correspondence to the non-negative tropical points of a reduced double Bruhat cell. We will also explore the combinatorics of this subclass of polytopes to show that the vertices can be labelled by the elements in the Weyl group which are less than w.

December 1: Noah Riggenbach. On the algebraic K-theory of double points

Algebraic K-theory is a very interesting and subtle invariant which shows up in algebra, geometry, and topology. For regular noetherian rings we have several comparison theorems which give us many compu$ and others to compute algebraic K-theory of some simple singular rings, such as double points.