Mathematics Department Colloquium 
The Ohio State University

  Year 2017-2018

Time: Thursdays 4:15 pm
Location: CH 240

Schedule of talks:


September 21st  
Th 4:15pm
Monica Vazirani 
(U.C. Davis) 
Combinatorics, Categorification, and Crystals
October 5th  
Th 4:15pm
Thomas Koberda 
(U. of Virginia) 
October 19th  
Th 4:15pm
Alan Reid  
(U. of Texas) 
October 26th  
Th 4:15pm
Renzo Cavalieri 
(Colorado State U.) 
November 2nd  
Th 4:15pm
Noah Snyder  
(Indiana U.) 
November 9nd  
Th 4:15pm
Deann Needell 


(M. Vazirani): Categorification attempts to replace sets or algebraic and geometric structures with more general categories. It has enjoyed amazing successes, such as Khovanov homology categorifying the Jones polynomial knot invariant, KLR algebras categorifying quantum groups, or Soergel bimodules categorifying Hecke algebras. Many of the algebras we see in categorification can be described diagrammatically, which is in its own way very combinatorial. This is related to an historic motivation for categorification: to construct knot and link invariants. The payoffs to finding these richer, higher categorical structures include not only constructing finer knot invariants, but proving positivity results and producing some fantastic mathematics. In this talk, I will focus on the second example, that is, on quantum groups. Their crystal bases or canonical bases exhibit the positivity and integrality that is a trademark feature of a decategorified structure. My launch point will be the type A combinatorics of Young diagrams or partitions. These encode the representation theory of the symmetric group, but they also form a crystal--the crystal graph of the basic representation of SL\infty. This is not a coincidence. The symmetric groups categorify the basic representation, with induction and restriction functors descending to raising and lowering operators. This phenomenon generalizes to all symmetrizable types replacing the symmetric groups with cyclotomic Khovanov-Lauda-Rouquier (KLR) algebras.

Past Colloquium

Ohio State University Mathematics Department Colloquium Year 2016-2017

This page is maintained by Facundo Mémoli.