We first revisit notions of interpretability and internality in a category-theoretical language (for first order theories), reframing work of Hrushovski and Kamensky in a formalism derived from Makkai's early work. We then describe the issue of recovering the bi-intepretability class of a theory in terms of the automorphism group of a saturated model, and the role of the "Small Index Property" (SIP) - a way of recovering the topology of a group action from purely algebraic information.
We then turn to abstract elementary classes, and discuss the same notions, in the opposite order: first, two situations where a Small Index Property holds (joint work with Ghadernezhad), and then some applications to the problem of interpretation and reconstruction, adapted to abstract elementary classes.
On the small index property for AECs with strong amalgamation properties
Mar. 15, 2022 2:30pm (MATH 2…
Lie Theory
Lucas Gagnon (CU Boulder)
X
Hopf algebras encode inductive structure. Two areas where this is useful are combinatorics and representation theory, and this talk will describe a Hopf algebraic connection between these areas. On the combinatorial side is the chromatic quasisymmetric function, a q-analogue of Stanley’s chromatic symmetric function; this function encodes details of the graph coloring problem and can be realized as a homomorphism between Hopf algebra structures on graphs and (quasi-)symmetric functions. On the representation theory side is a collection of GLn(Fq)-modules that I will show to be canonically equivalent to some chromatic quasisymmetric functions. Along the way, I will construct a new Hopf algebra of class functions for the unipotent upper triangular groups and use supercharacter theory to give a representation theoretic realization of the graph-to-symmetric-function Hopf algebra homomorphism.
Representation theoretic Hopf algebras and q-chromatic symmetric functions
Mar. 15, 2022 3pm (MATH 350)
Topology
Juan Moreno (CU Boulder)
X
This talk should outline the Turaev-Viro construction which takes in a spherical fusion cat- egory as input and outputs a TQFT. The related Reshetikhin-Turaev construction should also be covered.
The talks are in-person with a computer pointed at the blackboard zoom option: https://cuboulder.zoom.us/j/98979344859?pwd=K3ZrTWt4Wkk1cllOQ1pkcW4vZ1FCZz09 Meeting ID: 989 7934 4859 Passcode: 787231
On the small index property for AECs with strong amalgamation properties