In the 1970s Saharon Shelah initiated a program to develop classification theory for non-elementary classes, and eventually settled on the setting of abstract elementary classes. For over three decades, limited progress was made, most of which required additional set theoretic axioms. In 2001, Rami Grossberg and I introduced the model theoretic concept of tameness which opened the door for stability results in abstract elementary classes in ZFC. During the following 20 years, tameness along with limit models have been used by several mathematicians to prove categoricity theorems and to develop non-first order analogs to forking calculus and stability theory, solving a very large number of problems posed by Shelah in ZFC. Recently, Marcos Mazari-Armida found applications to Abelian group theory and ring theory. In this presentation I will highlight some of the more surprising results involving tameness and limit models from the past 20 years.
Twenty years of tameness
Mar. 08, 2022 2:30pm (MATH 2…
Lie Theory
Breeaan Wilson (CU Boulder) Practice Session for Comprenhensive Exam
Mar. 08, 2022 3pm (MATH 350)
Topology
Josh Grochow (CU Boulder)
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The goal is to make concrete some of the subtleties that have come up as questions in the previous talks about the definitions of modular tensor categories, by looking at them using some small finite groups as examples. In particular, we'll only need to look at groups of order at most 8, where we can compute everything explicitly by hand. We'll see: Why there is a GL(n)'s worth of choices of how to decompose a direct sum of n isomorphic simples. What the decomposition of a tensor product of simples looks like, and a fairly explicit example of the braiding and associativity isomorphisms In what way do MTCs generalize finite groups? There's some subtlety to the phrase "Rep(G) is an MTC", and this will take us into exactly what data about Rep(G) is enough to determine the group G, which will lead to a discussion of Tannaka duality, the representation ring (a decategorification of Rep(G)), and a result of Etingof & Gelaki characterizing what happens if you forget the symmetry/braiding isomorphisms. It'll be helpful for our calculations if you've seen character tables of finite groups before, but my goal is for the story to still make sense even if you haven't seen character tables.
Phenomena in modular tensor categories through the lens of small finite groups