Keller 303

Speaker: Isaac Goldbring, UC Irvine.

Title:  The Connes Embedding Problem, MIP*=RE, and the Completeness Theorem

Abstract:  The Connes Embedding Problem (CEP) is arguably one of the most famous open problems in operator algebras.  Roughly, it asks if every tracial von Neumann algebra can be approximated by matrix algebras.  In 2020, a group of computer scientists proved a landmark result in complexity theory called MIP*=RE, and, as a corollary, gave a negative solution to the CEP.  However, the derivation of the negative solution of the CEP from MIP*=RE involves several very complicated detours through C*-algebra theory and quantum information theory.  In this talk, I will present joint work with Bradd Hart where we show how some relatively simple model-theoretic arguments can yield a direct proof of the failure of the CEP from MIP*=RE while simultaneously yielding a stronger, Gödelian-style refutation of CEP.  No prior background in any of these areas will be assumed.

Keller 403.

Title:  Non-deterministic Automatic Complexity of Fibonacci words

Abstract:  Automatic complexity rates can be thought of as a measure of how random words can be for some given automaton (machine). By creating a scale between 0 and 1 that ranges from predictable to complex, if the rate of a given word is strictly between 0 and 1/2  then we call it indeterminate. In this paper we show that for an infinite Fibonacci word the non-deterministic automatic complexity can be no greater than 1/Φ^2.

Kuykendall Hall 210

Title: Low($Pi^0_1$-IM) = $Delta^0_1$

Abstract: My dissertation investigated $Pi^0_1$-immune sets, i.e. those
that have no co-enumerable subset. This talk continues that work, first
connecting it to modern notions of computability-theoretic lowness. Then
I settle (in the affirmative) a conjecture that my dissertation left
open: only the computable sets fail to co-enumerate a $Pi^0_1$-immune
set.