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.
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
Rukiyah Walker, Monday, November 13, 2:30 – 4:30 pm, Keller 403
Title: A Generalized Epidemiological Compartmental Model
Abstract: Epidemiological compartmental models are standard and important tools used to analyze the spread of infectious diseases. These models divide a population into distinct compartments, such as susceptible, infected, and recovered, based on their disease status. The behavior of the models can then be analyzed using methods from dynamical systems to better understand the spread of a particular disease. The ongoing COVID-19 pandemic highlighted the unpreparedness of societies worldwide in effectively responding to a devastating pandemic. Additionally, existing compartmental models often lack the capabilities to be applied to different diseases. Hence, the need to develop comprehensive models that can accurately predict the behavior of infectious diseases has become evident. In this work we propose a general compartmental model which has the potential to capture the behavior of specific diseases based on key parameters. We study the equilibria and stability of this model, as well as several submodels, by utilizing techniques from dynamical systems. Furthermore, we illustrate examples of varying fundamental parameters, such as transmission rates, latency periods, and more, with simulations for the different submodels that we will encounter.
Title: Wedge Product and Naturality in Discrete Exterior Calculus
Abstract: In exterior calculus on smooth manifolds, the exterior derivative and wedge product are natural with respect to smooth maps between manifolds, that is, these operations commute with pullback. In discrete exterior calculus (DEC), simplicial cochains play the role of discrete forms, the coboundary operator serves as the discrete exterior derivative, and the antisymmetrized cup product provides a discrete wedge product. We show that these discrete operations in DEC are natural with respect to abstract simplicial maps. A second contribution is a new averaging interpretation of the discrete wedge product in DEC. We also show that this wedge product is the same as Wilson’s cochain product defined using Whitney and de Rham maps. This may lead to existence of more accurate combinatorial wedge products that are associative in the limit. In any case, the combinatorial product may be useful in creating an A-infinity or C-infinity algebra. Joint work with Mark Schubel and Daniel Berwick-Evans.
Speaker: James Hyde, Copenhagen
Title: Groups, Orders and Dynamics.
Abstract: I will describe some of the history of combinatorial group theory, and its connections with algorithmic questions such as the decidability of the word problem.
I will then define ordered groups and describe their connection to group actions on the real line. Then I will state some of my recent results (in part joint with Lodha) resolving two central questions in the area.