**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.*

**:**Watanabe 112

__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.

https://math.hawaii.edu/home/depart/theses/MA_2023_Walker.pdf

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.

**Keller 303**

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.

Colloquium** in Keller 303**

The speaker is **Dr. Hailun Zheng**, from the University of Houston-Downtown.

Title: Polytope and spheres: the enumeration and reconstruction problems

Abstract: Consider a simplicial d-polytope P or a simplicial (d-1)-sphere P with n vertices. What are the possible numbers of faces in each dimension? What partial information about P is enough to reconstruct P up to certain equivalences?

In this talk, I will introduce the theory of stress spaces developed by Lee. I will report on recent progress on conjectures of Kalai asserting that under certain conditions one can reconstruct P from the space of affine stresses of P —- a higher-dimensional analog of the set of affine dependencies of vertices of P. This in turn leads to new results in the face enumeration of polytopes and spheres; in particular, a strengthening of (the numerical part of) the g-theorem.

Joint work with Satoshi Murai and Isabella Novik.