Abstract: In a landmark 1980 monograph, Ziegler studied finitely generated subgroups of existentially closed groups and uncovered a surprising connection of this purely algebraic question and computability. To this end, he defined a new reducibility which he called *-reducibility and which I will call Ziegler reducibility, which is a refinement of both Turing and enumeration reducibility. In particular, he showed that if a finitely generated group H is a subgroup of an existentially closed group G and the word problem of a finitely generated group H₀ is Ziegler reducible to the word problem of H, then H₀ is (isomorphic to) a subgroup of G as well.

In joint work with Isabella Scott and Josiah Jacobsen-Grocott, I studied the induced degree structure of the Ziegler degrees and was able to show that there is a minimal Ziegler degree, and indeed that any finite distributive lattice is isomorphic to an initial segment of the Ziegler degrees. As a consequence, the ∀∃∀-theory of the Ziegler degrees (as a partial order) is undecidable.

Abstract: Bakker, Brunebarbe, Tsimerman showed in 2022 that the definable structure sheaf O_Cⁿ of Cⁿ is a coherent O_Cⁿ-module as a sheaf on the site Cⁿ, where the coverings are finite coverings by definable open sets.

In general, let K be an algebraically closed field of characteristic zero. We give another proof of the coherence of O_Kⁿ as a sheaf of O_Kⁿ-modules on the site Kⁿ using spectral topology on the type space S_n(K). (Here S_n(K) just means S_2n(R) for some real closed field R.)

It also gives an example of how the intuition that sheaves on the type space are the same as sheaves on the site with finite coverings (see Proposition 3.2 in Edmundo (2006)) can be applied.

Abstract: How does the size of a collection of line segments in Rⁿ change if we extend each segment to a full line? In this talk, we investigate this problem in geometric measure theory using techniques from algorithmic information theory, specifically Kolmogorov complexity and the point-to-set principle of Lutz and Lutz. Working in the plane, we show that the packing dimension of any set does not increase under line segment extension. This allows us to prove the generalized Kakeya conjecture for packing dimension in the plane. Finally, we discuss versions of this problem in higher dimensions. This is based on joint work with Ryan Bushling.

Abstract: We consider combinatorial problems on the free parts of the Bernoulli shift actions of countable abelian groups, such as chromatic numbers, edge chromatic numbers, perfect matchings, etc. These problems can all be regarded as special cases of the problem whether there exist continuous equivariant maps from the free part of the Bernoulli shift action to a subshift of finite type. We prove a master theorem which in theory gives complete answers to the subshift problem. Furthermore, we show that the class of (codes for) all subshifts of finite type with a positive answer to the subshift problem is a complete c.e. set. This is joint work with Steve Jackson, Ed Krohne, and Brandon Seward.

Abstract: We investigate which algebra substructures A of reals are such that there is a real x for which A+xA=R.

Abstract: In 1961, Vaught proposed his conjecture on the number of countable models of a first-order theory. Despite some early progress and attempts from various areas of logic, the conjecture remains unsolved today. In this talk, we will survey certain results around the conjecture, including alternative formulations and solved special cases. In particular, we will discuss Shelah's proof of Vaught's conjecture for complete ω-stable first-order theories and how that relates to my own work in progress.

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Abstract: K-theory is a general method for associating countable Abelian groups with mathematical structures. These groups are invariants, but they are not always classifiers. Lately, Chris Eagle, Isaac Goldbring, Russell Miller, and I have been thinking about the computability of K-theory as it applies to operator algebras (i.e., C^*-algebras). I will try to explain the machinery of K-theory for operator algebras and explain at least the architecture of the following theorems. 1) If A is a computably presentable C^*-algebra, then the Abelian group K₀(A) has a c.e. (a.k.a. recursive) presentation. 2) If A is a computably presentable uniformly hyperfinite algebra, then K₀(A) has a computable presentation; moreover, the trace of A is computable and its supernatural number is lower semi-computable.

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