Overview

Date: March 23, 2024
Place: Dartmouth College (campus map)

Schedule

10:30–11am	Coffee & Snacks	300 Kemeny Hall
11am–12pm	Kameryn Williams (Bard College at Simon’s Rock)	041 Haldeman Center
12pm–12:30pm	Java Villano (University of Connecticut)	041 Haldeman Center
12:30–2pm	Lunch	300 Kemeny Hall
2–2:30pm	Jason Block (CUNY Graduate Center)	041 Haldeman Center
2:30–3pm	Heidi Benham (University of Connecticut)	041 Haldeman Center
3–3:30pm	Andrea Volpi (University of Udine)	041 Haldeman Center
(The room scheduled for talks is subject to change, so stay tuned for updates.)

Speakers & Abstracts

Heidi Benham (University of Connecticut)

The Ginsburg–Sands Theorem and Computability Theory

At present, topological theorems have a relatively small presence in the reverse mathematical zoo. One topological result that has turned out to be a fertile starting point for reverse mathematical analysis is a theorem due to Ginsburg and Sands. The theorem states that every infinite topological space has an infinite subspace that is homeomorphic to exactly one of the following five topologies on the natural numbers: indiscrete, discrete, initial segment, final segment, or cofinite. The original proof, which is nonconstructive and uses an application of Ramsey’s Theorem for pairs, was given in the context of topology. This left open the question of which axioms are necessary to prove this theorem. Using Dorais’s formalization of CSC spaces, we analyze the location of this theorem in the reverse math zoo, as well as the location of several related theorems. The Ginsburg–Sands Theorem for CSC spaces turns out to be equivalent to \(ACA_0\). We also look at the theorem restricted to certain types of topological spaces. On the weaker end of the spectrum, we have that the Ginsburg–Sands when restricted to Hausdorff spaces is equivalent to RCA_0. Interestingly, when restricting the theorem to T_1 CSC spaces, the strength is equivalent to none of the big five subsystems of second order arithmetic, but rather lies strictly between the strength of \(ACA_0\) and \(RT^2_2\), which is an unusual place for a natural mathematical theorem. This talk is based on work done jointly with Andrew DeLapo, Damir Dzhafarov, Reed Solomon, and Java Darleen Villano.

Jason Block (CUNY)

Complexities of Theories of Profinite Subgroups of \(S_\omega\) via Tree Presentations

Although \(S_\omega\) \(d\) (the group of all permutations of \(\mathbb{N}\) is size continuum, both it and a large class of its subgroups can be presented as the set of paths through a countable tree. The subgroups of \(S_\omega\) that can be presented this way with finite branching trees are exactly the profinite ones. We use these tree presentations to find upper bounds on the complexity of the existential and \(\exists\forall\) theories of profinite subgroups of \(S_\omega\), as well as to prove sharpness for some of these bounds. Additionally, these complexity results enable us to distinguish a simple subclass of profinite groups, those with \emph{orbit independence}, from the rest of the class.

Java Villano (University of Connecticut)

Computable Categoricity Relative to a C.E. Degree

A computable structure \(\mathcal{A}\) is said to be computably categorical relative to a degree \(\mathbf{d}\) if for all \(\mathbf{d}\)-computable copies \(\mathcal{B}\) of \(\mathcal{A}\), there exists a \(\mathbf{d}\)-computable isomorphism between \(\mathcal{A}\) and \(\mathcal{B}\). In a 2021 result by Downey, Harrison-Trainor, and Melnikov, it was shown that there exists a computable graph \(\mathcal{G}\) such that for an infinite increasing sequence of c.e.\ degrees \(\mathbf{x}_0 <_T \mathbf{y}_0 <_T \mathbf{x}_1 <_T \mathbf{y}_1\dots\), \(\mathcal{G}\) was computably categorical relative to each \(\mathbf{x}_i\) but not computably categorical relative to each \(\mathbf{y}_i\). In this talk, we will outline how to extend this result for partial orders of c.e.\ degrees, and discuss some possible future directions for this topic.

Andrea Volpi (Università di Udine)

A Transfinite Version of Ramsey Theorem

Finite Ramsey Theorem states that fixed \(n,m,k \in \mathbb{N}\), there exists \(N \in \mathbb{N}\) such that for each coloring of \([N]^n\) with \(k\) colors, there is a homogeneous subset \(H\) of \(N\) of cardinality at least \(m\). Starting with the celebrated Paris-Harrington theorem, many Ramsey-like results obtained by replacing cardinality with different largeness notions have been studied. I will describe the concept of largeness notion and I will give the definition of one of them using blocks and barriers. Then I will talk about how this can be used to study a more general Ramsey-like result. This is joint work with Alberto Marcone and Antonio Montalbán.

Kameryn Williams (Bard College at Simon’s Rock.)

Solid, Neat, Tight: Toward Charting the Boundary of Definability

Every logician knows that the Löwenheim–Skolem theorem rules out the possibility of absolute categoricity results for theories formulated in first-order logic. This is in contrast to the Dedekind and Zermelo theorems about the categoricity of arithmetic and set theory in second-order logic. Nevertheless, when something is impossible we still want to see how close we can get. The titular three properties, originating in work by Visser and named by Enayat, are notions of semantic completeness. Informally speaking, they capture the idea that a theory cannot be extended in the same signature to have new semantic content. Thus they form a categoricity-like property which is enjoyed by important foundational theories such as Peano arithmetic.

In this talk I will survey known results in this area, both positive and negative. A key empirical fact is that these categoricity-like properties seem to characterize canonical theories like PA or ZF. I will focus the examples for the talk on subsystems of second-order arithmetic.

Travel Support

Limited funding is provided from the NSF to support participation by students and others to NERDS. If interested, please email Russell Miller.

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