NERDS 20.0

Overview

Date: Saturday, December 11, 2021
Time: TBA (EST)
Place: Wellesley College (online)

Schedule

(All times in EST time zone.)

1:15pm General social time if anyone wants to log in early
1:30pm Jason Block (Grad Center CUNY) Categoricity Ordinals and Models of Presburger Arithmetic
2:00pm Break
2:10pm Rachael Alvir (University of Notre Dame) Scott Complexity and Finitely alpha-generated Structures
2:40pm Break
3:00pm Peter Gerdes (Oakland University) Computability and the Symmetric Difference Operator
3:30pm Break
3:40pm Josiah Jacobsen-Grocott (University of Wisconsin-Madison) Topological classification of classes of enumeration degrees

Abstracts

Jason Block (Grad Center CUNY)

Categoricity Ordinals and Models of Presburger Arithmetic

The categoricity ordinal of a structure M is a measure of how hard it is to compute an isomorphism between two copies of M. Presburger arithmetic is the theory of (Z,+,<). We will define precisely what a categoricity ordinal is, examine computable theoretic properties of Presburger arithmetic, and examine which ordinals can be the categoricity ordinal for a model of Presburger arithmetic.

Rachael Alvir (University of Notre Dame)

Scott Complexity and Finitely alpha-generated Structures

In this talk, we present a generalization of a finitely generated structure called a finitely α-generated structure. We use this generalization to lift results about Scott sentences earlier known only for finitely generated structures. We will show how these results can be used to the connect some of the existing non-equivalent definitions of Scott rank.

Peter Gerdes (Oakland University)

Computability and the Symmetric Difference Operator

Combinatorial operations on sets are almost never well-defined on Turing degrees, a fact so obvious that counterexamples are worth exhibiting. The case I will focus on in this talk is the symmetric difference operator. There are pairs of (nonzero) degrees for which the symmetric-difference operation is well-defined. Some examples can be extracted from the literature, for example, from the existence of nonzero degrees with strong minimal covers. In this talk, I will focus on the case of incomparable r.e. degrees for which the symmetric-difference operation is well-defined and end with some interesting open questions.

This talk is based on joint work with Uri Andrews, Steffen Lempp, Joseph Miller and Noah Schweber.

Josiah Jacobsen-Grocott (University of Wisconsin-Madison)

Topological classification of classes of enumeration degrees

A point in a represented second-countable T 0 -space can be identified with the set of basic open sets containing that point. Using this coding, we can consider the enumeration degrees of the points in a second-countable T 0 -space. For example, the ω-product of the Sierpiński space is universal for second-countable T 0 -spaces and gives us all enumeration degrees and the Hilbert cube gives us all continuous degrees. Kihara, Ng, and Pauly have studied various classes that arise from different spaces. They show that any enumeration degree is contained in a class arising from some decidable, sub-metrizable space, and that no T 1 -space contains all enumeration degrees. Similarly they separate T 2 classes from T 1 classes and T 2.5 classes from T 2 classes by showing that no T 2 class contains all the cylinder-cototal degrees and no T 2.5 class contains all degrees arising from (Nrp )^ω . (We answer several questions posed in their article:) We extend their results to show that the cylinder-cototal degrees are T 2 -quasi-minimal and the (Nrp )^ω degrees are T 2.5 -quasi-minimal. We then give separations of the T 2.5 degrees from the submetrizable degrees using the Arens co-d-CEA degrees and the Roy halfgraph above degrees.