### Overview

**Date:** Sunday, October 29, 2017

**Time:** 10am-3:20pm

**Place:** Wellesley College, Room TBA (Directions).

### Schedule

10:00-10:30 | Coffee and pastries | |

10:30-11:30 | Timothy H. McNicholl (Iowa State) | |

11:40-12:10 | James Barnes (Cornell) | |

12:10-1:40 | Lunch (Bae Pao Lu Chow Dining Room) | |

1:40-2:40 | Matthew Harrison-Trainor (Waterloo) | |

2:50-3:20 | David Nichols (UConn) | |

3:30-4:30 | Russell Miller (Queens College and Graduate Center CUNY) |

### Abstracts

#### Timothy H. McNicholl

##### Computable analysis and computable structure theory

I will discuss recent work on extending the computable structures program to metric structures by means of the framework of computable analysis. I will focus on Banach spaces, and in particular recent results on computable categoricity and degrees of categoricity of L^p spaces. The solutions of some of the resulting problems involve am interesting blend of methods from functional analysis and classical computability theory.

#### James Barnes

##### The complexity of fragments of the theory of the hyperarithmetic degrees

We say a real X is hyperarithmetic in a real Y if X is Turing reducible to the alpha-th jump of Y for some ordinal alpha with a Y-recursive representation. This reducibility induces a degree structure, and, as with many degree structures, one way we understand it is through the complexity of its theory. I will outline what is known about this topic and discuss the techniques involved.

#### Matthew Harrison-Trainor

##### Structures of Scott Rank Omega_1^{CK}

I will talk about a few results about computable structures of Scott rank Omega_1^{CK}. We will start with a few different constructions of structures of this Scott rank with different properties, such as one whose infinitary theory is not countably categorical. Then we will argue that there is no natural/canonical construction of a computable structure of Scott rank Omega_1^{CK}.

#### David Nichols

##### Strong Computable Reductions between Stable Versions of Ramsey’s Theorem

We analyze four variants of the stable Ramsey’s theorem for pairs. These four principles live in the reverse mathematics “zoo,” the collection of principles which are logically (strictly) stronger than RCA_0 but logically (strictly) weaker than ACA_0. Moreover, these four principles are provably equivalent to one another over RCA_0. For a finer reverse mathematical analysis which captures intuitions that these principles have different strengths, we compare the computable content of the principles under strong computable reductions. Using a new tree-labeling construction, we show that in fact each of these four variants of the stable Ramsey’s theorem for pairs has a different computable strength and that we can rank them in order of strength in a way that agrees with intuition.

#### Russell Miller

##### Classification and Measure for Algebraic Fields

The algebraic fields of characteristic 0 are precisely the subfields of the algebraic closure of the rationals, up to isomorphism. We describe a way to classify them effectively, via a computable homeomorphism onto Cantor space. This homeomorphism makes it natural to transfer Lebesgue measure from Cantor space onto the class of these fields, although there is another probability measure on the same class which seems in some ways more natural than Lebesgue measure. We will discuss how certain properties of these fields — notably, relative computable categoricity — interact with these measures: the basic result is that only measure-0-many of these fields fail to be relatively computably categorical. (The work on computable categoricity is joint with Johanna Franklin.)