Fall 2015 Schedule

Fridays 5th period (11:45 am-12:35 pm) in Little 368

• • Friday, August 28: Chris Porter, The preservation of algorithmic randomness, Part 1
    Abstract: In this talk, I will introduce the basics of algorithmic randomness, with an eye towards discussing a phenomenon known as the preservation of randomness. The rough idea behind randomness preservation is this: given an object that is algorithmically random with respect to a computable probability measure, if we effectively transform this object via map that is defined on almost every member of our probability space, the resulting object will be algorithmically random with respect to the pushforward measure induced by this transformation. I will discuss applications of randomness preservation to the study of interactions between algorithmically random closed subsets of Cantor space and algorithmically random continuous functions on Cantor space. This is joint work with Quinn Culver.
  • Friday, September 04: Chris Porter, The preservation of algorithmic randomness, Part2
    Abstract: After a brief review of Part 1, I will give a proof of a partial converse of the randomness preservation theorem, referred to as the no randomness ex nihilo principle, and I will consider a possible weakening of the randomness preservation theorem. I will then turn to applications of the machinery of randomness preservation to the study of random closed subsets, random continuous functions, and random measures (in the context of Cantor space).
  • Friday, September 11: Jindra Zapletal, Interpretations of topological spaces in models of set theory
    Abstract: I describe the final version of my work on the subject. There is a broad category of interpretable topological spaces such that given two wellfounded models Msubset N of set theory, there is an “interpretation” functor from the category as evaluated in M to the category as evaluated in N. The functor preserves a good number of topological and algebraic properties of topological structures. This work places the traditional informal interpretation calculus in forcing on firm footing.
  • Friday, September 18: Jindra Zapletal, Interpretations of topological spaces, II in Little 305
  • Friday, September 25: Doug Cenzer, Reducibility on Computably Enumerable Equivalence Relations
    Abstract: Given equivalence relations R and S over the set of natural numbers, define R as reducible to S if there exists a computable function f such that x R y iff f(x) S f(y) for all x,y. This can be viewed as a natural computable analogue of Borel reducibility, widely studied in descriptive set theory to measure the relative complexity of Borel equivalence relations on Polish spaces. The computable version has been studied by many, going back to Ershov (1973), Sorbi (1983), Lachlan (1987) and continuing up to now with Friedman, Gao, Nies, Miller and others, see for example 6-author paper in JSL 2014. We will examine universal ceers (computably enumerable equivalence relations).
  • Friday, October 02: Bill Mitchell, Inconsistency of Arithmetic
  • Friday, October 09: Rupert Holzl (Univeristät der Bundeswehr München), Vitali Covering in the Weihrauch Degrees
    Abstract: In a 2002 paper, Brown, Giusto, and Simpson studied the role of the Vitali Covering Theorem in reverse mathematics. Among other results they showed that it is equivalent to Weak Weak König’s Lemma.
    In this talk we present results from an ongoing research project on the Weihrauch lattice, a framework that allows classifying the computational content of theorems from classical mathematics. This topic is closely related to reverse mathematics, but the Weihrauch lattice is more fine-grained in the sense that it can reveal structural differences between principles that fall together in reverse mathematics.

We will first introduce the Weihrauch lattice and then formalise the Weihrauch degree of the Vitali Covering Theorem in several different ways. Next we analyse their respective computational content, and show that all formalisations have different strength in this sense. This will contrast with the previous results in reverse mathematics. Finally we show that some of the above principles interact in interesting ways with choice principles for sets on certain levels of the Borel hierarchy.(Based on joint work with Vasco Brattka and Guido Gherardi.)

• Friday, October 16: Francis Adams, Extending Anticliques in Definable Graphs 1
  Abstract: In a graph, an anticlique A is a collection of vertices such there are no edges between elements of A. In these talks I will look at the problem of extending anticliques in definable graphs on Polish spaces. This problem includes the important special case of extending partial selectors for Borel equivalence relations. For a graph G, the smallest cardinality of an anticlique contained in no Borel anticlique is a natural cardinal characteristic associated with G. After an introduction to cardinal characteristics and to the theory of Borel equivalence relations I will discuss both provable relations among these graph characteristics and independence results for them.
• Friday, October 23: Francis Adams, Extending Anticliques in Definable Graphs 2
• Friday, October 30:
• Friday, November 06: Homecoming (no seminar)
• Friday, November 13:
• Friday, November 20:
• Friday, November 27: Thanksgiving (no seminar)
• Friday, December 04: Jean Larson (last Friday of Classes)