Midwest Computability Seminar

X


The Midwest Computability Seminar meets twice in the fall and twice in the spring at University of Chicago.  Researchers in computability theory and their students and postdocs from University of Chicago, University of Notre Dame, and University of Wisconsin-Madison plus some others throughout the area regularly attend. Normally we have half hour talks and 1-hour talks, though this varies, and a few hours to talk and collaborate with each other.  The seminar started in the fall of 2008.

DATE: Tuesday, November 1, 2011.
PLACE: Ryerson, University of Chicago.
1100 East 58th Street, Chicago, IL 60637.


Speakers:

Schedule:

Abstracts:

Speaker: Mingzhong Cai
Title: Definability problem of domination properties
Abstract: We discuss the definability problem for hyperimmunity and array nonrecursiveness. Both notions are defined from domination notions regarding the rates of growth of functions recursive in a degree. A positive result is that array nonrecursiveness can be defined in the Turing degrees with order and relative r.e. relation. A negative result, answering a question by Miller and Martin, is that there is no quantifier-free definition for hyperimmunity or array nonrecursiveness in the language of order and jump. Both results are joint work with Shore.

Speaker: Chris Conidis
Title: Chain Conditions in Computable Rings, Part II
Abstract: Three years ago in November 2008 I presented the first part of this talk at this meeting.  In particular, I then set out to examine the reverse mathematical strength of the statements
  1. Every Artinian ring is Noetherian, and
  2. Every Artinian ring has finite length (as a module over itself).
More specifically, I showed that statement 1 is implied by ACA_0 and implies WKL_0 over RCA_0+I\Sigma_2, while statement 2 is equivalent to ACA_0 over RCA_0+\BSigma_2.  In this talk I will shed considerably more light on the strength of statement 1, showing that it is in fact equivalent to WKL_0 over RCA_0+\ISigma_2. The key to this result lies in a new algebraic approach to proving that the Jacobson radical of an Artinian ring is nilpotent.

Speaker: Stephen Flood
Title: Reverse mathematics and a packed Ramsey's theorem.
Abstract: Ramsey's theorem states that each coloring has an infinite homogeneous set, but these sets can be arbitrarily spread out.  Paul Erdos and Fred Galvin proved that for each coloring f, there is an infinite set which is not ``too spread out'' that is not given ``too many'' colors by f.  In this talk, I will give the precise statement of this packed Ramsey's theorem and discuss my work on its reverse mathematical strength.  In particular, I have shown that this theorem is equivalent to Ramsey's theorem for each exponentn=\=2, and that it implies Ramsey's theorem for n=2.

Speaker: Jeff Hirst
Title: Reverse mathematics and persistent reals
Abstract: The dichotomy principle asserts that each real number is either less than or equal to 0 or greater than or equal to 0.  The reverse mathematics subsystem RCA0 can prove dichotomy.  On the other hand, given a sequence of reals, the existence of a set separating them into disjoint collections of non-positive and non-negative reals is equivalent to the subsystem WKL0 over RCA0. Thus, this sequential form of dichotomy fails in computable analysis.  We will show how a computable restriction of the sequential form can motivate the formulation of a constructive restriction of the original principle.  The results in this talk are joint work with François Dorais and Paul Shafer.

Speaker: Asher Kach
Title: Orders on Computable Torsion-Free Abelian Groups
Abstract: In this talk, we will discuss the (Turing) degrees of orders on computable presentations of computable torsion-free abelian groups. After providing the requisite algebraic background, we will review known results and discuss new results.  In particular, we show the set of degrees of orders for a fixed computable presentation need not be upward closed.  This new work is joint with Karen Lange and Reed Solomon.

Previous Seminars:
   If you haven't been receiving the announcements, and you would like to be included in the list, send an email to drh@math.uchicago.edu.