Midwest
Computability Seminar
XIII
The Midwest Computability
Seminar is a joint venture between the University of Chicago, the University
of Notre Dame, and the University of Wisconsin-Madison. It
meets once or twice per semester at the
University of Chicago, and is attended by faculty and students from these
universities and others
in the area. The seminar started in the
fall of 2008.
DATE: Tuesday, October 1, 2013.
PLACE: Ryerson Hall 352 (the
Barn), The University of
Chicago.
1100 East 58th Street, Chicago, IL 60637.
Speakers, with links to slides if available:
Schedule:
- 12:00 - 1:00: Lunch
- 1:00 - 1:35: Mushfeq Khan
- 1:45 - 2:20: Jonathan Stephenson
- 2:30 - 3:15: Coffee Break
- 3:15 - 3:50: Víctor Ocasio-González
- 4:00 - 5:00: Peter Cholak
- 5:45: Dinner: The Sit Down, 1312 E. 53rd Street
Abstracts:
Peter Cholak
Title: On uniform splits of r.e. sets
Abstract: We discuss various splits of r.e. sets starting with the
Friedberg split. It turns out that there are many other types of
splits. However given an r.e. set W we can uniformly find a
Friedberg split of W. We show this uniformity fails for all
non-Friedberg splits.
Mushfeq Khan
Title: Density-one points of Π^{0}_{1} classes
Abstract: Recent work on effective versions of the Lebesgue Density Theorem has
brought much attention to the class of density-one points: the reals X such
that for every effectively closed (i.e., Π^{0}_{1}) set C that contains X, the
lower Lebesgue density of C at X is 1. As a result, we know a lot about how
the 1-random density-one points behave. For example, they comprise a proper
subclass of the difference random reals, and are therefore incomplete. This
work has had unexpected side effects, such as the recent solution by Day,
Miller, et al., of the ML-cupping problem.
However, not much is known about the behavior of density-one points in the
absence of randomness. The property of being density-one can be viewed as a
generalization of 1-genericity, and this leads to many interesting questions
about them. We survey some results that are part of ongoing work in this
area. For example, we now know that density-one points can, in general, be
complete. We also investigate the thorny issue of dyadic density vs full
density.
Víctor A. Ocasio-González
Title: Computability in the class of real closed fields
Abstract: The class of Real Closed Fields (RCF) is known to have very
nice model theoretic properties, among them o-minimality and
quantifier elimination. In our work, we consider some non-elementary
subclasses of RCF and explore their computability theoretic
properties. We locate the class of Archimedean Real Closed Fields
using Turing computable embeddings and compare it with other
non-elementary first order subclasses of RCF. We also explore relative
categoricity and show that under some conditions one can obtain a
sharp result on the complexity of the relative categoricity of a real
closed field that is constructed using a linear order as an oracle.
Jonathan Stephenson
Title: Isomorphisms between low_{n} and computable boolean algebras
Abstract: A long-standing problem in the area of computable structures is to
determine for which n every low_{n} boolean algebra is
isomorphic to a computable boolean algebra. For those n where such
isomorphisms exist, a related problem is that of determining how
complicated the isomorphisms between low_{n} boolean algebras and
computable algebras must be.
Harris and Montalbán have demonstrated the existence of a
low_{5} boolean algebra that is not isomorphic to any
computable boolean algebra via a ∅^{(7)}-computable map.
Using Montalbán's machinery for diagonalizable structures, we can
show that proofs in the style of Harris and Montalbán's exist for
low_{n+2} boolean algebras whenever they exist for
low_{n} boolean algebras.
Previous Seminars:
- Sept 23th 2008. Antonio
Montalbán - Logan Axon - Joe Miller
- Nov 11th 2008. Chris
Conidis - Keng Meng (Selwyn) Ng - Peter Gerdes
- Feb 3rd 2009. David
Diamondstone - Bart Kastermans - Richard A. Shore
- April 21th 2009. Dan Turetsky
- Julia Knight - Ted Slaman
- Sept 29th 2009. Carl Jockusch
- Rachel Epstein - Rebecca Weber
- Jan 26th 2010. Sara Quinn -
John Wallbaum - Steffen Lempp - Reed Solomon
- May 11th 2010. Adam Day -
Liang Yu - Rod Downey - Boris Zilber
- Sept 28th 2010. Maurice
Chiodo - Peter Gerdes - Damir Dzhafarov - Andy Lewis
- Feb 15th 2011. Uri Andrews -
Paola D'Aquino - David Diamondstone - Christopher Porter -
Rebecca Steiner.
- Nov 1st 2011. Mingzhong Cai -
Chris Conidis - Stephen Flood -
Jeff Hirst - Asher Kach.
- Nov 15th 2012 Achilles Beros
- Rod Downey - Jesse Johnson - Sam Sanders - Steven VanDendriessche -
Matthew Wright.
- April 2nd 2013 Howard
Becker - Denis Hirschfeldt - Paul Schupp.
If you haven't
been receiving the announcements and would like to be included
in the list, send an email to drh@math.uchicago.edu.