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, April 17th, 2018.

PLACE: Ryerson Hall 352 (the Barn), The University of Chicago.

1100 East 58th Street, Chicago, IL 60637.

Speakers:

- Peter Cholak - University of Notre Dame

- Meng-Che "Turbo" Ho - Purdue University

- Ethan McCarthy - University of Wisconsin

- Joe Miller - University of Wisconsin

- 2:00 - 2:45: Peter Cholak
- 2:50 - 3:35: Ethan McCarthy

- 3:40 - 4:20: Break

- 4:20 - 5:05: Joe Miller

- 5:10 - 5:55: Meng-Che "Turbo" Ho

- 6:15 Dinner at Jolly Pumpkin, 5215 S. Harper Ave.

Title: Encodable by thin sets

Abstract: Let

Meng-Che "Turbo" Ho

Title: Finitely-generated groups are universal

Abstract: It is well-known that any structure can be coded in a graph. Hirschfeldt, Khoussainov, Shore, and Slinko showed that any structure can be coded in a partial ordering, lattice, integral domain, or 2-step nilpotent group, and showed that certain computability properties are preserved under this coding. Recently, Miller, Poonen, Schoutens, and Shlapentokh added fields to this list, and they gave a category-theoretic framework to establish their result. Around the same time, Montalbán introduced the notion of effective interpretation and showed that effectively bi-interpretable structures share many computability-theoretic properties. Thus, using either of these notions, we may define the notions of a class of structures being universal. Harrison-Trainor, Melnikov, Miller, and Montalbán showed that these two notions are indeed equivalent. In joint work with Harrison-Trainor, we use these ideas to define a class of structures being universal within finitely-generated structures. We then use small cancellation techniques from group theory to show that the class of finitely-generated groups with finitely-many constants named is universal within finitely-generated structures. One application of this is to the study of Scott sentences of finitely-generated groups. Julia et al. showed that a finitely-generated structure always have a Σ

Ethan McCarthy

Title: Cototal enumeration degrees and the degree spectra of minimal subshifts

Abstract: A set of natural numbers is cototal if it is enumeration-reducible to its complement. The degrees of cototal sets, the cototal degrees, form an intermediate structure between the Turing degrees and the enumeration degrees. We characterize the cototal degrees as the degrees of maximal anti-chain complements on ω

Joe Miller

Title: A structural dichotomy in the enumeration degrees

Abstract: The continuous degrees measure the computability-theoretic content of elements of computable metric spaces. They properly extend the Turing degrees and naturally embed into the enumeration degrees. Although nontotal (i.e., non-Turing) continuous degrees exist, they are all very close to total: joining a continuous degree with a total degree that is not below it always results in a total degree. We call this property "almost totality". Recently, Andrews, Igusa, M., and Soskova proved that almost totality characterizes the continuous degrees inside the enumeration degrees.

In this talk, I will describe another natural structural characterization of the continuous degrees. Kalimullin introduced the notion of a K-pair in order to define the enumeration jump. K-pairs, which can be thought of as "robust minimal pairs", play a part in almost all known definability results in the enumeration degrees. Indeed, they were used to define totality, and hence almost totality. We show that the non-continuous enumeration degrees are exactly the halves of nontrivial K-pairs relative to some (total) degree. Along with the earlier definition of the continuous degrees, this gives us a structural dichotomy in the enumeration degrees.

This talk is on joint work with Ganchev, Kalimullin, and Soskova.

- Sept 23rd 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 21st 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
- October 1st 2013 Peter Cholak - Mushfeq Khan - Victor Ocasio-González - Jonathan Stephenson
- April 29th, 2014 Rod Downey - Noam Greenberg - Gregory Igusa - Alexander Melnikov - Kyle Riggs
- September 30th, 2014 Eric Astor - Quinn Culver - Jack Lutz - Timothy McNicholl
- February 17th, 2015 Carl Jockusch - Julia Knight - Steffen Lempp
- January 28th, 2016 Reese Johnston - Rutger Kuyper - Mariya Soskova - Mars Yamaleev
- October 22nd and 23rd, 2016 Special Meeting in Honor of Carl Jockusch's 75th Birthday
- March 16th, 2017 Greg Igusa - Jack Lutz - Sasha Melnikov - Reed Solomon
- October 24th, 2017 Noah Schweber - Don Stull - Dan Turetsky - Rose Weisshaar

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.