MidWest Model Theory Day

Tuesday, November 1, 2016 at UIC

Spring 2017 MWMT is tentatively scheduled for April 4.

: Anand Pillay, Alex Kruckman, Noah Schweber

Schedule: All talks are about an hour long, in SEO 636. There will also be coffee & cookies in 636.
It is probably easiest to park in the university parking lot on Morgan St. between Roosevelt and Taylor.

Let us know (jfreitag at uic dot edu) if you are planning to come to lunch and/or dinner so we can make approximately correct reservations!

Titles & Abstracts:

Anand Pillay
Title: The canonical base property.
Abstract. I will discuss the CBP.  And prove that in a certain finite rank environment, rigidity of definable automorphism groups is equivalent to a strong version of the CBP.
The proof is related to old results about 1-based groups. This is joint with Daniel Palacin.

Noah Schweber
Title: Strong reducibilities and counterexamples to Vaught’s conjecture
Abstract: In computable structure theory, the most common method for comparing the complexity of two (countable) structures is Muchnik reducibility: A is Muchnik reducible to B if every copy of B computes a copy of A. This reducibility has been extensively studied; however, its uniform version, Medvedev reducibility — where we demand that the computation be independent of the copy of B chosen — has received much less attention. In this talk, I will present some properties of Medvedev reducibility in the context of counterexamples to Vaught’s conjecture: broadly speaking, if A is a (model of a) counterexample to Vaught’s conjecture, then uniform computations from A have a number of nice combinatorial properties. Moreover, these properties hold for any "reasonably definable” uniform reducibility, not just Medvedev reducibility; and the proofs of these properties rely on large cardinals.

Alex Kruckman
Title: Foundations of cologic
Abstract: The existence of a robust categorical dual to first-order logic is hinted at in (at least) four independent bodies of work: (1) The cologic of profinite groups (e.g. Galois groups), which plays an important role in the model theory of PAC fields [Cherlin - van den Dries - Macintyre, Chatzidakis]. (2) Projective Fraïssé theory [Solecki & coauthors, Panagiotopolous]. (3) Universal coalgebra and coalgebraic logic [Rutten, Kurz - Rosicky, Moss, others]. (4) Ultracoproducts and coelementary classes of compact Hausdorff spaces [Bankston]. In this talk, I will propose a natural syntax and semantics for such a dual "cologic", in which "coformulas" express properties of partitions of "costructures", dually to the way in which formulas express properties of tuples from structures. I will show how the basic theorems and constructions of first-order logic (completeness, compactness, ultraproducts, Henkin constructions, Löwenheim-Skolem, etc.) can be dualized, and I will discuss some possible extensions of the framework.

Other MWMTDs:
April 5th, 2016
October 28th, 2014
October 22nd, 2013
April 18th, 2013
October 23rd, 2012
April 26th, 2012
October 11th, 2011
April 7th, 2011
October 26th, 2010