Schedule

Directions

Location
Evans Hall, Room 1015

See below for abstracts.

Monday

9:30-10:00: Coffee
10:00 - 10:40:  Tutorial in Computability Theory: Uri Andrews
10:55 - 11:35: Tutorial in Model Theory: Chris Laskowski
11:50 - 12:30: Tutorial in Descriptive Set Theory: Su Gao
Lunch break
2:20 - 3:00: Nate Ackerman (Vaught's conjecture and Grothendieck toposes)
Coffee break
3:30 - 4:10: Slawomir Solecki

Tuesday
9:30-10:00: Coffee
10:00 - 10:40:  Tutorial in Computability Theory: Uri Andrews
10:55 - 11:35: Tutorial in Model Theory: Chris Laskowski
11:50 - 12:30: Tutorial in Descriptive Set Theory: Su Gao
Lunch break
2:20 - 3:00: Open questions session
Coffee break
3:30 - 4:10: Work on open questions
4:20 - 5:00: Samuel Coskey

Wednesday
9:30-10:00: Coffee
10:00 - 10:40:  Tutorial in Computability Theory: Uri Andrews
10:55 - 11:35: Tutorial in Model Theory: Chris Laskowski
11:50 - 12:30: Tutorial in Descriptive Set Theory: Su Gao
Lunch break
2:20 - 3:00: Cameron Freer (A "Vaught conjecture" for measures invariant under the logic action)
Coffee break
3:30 - 4:10: Robin Knight
4:20 - 5:00: Sy Friedman (Vaught's conjecture and absoluteness)

Thursday
9:30-10:00: Coffee
10:00 - 10:40: Iouannis Souldatos (Absolute indescernibles, characterizing cardinals and Vaught's Conjecture)
10:55 - 11:35: Paul Larson
11:50 - 12:30: Richard Rast (The Borel complexity of some ordered theories)
Lunch break
2:20 - 3:00: Open questions session
Coffee break
3:30 - 4:10: Work on open questions
4:20 - 5:00: Howard Becker

Friday:
9:30-10:00: Coffee
10:00 - 10:40: David Marker
10:55 - 11:35: Ludomir Newelski (Multiplicity)
11:50 - 12:30: John Baldwin

Nate Ackerman

Title: Vaught's conjecture and Grothendieck toposes

Abstract: Grothendieck toposes, which were originally discovered as an abstract notion of "space", have been found to play a fundamental role in areas ranging from type theory to algebraic topology to intuitionistic set theory. The goal of this talk is to find a version of Vaught's conjecture which makes sense for structures inside a Grothendieck topos, and to discuss some results in this area.

We will begin by reviewing the basics of categorical logic as well as the notion of what a Grothendieck topos is. We will then address one of the most significant hurdles to stating a version of Vaught's conjecture for a Grothendieck topos, namely the question: "What does it mean for a structure in a Grothendieck topos to be countable?"

We will consider three possible answers to this question along with their advantages and disadvantages after which we will discuss what is known about the number of countable models in a Grothendieck topos for each notion of countable.

Cameron Freer

Title: A "Vaught conjecture" for measures invariant under the logic action

Abstract: Consider graphs with underlying set the natural numbers. Erdős and Rényi described an invariant measure on the set of such graphs, where each pair of vertices is assigned an edge (or not) according to an independent fair coin flip. This measure concentrates on the isomorphism class of the Rado graph. A natural generalization of this invariant measure to a language with countably infinitely many binary relations is obtained by using independent coins for each relation. In contrast to the Erdős–Rényi measure, this measure does not concentrate on any single isomorphism class, and any Borel set on which it concentrates that is also the union of isomorphism classes must consist of continuum many isomorphism classes.

This raises the following question: Suppose L is a countable relational language and μ is a (Borel) probability measure on the collection of L-structures with underlying set the natural numbers such that μ is invariant under the logic action. If μ does not concentrate on the union of a countable collection of isomorphism classes, must any collection of isomorphism classes on whose union it concentrates have size the continuum? We can think of an affirmative answer to this question as a "Vaught Conjecture" for invariant measures.

This conjecture was proven independently by Kechris and Kruckman. In this talk I will present background on and further motivation for this conjecture, and describe its proof and related joint work with Ackerman, Kruckman, and Patel.

Sy-David Friedman

Title: Vaught's Conjecture and Absoluteness

Abstract: Let's say that an analytic equivalence relation E is scattered if there is no perfect set of E-classes and is trivial if there are only countably many E-classes. So $\varphi$ is a counterexample to Vaught's Conjecture exactly if the isomorphism relation on the countable models of $\varphi$ is scattered and nontrivial. Although scatteredness
is $\Pi^1_2$ and therefore absolute, it is not immediately clear that nontriviality is absolute, as it appears to be $\Pi^1_3$. In fact the Scott-Morley analysis of rank shows that it is $\Pi^1_2$ and therefore absolute as well. In this talk I extend this to all analytic equivalence relations with just Borel classes (and in particular to all equivalence relations induced by a Borel action of a Polish group on a Polish space): Triviality is absolute for all such equivalence relations. Combining this with a theorem of Stern we conclude that Silver's dichotomy holds for all analytic equivalence whose classes are Borel of bounded rank. There are counterexamples to this if even just one class is not Borel and there is a bound on the Borel ranks of the remaining classes, as first observed by Sami.

Ludomir Newelski

Title: Multiplicity

Abstract: After Shelah's proof of Vaught's conjecture for omega-stable
theories, multiplicity of types became the central point of research on
Vaught conjecture for superstable theories. In particular, it was crucial
in the proof of Vaught's conjecture for superstable theories of finite
rank (Buechler). I will survey my results on this subject.

Richard Rast

Title: The Borel complexity of some ordered theories

Abstract: We examine some old solutions to special cases of Vaught's Conjecture, in particular Mayer's solution for o-minimal theories and Rubin's solution for colored linear orders.  We extract information about the Borel complexity of isomorphism, and refine these results significantly.  In particular, we show that in both cases, the interesting dichotomy is not when the theory has uncountably many models.  Time permitting, we will also discuss some open questions related to these theorems.

Iouannis Souldatos

Title: Absolute indescernibles, characterizing cardinals and Vaught's Conjecture

Abstract: After the first Vaught Conference, Greg Hjorth proved (cf. [2]):

Theorem. If there exists a counterexample to Vaught's Conjecture, then there exists one with no model in $\aleph_2$.

The proof uses Descriptive Set Theory. This result was later improved to (cf. [1])

Theorem. If there exists a counterexample to Vaught's Conjecture, then there exists one with only maximal models in $\aleph_1$.

The second proof uses Model Theory. During the talk we will present the machinery behind the second result (absolute indiscernibles plus characterizing cardinals),
survey related results, and present recent developments.

[1] John Baldwin, Sy Friedman, Martin Koerwien, Chris Laskowski. "Three red herrings around Vaught's conjecture. Preprint at http://homepages.math.uic.edu/~jbaldwin/model11.html

[2] Greg Hjorth. "A note on counterexamples to the Vaught conjecture. Notre Dame J. of Formal Logic, 48(1):49-51, 2007