University of California - Berkeley

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 (

Coffee break

3:30 - 4:10: Slawomir Solecki

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

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 (

Coffee break

3:30 - 4:10: Robin Knight

4:20 - 5:00: Sy Friedman (

9:30-10:00: Coffee

10:00 - 10:40: Iouannis Souldatos (

10:55 - 11:35: Paul Larson

11:50 - 12:30: Richard Rast (

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

9:30-10:00: Coffee

10:00 - 10:40: David Marker

10:55 - 11:35: Ludomir Newelski (

11:50 - 12:30: John Baldwin

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.

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.

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.

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.

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