# Thursday, April 18th, 2013 at UIC

Speakers: Martin Koerwien, Uri Andrews, Rehana Patel

Schedule:
• 11am: Meet on the first floor of SEO if you're already here. We will leave to lunch at 11:20.
• 11:30am: Lunch at Joy Yee's
• 1pm: One talk
• 2:30: Two talk
• 4pm: Three talk
• 5:30pm: Dinner at Jaks Tap
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 (@math.wisc.edu">andrewsmath.wisc.edu) if you are planning to come to lunch and/or dinner so we can make approximately correct reservations!

Titles & Abstracts:

Speaker: Martin Koerwien
Title: Around characterizing $\aleph_1$

Abstract: An $L_{\omega_1,\omega}$ sentence characterizes a cardinal $\kappa$ if it has a model of size $\kappa$ but no model in $\kappa^+$. We study the known examples of complete sentences that characterize $\aleph_1$ and observe several notable phenomena about them. Our goal is to understand the mechanisms that make a sentence characterize $\aleph_1$. This is related to some recent developments: (1) Hjorth showed that if there is a counterexample to Vaught's conjecture, there is also one that characterizes $\aleph_1$. So it is tempting to try proving Vaught's conjecture by showing that every counterexample must have a model in $\aleph_2$ (which a result of Harrington's might make appear plausible). This however turns out to be a red herring. (2) While we know the notion of a complete sentence having a model in $\kappa$ is absolute for $\kappa=\aleph_1$ and non-absolute for $\kappa=\aleph_3$, even assuming GCH, this is still an open issue for $\kappa=\aleph_2$.

Speaker: Uri Andrews
Title: A hands on approach to definability in randomizations

Abstract: The randomization of a complete first order theory $T$ is the complete continuous theory $T^R$ with two sorts, a sort for random elements of models of $T$, and a sort for events in an underlying probability space. I'll talk about a very hands-on way to understand algebraicity and definability of elements in models of $T^R$. This is joint work with Isaac Goldbring and H. Jerome Keisler.

Speaker: Rehana Patel
Title: Countable structures from symmetric probabilistic constructions

Abstract: Which countable structures admit symmetric probabilistic constructions? More specifically, given a countable structure $M$ in a countable language $L$, is there a probability measure on the space of $L$-structures with the same underlying set as $M$ that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of $M$? The Rado graph is the classic example of such a structure, as it is generated almost surely by the Erdõs-Rényi random graph construction; the rational Urysohn space too admits an invariant measure, by a result of Vershik (2002). However, until as recently as 2010, it was not known whether Henson's countable homogeneous-universal triangle-free graph could arise from such a symmetric random process. Petrov and Vershik (2010) showed that it could, by constructing a continuum-sized graph that, upon sampling, produces Henson's graph almost surely. The continuum-sized graph that they construct is in fact a 'graph limit', in the sense of Lovasz and Szegedy (2006).

In this talk, I will provide a characterisation of countable structures that admit invariant measures: A countable structure $M$ admits an invariant measure if and only if $M$ has trivial (group-theoretic) definable closure. When $M$ is a Fraïssé limit, this is equivalent to requiring that the age of $M$ exhibit strong amalgamation. The proof combines ideas from infinitary logic with methods from Petrov and Vershik's construction. I will discuss the proof and its connections with the theory of graph limits, as well as give some consequences of the result. This is joint work with Nathanael Ackerman and Cameron Freer.

Organizers:
Uri Andrews
Lynn Scow

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