# Simon Thomas: the first 60 years, Rutgers, September 15-17, 2017

The conference Simon Thomas: the first 60 years will take place this September 15–17 at Rutgers University, New Brunswick, New Jersey.

List of speakers

• Clinton Conley
• Ilijas Farah
• Matt Foreman
• Alekos Kechris
• Justin Moore
• Andrew Marks
• Itay Neeman
• Anush Tserunyan
• Robin Tucker-Drob
• Saharon Shelah
• Hugh Woodin (canceled)

Organizer
Grigor Sargsyan

1. Schedule.

Each talk is 50 minutes with 10 minutes devoted to questions and extras.

Day 1. Friday September 15.

All Friday talks will take place in Hill 705.

9:00am–10:00am Saharon Shelah (Jerusalem, Rutgers)
Title: ZFC constructing somewhat free Abelian group with prescribed endomorphism ring

10:30am-11:30am Robin Tucker-Drob (Texas A&M)
Title: Invariant means and lattices in totally disconnected locally compact groups

2:30pm–3:30pm Clinton Conley (CMU)
Title: Borel, mu-measurable, and Baire measurable matchings.

4:00pm–5:00pm Matthew Foreman, cross listed as the colloquium talk. (UCI)
Title: Classifying diffeomorphisms of surfaces
Abstract: In 1932, von Neumann proposed classifying the statistical behavior of diffeomorphisms of manifolds. In modern language this means classifying diffeomorphisms that preserve a smooth volume element up to measure theoretic isomorphism. Despite important progress using entropy and spectral invariants, the general problem remained open. This talk proves that a complete classification is impossible in a rigorous sense—even on compact surfaces. The proof of the theorem involves producing new examples of diffeomorphisms with strong structural properties such as high distal rank.

5:10pm–6:00pm (Special Talk) Francis Urquhart (Rutgers)
Title: The classification problem of FU groups is positively complex.

Day 2. Saturday September 16.

All Saturday talks will take place at Rutgers Conference Center.

9:00am–10am Itay Neeman (UCLA)
Title: Embedding theorem and regularity properties under $AD^+$
Abstract: We present an absoluteness theorem under $AD^+$, showing approximately that proper forcing extensions of sufficiently elementary countable submodels can be embedded back into the universe. We use this embedding theorem to prove, under $AD^+$, that all sets of reals have the Ramsey property, and in fact are $H$-Ramsey for every happy family $H$. This in particular implies that there are no infinite MAD families under $AD^+$. We also use the embedding theorem to prove, again under $AD^+$, that for any equivalence relation $E$ whose equivalence classes belong to a pointclass $\Gamma$ closed under Borel substitutions (respectively to $\Gamma\cap\check{\Gamma}$), and any nice enough $\sigma$-ideal $I$ on $\omega^\omega$, there are $I$-positive sets $C$ so that $E\restriction C$ belongs to $\Gamma$ (respectively $\Gamma\cap\check{\Gamma}$). This is joint work with Zach Norwood. The embedding theorem extends a result of Neeman-Zapletal. The application to mad families extends a result of Tornquist restoring methods of Mathias. The application to equivalence relations extends results of Chan-Magidor.

10:30am–11:30am Anush Tserunyan (Urbana)
Title: Hyperfinite ergodic subgraphs
Abstract: Using the work of Hutchcroft and Nachmias on indistinguishability of the Wired Uniform Spanning Forest, R. Tucker-Drob proved a powerful theorem last year: any probability measure preserving (p.m.p.) locally countable ergodic Borel graph admits an ergodic hyperfinite subgraph. By completely different and purely descriptive set theoretic methods, we prove this theorem without the p.m.p. requirement, thus obtaining the following generalization: any locally countable ergodic Borel graph on a standard probability space admits an ergodic hyperfinite subgraph. In this talk, we will discuss the main result and one or two new gadgets involved in the proof.

2:30pm–3:30pm Ilijas Farah (York)
Title: Approximately matricial C*-algebras Abstract: A unital C*-algebra A is AM (approximately matricial) if it is an inductive limit of a directed system of full matrix C*-algebras. Separable AM algebras were classified by Glimm (the equivalence relation is smooth) and they form the simplest nontrivial class of simple (no pun intended) C*-algebras, also known as the UHF (uniformly hyperfinite) C*-algebras. Some years ago it was proved by Katsura and myself that not every nonseparable AM algebra is UHF. I’ll talk about more recent interesting examples of AM algebras and the role of set theory in constructing them.

4:00pm-5:00pm Justin Moore (Cornell)
Title: Subgroups of Thompson’s group F

Day 3. Sunday September 17.

All Sunday talks will take place at Rutgers Conference Center.

9:30am–10:30am Aleksander Kechris (Caltech)
Title: Borel equivalence relations, cardinal algebras and structurability.
Abstract: The theory of Borel equivalence relations has been a very active area of research in descriptive set theory during the last 25 years. In this talk, I will discuss how Tarski’s concept of cardinal algebras, going back to the 1940’s, appears naturally in this theory and show how Tarski’s theory can be used to discover new laws concerning the structure of Borel equivalence relations, which, rather surprisingly, have not been realized before. In addition, I will discuss the concept of structurability for equivalence relations and explain some of its implications concerning the algebraic structure of the reducibility order among such equivalence relations. (This is joint work with H. Macdonald and R. Chen.)

11am–12pm Andrew Marks (UCLA)
Title: Martin’s measure and countable Borel equivalence relations
Abstract: Thomas has shown that Martin’s conjecture on Turing invariant functions would have a wealth of consequences for the structure of countable Borel equivalence relations. Many of these consequences derive from the strong ergodicity properties that the conjecture implies for Martin measure. We discuss some results related to Martin measure and countable Borel equivalence relations. This is joint work with Adam Day, and also Clinton Conley, Steve Jackson, Brandon Seward and Robin Tucker-Drob.