A number of "stubbornly open" problems about countable Borel equivalence relations concern hyperfiniteness. For instance, the increasing union problem asks if the increasing union of a sequence of hyperfinite equivalence relations is still hyperfinite. In the past decade or so, the only progress on hyperfiniteness problems has been the proof of hyperfiniteness for orbit equivalence relations of countable abelian group actions (Gao and Jackson, "Countable abelian group actions and hyperfinite equivalence relations", Inventiones Mathematicae, 2015) and then the extension of this result to locally nilpotent groups (Schneider and Seward, "Locally nilpotent groups and hyperfinite equivalence relations", to appear).

The hyperfiniteness proofs are based on an elaborate theory of Borel marker structures with regularity properties. Now researchers have a good understanding of which regularity properties are possible and which are beyond hope. For the proofs of negative results two new concepts and methods have been playing a key role. One of them is the introduction and construction of hyperaperiodic elements with various additional properties. The other is the introduction of new forcing notions that are special cases of the so-called orbit forcing. The workshop will be roughly divided into four lectures:

- In the first lecture we will construct some basic regular marker
structures for the Bernoulli shift of ℤ
^{n}. Using such marker structures, we will give an outline of the proof of hyperfiniteness for orbit equivalence relations of countable abelian group actions. - In the second lecture we will give some advanced constructions of
regular marker structures and use them to illustrate a proof that
the Borel chromatic number for the free part of 2
^{ℤn}is 3. - In the third lecture we will construct some hyperaperiodic elements
for 2
^{ℤn}and use them to show that the continuous chromatic number for the free part of 2^{ℤn}is 4. - In the fourth lecture we will consider some forcing constructions and use them to show that certain regular Borel marker structures do not exist.

- Basics:
- Kechris, Classical Descriptive Set Theory, Springer-Verlag, 1995. Chapter I and II.

- Intermediate:
- Gao, Invariant Descriptive Set Theory, CRC Press, 2009. Chapters 5-7.
- Dougherty, Jackson, Kechris, The structure of hyperfinite Borel equivalence relations, Transactions of the American Mathematical Society 341 (1994), no. 1, 193-225.

- Advanced:
- Kechris, Solecki, Todorcevic, Borel chromatic numbers, Advances in Mathematics 141 (1999), 1-44.
- Jackson, Kechris, Louveau, Countable Borel equivalence relations, Journal of Mathematical Logic 2 (2002), no. 1, 1-80.
- Gao, Jackson, Countable abelian group actions and hyperfinite equivalence relations, Inventiones Mathematicae 201 (2015), no. 1, 309-383.

Information about travel to Pittsburgh can be found here.

The CMU mathematics department is located in Wean Hall on the CMU campus. Wean Hall is building 33 on the campus map.

VERY IMPORTANT NOTE ABOUT LODGING: A block of rooms earmarked for attendees has been set aside at a local hotel (the Shadyside Inn). If we are covering your lodging expenses then we will need to make a reservation for you. Please don't make your own reservation if we have promised you support, this will cause confusion and may make it impossible for us to reimburse you.

Funds provided by the National Science Foundation will be used to reimburse some participant transportation and lodging expenses. Priority will be given to students and faculty who do not hold federal research grants. Please request such funds as far in advance of the meeting as possible by sending the following information to the email address appalachiansettheory@gmail.com

- Your name, university affiliation, mailing address, phone number and email address
- Your nationality and visa status
- Your professional status and some additional information:
- Undergraduate students: please describe your background in set theory
- Graduate students: please tell us your year and the name of your thesis advisor if you have one
- Faculty: please tell us whether you hold a federal research grant

- A brief statement about your interest in the workshop