Category Archives: Appalachian Set Theory

Appalachian Set Theory workshop: Benjamin Miller, January 21, 2017

Appalachian set theory

Saturday, January 21, 2017

9:30 a.m. – 6 p.m. with coffee and lunch breaks

Carnegie Mellon University

Benjamin Miller : “Applications of the open graph dichotomy”


The open graph dichotomy is a generalization of the perfect set theorem, ensuring that every open graph on an analytic set has either a countable coloring or a perfect clique. As the proof of this result is essentially the same as that of the perfect set theorem, it can be viewed as one of the very simplest descriptive set-theoretic dichotomy theorems. Nevertheless, there is an infinite-dimensional analog of the open graph dichotomy (whose proof is essentially the same) that has recently proven particularly useful in studying Borel functions, graphs, and sets of low complexity.

We will begin by stating and proving the infinite-dimensional analog of the open graph dichotomy. We will then describe how it can be used to give particularly simple proofs of several well-known facts, such as the Hurewicz dichotomies, the Jayne-Rogers theorem, and Lecomte’s characterization of the existence of countable Borel colorings of low complexity. Finally, we will turn our attention to the new result that there is a twenty-four element basis, under closed continuous embeddability, for the class of Borel functions that are not Baire class one.


Appalachian Set Theory workshop: Maryanthe Malliaris, November 5, 2016

The next meeting of the Appalachian Set Theory workshop series will take place
at CMU and may be of interest to the Pittsburgh logic community.

Maryanthe Malliaris will lead a workshop on “Cofinality spectrum
problems: p, t, and model theory”.

For details please see the workshop web page at

The Appalachian Set Theory workshop series is supported by the National
Science Foundation.

Appalachian Set Theory workshop: Itay Neeman, April 9, 2016

Appalachian set theory

Saturday, April 9, 2016

9:30 a.m. – 6 p.m. with coffee and lunch breaks

Virginia Commonwealth University

Lectures in Temple 1165

Itay Neeman : “Forcing with countable conditions”


The workshop will cover forcing techniques used in the proof that consistently every two ℵ2-dense real order types are isomorphic.

The question is related to the quest for higher analogues for applications of the proper forcing axiom (PFA). It was raised by Baumgartner in the early 1970s, when he proved the analogous result at ℵ1. Baumgartner’s result at ℵ1 is a central consequence of PFA (though it precedes the formulation of PFA by almost a decade) and moreover the methods of his proof have been instrumental for other key applications of PFA.

Projected topics include countable reflection of clubs, sparse sets, and iteration theory for countably closed forcing with side conditions.


Appalachian Set Theory workshop: Su Gao, October 24, 2015

Appalachian set theory

Saturday, Oct 24, 2015

9:30 a.m. – 6 p.m. with coffee and lunch breaks

Carnegie Mellon University

Su Gao : “Countable abelian group actions”


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 ℤn. 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 2n is 3.
  • In the third lecture we will construct some hyperaperiodic elements for 2n and use them to show that the continuous chromatic number for the free part of 2n 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.


Appalachian Set Theory workshop: John Steel

Appalachian set theory

Saturday, May 9, 2015

Carnegie Mellon University

John Steel : “Iteration Trees”


The workshop will cover the basic theory of iteration trees, and some of its applications. It will start at a basic level, defining ultrapowers of models of ZFC and their properties, and is intended to be accessible to anyone who has taken a graduate-level course in set theory. In particular, no background in inner model theory will be assumed.

Appalachian Set Theory workshop: Slawomir Solecki

November 22, 2014, at the University of Illinois at Chicago

Slawomir Solecki

Ultrafilter space methods in Infinite Ramsey Theory

Professor Solecki will present a part of Infinite Ramsey Theory by explaining the methods and results stemming from Ellis’ idempotent lemma. These methods rely heavily on ultrafilters and their spaces and are far reaching generalizations of the initial argument due to Galvin and Glazer. The lectures will include proofs of older and more recent results, including the Furstenberg-Katznelson and Gowers theorems. The lectures will present a a new unified treatment of this part of Infinite Ramsey Theory, including some applications if time allows.

Here is a reading list containing some background material for the workshop:


  • Basics:
    • The ultrafilter entry on Wikipedia.
    • Pages 409-413 in W. Rudin, Homogeneity problems in the theory of Čech compactifications, Duke Math. J. 23 (1956), 409-419.
  • Intermediate:
    • Sections 2 and 3 in V. Bergelson, A. Blass, N. Hindman, Partition theorems for spaces of variable  words, Proc. London Math. Soc. 68 (1994), 449-476.
  • Advanced:
    • H. Furstenberg, Y. Katznelson, Idempotents in compact semigroups and Ramsey theory,   Israel J. Math. 68 (1989), 257-270.
    • Chapter 2 in S. Todorčević, Introduction to Ramsey Spaces, Annals of Mathematics Studies, 174. Princeton University Press, 2010.


  • Supplementary:
    • Chapters 1-5 in  N. Hindman, D. Strauss, Algebra in the Stone-Čech Compactification, de Gruyter  Expositions in Mathematics 27, Walter de Gruyter & Co., 2012.

Appalachian Set Theory workshop: Caleb Eckhardt and Andrew Toms

Appalachian Set Theory Workshop on
C*-algebras, classification, and descriptive set theory

September 8-9, 2012, at the Fields Institute in Toronto, Canada

The theme of these lectures will be that there are at least two quite different ideas of what it means to “classify” the objects of a category up to isomorphism. One of these is classification by invariants with good computational properties, while the other, surely familiar to regular attendees of the AST series, is the notion of Borel reducibility. We will examine how they both contrast and complement each other through the lens of C*-algebra theory. What follows is a rough outline of the lecture series. Continue reading

Appalachian Set Theory workshop: Hugh Woodin

Hugh Woodin will be giving a workshop at Cornell on April 7, 2012. The title of the talk is “The HOD dichotomy”.

See the web site for the talk here:

Appalachian Set Theory workshop: Asger Tornquist

Asger Tornquist will be giving a workshop at Carnegie Mellon University on March 3, 2012. The title of the talk is “Set theory and von Neumann algebra”. See the web site for the talk here:

Appalachian Set Theory workshop: Boban Velickovic

There will be an all-day Appalachian Set Theory workshop by Boban Velickovic at University of Illinois, Chicago, on Saturday, October 15, 2011.

The title is “Proper forcing remastered”. For a description and additional information, please see