Research

My work has been primarily in descriptive set theory, specifically on Borel equivalence relations, classification problems, and Polish group actions. My work often involves forcing, and other techniques from axiomatic set theory. I have also worked on combinatorial set theory.

Publications and Preprints

• Intermediate models with deep failure of choice (with Y. Hayut)
  [pdf][arXiv]

• The finite Friedman-Stanley jumps: generic dichotomies for Borel homomorphisms.
  [pdf][Slides][arXiv]

• Strong ergodicity for the Gamma-jump operator and for actions of Polish wreath products. arXiv:2203.14427
  [pdf] [Slides 25 min]

• Classifying invariants for E_1: A tail of a generic real. arXiv:2112.12881
  Notre Dame Journal of Formal Logic, to appear.
  [pdf] [Slides 50 min]

• Strong ergodicity around countable products of countable equivalence relations.
  Israel Journal of Mathematics, to appear.
  [pdf]
  [Slides: On the Gamma-jumps of Clemens and Coskey, 25 min]
  [Slides: Strong ergodicity between countable products of countable equivalence relations, 20 min]
  [Slides: Classification using countable sequences of countable sets of reals, 25 min]

• Strong ergodicity phenomena for bernoulli shifts of bounded algebraic dimension (with A. Panagiotopoulos).
  Annals of Pure and Applied Logic, Vol. 175, Issue 5 (2024).
  [pdf] [doi]

• Actions of tame abelian product groups (with S. Allison).
  Journal of Mathematical Logic, Vol. 23, No. 03 (2023).
  [pdf] [doi] [Slides 50 min]

• Anti-classification results for groups acting freely on the line (with F. Calderoni, D. Marker, L. Motto Ros).
  Advances in Mathematics, Vol. 418 (2023).
  [pdf] [doi] 

• Borel reducibility and symmetric models.
  Transactions of the American Mathematical Society, Vol. 374 (2021).
  [pdf] [doi] [Slides 50 min] 

• Fresh Subsets of ultrapowers.
  Archive for Mathematical Logic, vol. 55 (2016), pp. 835-845.
  [pdf] [doi]

• Ultrapowers of forcing notions. Master's thesis, Hebrew University, 2013.
  [pdf]

Other Slides

• ASL invited address, JMM, Boston, January 2023. [Slides 50 min]

Unpublished notes

• Unpinned actions via metric Scott analysis, 2022.
  A short and streamlined proof is given for a theorem of Thompson: that a non-CLI group has an unpinned (and so not essentially countable) action. The proof relies on the metric Scott analysis due to Ben Yaacov, Doucha, Nies, and Tsankov. 

• On b and add(B), 2021.
  A characterization of add(B) is given, as a Baire-category variant of the bounding number b. A theorem of Cichon and Pawlikowsky is recovered.

• A note on E_1 and orbit equivalence relations, 2018.
  The note presents a proof of a theorem of Kechris and Louveau, using intersections of models of set theory. This theme is expanded upon in my paper "Classifying invariants for E_1: A tail of a generic real".

• On Moore's partition, 2016.
  This note provides a negative answer to Conjecture 4 in [Moore - A solution to the L space problem]. After sending this out on September 2016, I learned that a solution was announced earlier by Peng and Wu.

• On the proof that a tree with an ascent path is not special, 2016.
  A simple conceptual proof is given, using an ultrapower, to a result of Shelah and Stanley. (Working with an ill-founded ultrapower, in contrast to the reduced products used by [Devlin].)

• Zero sharp implies all (branchless, fat) trees in L are special, 2015.
  An equivalence with zero sharp is proved.

• Separating weak squares, 2013.
  This note presents the most basic separation results for weak squares, with simpler proofs than in [Magidor, Lambie-Hanson - On the strengths and weaknesses of weak squares]. This simplified approach for separation is further pursued in my paper "Fresh subsets of ultrapowers".