I'm an assistant professor at UCLA. My research interests lie in descriptive set theory and its connections to related areas such as computability theory, combinatorics, ergodic theory, probability, and operator algebras.
Office: MS 6228.
Los Angeles, CA 90095-1555
Publications and preprints:
- Folner tilings for actions of amenable groups (with Clinton Conley, Steve Jackson, David Kerr, Brandon Seward, and Robin Tucker-Drob). [ arXiv | pdf ]. Submitted.
- Borel circle squaring (with Spencer Unger). Submitted. [ arXiv | pdf ].
- Descriptive graph combinatorics (with Alekos Kechris). Preprint [ pdf ].
- Hyperfiniteness and Borel combinatorics (with Clinton Conley, Steve Jackson, Brandon Seward, and Robin Tucker-Drob). Submitted. [ arXiv | pdf ]
- Topological generators for full groups of hyperfinite pmp equivalence relations. Submitted. [ arXiv | pdf ]
- Uniformity, universality, and computability theory. To appear in J. Math. Logic. [ arXiv | pdf ].
- Jump operations for Borel graphs (with Adam Day). Submitted. [ arXiv | pdf ].
- The universality of poly-time Turing equivalence. Mathematical Structures in Computer Science (2016) [ arXiv | pdf | doi ].
- Brooks's theorem for measurable colorings (with Clinton Conley and Robin Tucker-Drob). Forum of Math. Sigma 4 (2016) [ arXiv | pdf | doi ].
- Baire measurable paradoxical decompositions via matchings (with Spencer Unger). Advances in Mathematics 289 (2016), 397-410.
[ arXiv | pdf | doi ].
- A determinacy approach to Borel combinatorics. J. Amer. Math. Soc. 29 (2016), 579-600. [ arXiv | pdf | doi ]
- Martin's conjecture, arithmetic
equivalence, and countable Borel equivalence
relations (with Theodore Slaman and John Steel). Ordinal definability and recursion theory: The
cabal seminar volume III, Lecture Notes in Logic 43, Cambridge University Press, 2016, 200-219. [ arXiv | pdf ]
- Minimal Betti Numbers (with Christopher Dodd, Victor Meyerson, and Ben Richert). Communications in Algebra Vol 35 (3), 2007, pp 759-772. [ arXiv | doi ]
Research notes (not intended for publication):
- A Baire category proof of the Ackerman-Freer-Patel Theorem. May 2016. [ pdf ]
Structure in complete sections of
the shift action of a residually finite group. November 2013. [ pdf ]
- Distance from marker sequences in locally finite Borel graphs (with Clinton Conley). October 2013. [ pdf ]
- A short proof that an acyclic n-regular Borel graph may have Borel chromatic number n+1. May 2013. [ pdf ]
- Is the Turing jump unique? : Martin's conjecture and countable Borel equivalence relations. December 2011. [ pdf ]
- Borel circle squaring. Slides from the Czech winter school, 2017.
[ pdf ]