My research is mainly in descriptive set theory as applied to ergodic theory and Borel dynamics, measured group theory, descriptive combinatorics, and Borel equivalence relations in general. My work also includes finite combinatorics and applications of logic to other areas of mathematics such as Ramsey theory and arithmetic combinatorics.
Publications and preprints
Pointwise ergodic theorem for locally countable quasi-pmp graphs
A short nonalgorithmic proof of the containers theorem for hypergraphs
(with Terence Tao) Banff International Research Station, Canada (July, 2015)
The Effros space of a σ-Polish space is standard Borel
UIUC (January, 2014)
Countable compact Hausdorff spaces are Polish
UCLA (March, 2013)
Segal's effective witness to measure-hyperfiniteness
Caltech (Fall, 2012)
Hjorth's proof of the embeddability of hyperfinite equivalence relations into E0
UCLA (Fall, 2010)
Contributions in others' works
Y. Moschovakis, Abstract Recursion and Intrinsic Complexity, to appear in the ASL Lecture Notes in Logic (2018)
Section 3B consists of the results of Part 3 of my Ph.D. thesis on complexity measures for recursive programs.
UCLA (Fall, 2009)
A. Kechris, The spaces of measure preserving equivalence relations and graphs, preprint (2017)
Sections 3, 6, 19 contain some of my results regarding the weak and strong topologies, the discontinuity of the map from actions to equivalence relations, and subtreeings of treeings of an equivalence relation.
Caltech (Spring, 2013)
M. Lupini, Polish groupoids and functorial complexity, Trans. of the Amer. Math. Soc., 369 (2017), no. 9, 6683–6723
Appendix consists of my proof of the Effros space of a σ-Polish space being standard Borel. [pdf]