I am broadly interested in the descriptive set theory, dynamics, and geometry of topological groups, especially Polish groups. Groups arise naturally as the collections of transformations of familiar mathematical objects, e.g. permutations of a set, symmetries of a geometric structure, homeomorphisms of a topological space, diffeomorphisms of a smooth manifold, etc. It is a non-trivial fact that many groups (especially those I study) come with uniquely determined (Polish) topologies which make the group operations continuous. So the study of such sets of transformations is both algebraic and topological.
I have focused lately on groups of homeomorphisms of compact one-dimensional manifolds: the interval and the circle. I am especially interested in contrasting the structures of groups which satisfy smoothness conditions arising in classical real analysis, e.g. continuous differentiability; piecewise linearity; Lipschitz and Hölder conditions; absolute continuity; and bounded variation.
Here is a list of research papers:
A Kuratowski closure-complement variant whose solution is independent of ZF (joint with Todd Johnson, Adam Kral, Aaron Li, and Justin Soll), preprint.
Arxiv link.
The closure-complement-frontier problem in saturated polytopological spaces (joint with Sara Canilang, Nicolas Graese, and Ian Seong), to appear in New Zealand Journal of Mathematics.
Arxiv link.
Maximal pseudometrics and distortion of circle diffeomorphisms, to appear in the Journal of the London Mathematical Society.
Arxiv link.
Polishability of some groups of interval and circle diffeomorphisms, Fundamenta Mathematicae, Vol. 248 (2020), pp. 91--109.
Arxiv link.
Profinite branch groups give small actions (joint with Phillip Wesolek), an appendix to Geometric stability theory for μ-structures by Junguk Lee, Annals of Pure and Applied Logic, Vol. 170(8) (2019), pp. 843--866.
Arxiv link.
On the large-scale geometry of diffeomorphism groups of 1-manifolds, Forum Mathematicum, Vol. 30(1) (2018), pp. 75--86. doi:10.1515/forum-2016-0149.
Arxiv link.
Existence and genericity of finite topological generating sets for homeomorphism groups (joint with Azer Akhmedov), Indiana University Mathematics Journal, Vol. 68(6) (2019), pp. 1833--1848.
Arxiv link.
PL+(I) is not a Polish group (joint with Robert R. Kallman),
Ergodic Theory and Dynamical Systems, Vol. 36(7) (2016), pp. 2121--2137. doi:10.1017/etds.2015.13.
Arxiv link.
Some applications of Hölder's theorem in groups of analytic diffeomorphisms of 1-manifolds (joint with Azer Akhmedov),
Topology and Its Applications, Vol. 180 (2015), pp. 85--90. doi:10.1016/j.topol.2014.11.002.
Arxiv link.
Openly Haar null sets and conjugacy in Polish groups (joint with Robert R. Kallman),
Israel Journal of Mathematics, Vol. 215(1) (2016), pp. 1--30. doi:10.1007/s11856-016-1374-y
View preprint.
A conjecture of Gleason on the foundations of geometry (joint with Robert R. Kallman),
Topology and its Applications, Vol. 161 (2014), pp. 279--289. doi:10.1016/j.topol.2013.10.027.
View preprint.
The descriptive complexity of series rearrangements,
Real Analysis Exchange, Vol. 38(2) (2012/13), pp. 337--352.
View preprint.