Research

My research interests are in Analysis and Geometry on Metric Spaces.

• • Geometric function theory: quasiconformal, quasisymmetric and bi-Lipschitz mappings.

  • Sub-Riemannian geometry.

  • Geometric measure theory.

  • Potential theory and Harmonic measure.

  • One variable complex analysis

My research is currently supported by National Science Foundation under Grant No. DMS-1800731, Grant No. DMS-1952510, and Grant No. DMS-2154918. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Papers and Preprints

• • Bi-Lipschitz arcs in metric spaces with controlled geometry [arxiv]

(with J. Honeycutt and S. Zimmerman) to appear in Rev. Mat. Iberoam.

• • Parametrizability of infinitely generated attractors [arxiv] [journal]
    (with E. Shaw) Ann. Fenn. Math., 49 (2024), no.1, 81-97.

  • Genus g Cantor sets and Germane Julia sets [arxiv]

(with A. Fletcher and D. Stoertz) 2022

• • Time complexity of the Analyst's Traveling Salesman algorithm [arxiv] [journal]
    (with A. Ramirez) J. Log. Anal., 16 (2024) no.2, 1-17.

  • Bi-Lipschitz embeddings of quasiconformal trees [arxiv] [journal]
    (with G. C. David and S. Eriksson-Bique) Proc. Amer. Math. Soc., 151 (2023), no. 5, 2031-2044.

  • Decomposing multitwists [arxiv] [journal]
    (with A. Fletcher) to appear in J. Anal. Math.

  • Bi-Lipschitz geometry of quasiconformal trees [arxiv] [journal]
    (with G. C. David) Illinois J. Math., 66 (2022) no.2, 189-244.

  • Uniformization of Cantor sets with bounded geometry [arxiv] [journal]
    Conform. Geom. Dyn., 25 (2021), 88-103.

  • On uniformly disconnected Julia sets [arxiv] [journal]
    (with A. Fletcher) Math. Z., 299 (2021), 853-866.

  • Hölder parameterization of iterated function systems and a self-affine phenomenon [arxiv] [journal]
    (with M. Badger) Anal. Geom. Metr. Spaces, 9 (2021), 90-119.

  • Bi-Lipschitz embeddings of Heisenberg submanifolds into Euclidean spaces [arxiv] [journal]
    (with V. Chousionis, S. Li and S. Zimmerman) Ann. Acad. Sci. Fenn. Math., 45 (2020), 931–955.

  • Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem [arxiv] [journal]
    (with M. Badger and L. Naples) Adv. Math., 349 (2019), 564-647.

  • Geometry of measures in real dimensions via Hölder parameterizations [arxiv] [journal]
    (with M. Badger) J. Geom. Anal., 29 (2019), no. 2, 1153–1192.

  • Quasisymmetric extension on the real line [arxiv] [journal]
    Proc. Amer. Math. Soc., 146 (2017), no. 6, 2435-2450.

  • Quasiconformal non-parametrizability of almost smooth spheres [arxiv] [journal]
    (with P. Pankka) Selecta Math., 23 (2017), 1121–1151.

  • Bi-Lipschitz embedding of the generalized Grushin plane in Euclidean spaces [arxiv] [journal]
    (with M. Romney) Math. Res. Lett. 24 (2017), no. 4, 1179-1205.

  • Weak chord-arc curves and double-dome quasisymmetric spheres [arxiv] [journal]
    Anal. Geom. Metr. Spaces 4 (2016), 54-67.

  • Quasisymmetric spheres over Jordan domains [arxiv] [journal]
    (with J.-M. Wu) Trans. Amer. Math. Soc. 368 (2016), 5727-5751.

  • Sets of constant distance from a jordan curve [arxiv] [journal]
    (with J.-M. Wu) Ann. Acad. Sci. Fenn. Math. 39 (2014), 211-230.

Thesis and other

• 1. Very special snowflakes [article]
     A UConn Today article written by Kim Krieger

  2. Quasisymmetric spheres constructed over quasidisks [thesis]
     Thesis (Ph.D.)–University of Illinois at Urbana-Champaign. 2014. 107 pp. ISBN: 978-1321-51488-9, ProQuest LLC

  3. Constructions on Rohde snowflakes [preprint]
     Preliminary examination paper