Research

Shadows Can Be Deceiving          While a single projection may not tell much about a set,  the more angles we consider, the more we may know about it. (image courtesy of J. Zahl)

Direct link to my CV and list of research talks

Published Research and pre-prints:

• • Prescribed projections and efficient coverings by curves in the plane, (with Alan Chang & Alex McDonald) (arXiv:2310.08776)
  • Realizing trees of hypergraphs in thin sets,  (with Allan Greenleaf & Alex Iosevich) (arXiv:2401.11597)
  • Infinite constant gap length trees in products of thick Cantor sets,  (with Alex McDonald) Proc. Roy. Soc. Edinburgh Sect. A 154 (2024), no. 5, 1336–1347.
  • Nonempty interior of configuration sets via microlocal partition optimization, (with Allan Greenleaf & Alex Iosevich) Math. Z. 306 (2024), no. 4, Paper No. 66.
  • Finite Point configurations in Products of Thick Cantor sets and a Robust Nonlinear Newhouse Gap Lemma, (with Alex McDonald) (Math. Proc. Cambridge Philos. Soc., 175 (2023), no. 2, 285-301)

• • Lattice Points Close to the Heisenberg Spheres, (with Elizabeth Campolongo), Matematica, 2 (2023), no. 1, 156–196.

• • Dimension and measure of sums of planar sets and curves – Simon – 2022 – Mathematika – Wiley Online Library, (with K. Simon), (Mathematika 68) (2022), no. 4, 1364–1392.

• • Transversal families of nonlinear projections and generalizations of Favard length, (with R. Bongers), (Anal. PDE 16 (2023), no.1, 279–308.).

  • Finite Point Configurations and the Regular Value Theorem in a Fractal setting, (with Yumeng Ou), (Indiana Univ. Math. J.)  71 (2022), no. 4, 1707–1761.

  • Upper and lower bounds on the rate of decay of the Favard curve length for the Cantor four-corner set , (with L. Cladek, B. Davey), (Indiana Journal of Math), 71 (2022), no. 3, 1003- 1025.

• • A quantitative version of the Besicovitch projection theorem via multiscale analysis, (with Blair Davey), (The Journal of Geometric Analysis) vol.  32, no. 4, 138, (2022).

• • On k-point Configurations with Nonempty Interior, (with Allan Greenleaf, Alex Iosevich), (Mathematika) 68 (2022), no. 1, 163-190.

  • Book Review: The Finite Field Distance Problem by David Covert,
    (The American Mathematical Monthly) 32 (2022), https://doi.org/10.1080/00029890.2022.2072654

  • On the Fourier dimension of Sums and Products of subsets of Euclidean Space, (with K. Hambrook) (Proc. Amer. Math. Soc.) (2021).

  • Configuration Sets with Nonempty Interior, (with A. Greenleaf, A. Iosevich) (The Journal of Geometric Analysis) vol. 31, no. 7, 6662–6680(2021).

• • Pinned geometric configurations in Euclidean space and Riemannian Manifolds, (with A. Iosevich, and I. Uriarte-Tuero), (Mathematics) (2021)

• • Interior of Sums of Planar sets and Curves, (with K. Simon) (Math. Proc. Cambridge Philos. Soc.)  (2020)

• • Finite trees inside thin subsets of R^d, (with A. Iosevich), (Springer Proc. Math.) (2019)
  • Maximal operators: scales, curvature, and the fractal dimension , (with A. Iosevich, E. Sawyer, and I. Uriarte-Tuero), Anal Math (2018).
  • Finite chains inside thin subsets of Euclidean space, (with M. Bennett, A. Iosevich), Analysis and PDE, (2016).
  • Intersections of sets and Fourier analysis , (with S. Eswarathasan and A. Iosevich), Journal d’Analyse Mathematique, (2016).
  • The lattice point counting problem on the Heisenberg groups, (with R. Garg, A. Nevo), Annales de l’Institut Fourier, (2015).
  • On the Mattila-Sjolin Theorem for distance sets , (with A. Iosevich and M. Mourgoglou), Annales Academia¦ Scientiarum Fennica Mathematica) (2012).
  • Lattice points close to families of surfaces, nonisotropic dilations and regularity of generalized Radon transforms, (with A. Iosevich), (New York Journal of Mathematics) (2011).
  • Fourier integral operators, fractal sets, and the regular value theorem , (with S. Eswarathasan and A. Iosevich), (Advances in Mathematics) (2011)
  • Ph.D. thesis: Applications of generalized Radon transforms to problems in harmonic analysis, geometric measure, and analytic number theory, Thesis work for Ph.D. in Mathematics, University of Rochester; Advisor: Alex Iosevich, (2012).