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My research lies mainly in the areas of classical harmonic analysis and discrete geometry. In particular, I study questions relating to Kakeya-type maximal functions and questions in incidence geometry such as the unit and distinct distance problems, and point-curve or point-surface incidence bounds.

Papers and Preprints

Combinatorial geometry

  • Zahl. Breaking the 3/2 barrier for unit distances in three dimensions. 2017. Submitted. [arXiv]
  • Sharir, Zahl. Cutting algebraic curves into pseudo-segments and applications. 2016. JCTA. 150:1-–35, 2017. [arXiv] [Journal]
  • Guth, Zahl. Curves in ℝ4 and two-rich points. 2015. Discrete. Comput. Geom. 58(1): 232–253, 2017. [arXiv] [Journal]
  • Ellenberg, Solymosi, Zahl. New bounds on curve tangencies and orthogonalities. 2015. Discrete Analysis. 22:1–22, 2016 [arXiv] [Journal]
  • Guth, Zahl. Algebraic curves, rich points, and doubly-ruled surfaces. 2015. Accepted, Am. J. Math. [arXiv]
  • Zahl. A note on rich lines in truly high dimensional sets. 2015. FoM, Sigma. 4(e2):1–13, 2016 [arXiv] [Journal]
  • Sheffer, Szabó, Zahl. Point-curve incidences in the complex plane. 2015. Accepted, Combinatorica. [arXiv] [Journal]
  • Fox, Pach, Sheffer, Suk, Zahl. A semi-algebraic version of Zarankiewicz's problem. 2014. JEMS. 19(6): 1785–1810, 2017. [arXiv] [Journal]
  • Sheffer, Zahl, de Zeeuw. Few distinct distances implies no heavy lines or circles. 2013. Combinatorica 36(3), 349--364, 2016. [arXiv] [Journal]
  • Sheffer, Sharir, and Zahl. Incidences between points and non-coplanar circles. 2012. Combin. Probab. Comput. 24(3), 490–520, 2015.[arXiv] [journal]
  • Zahl. A Szemeredi-Trotter type theorem in ℝ4. 2012. Discrete. Comput. Geom. 54(3):513–572, 2015. [arXiv] [Journal]
  • Zahl. An improved bound on the number of point-surface incidences in three dimensions. 2011. Contrib. Discrete Math. 8(1):100–121, 2013. [arXiv] [Journal] [errata]

Harmonic analysis and Kakeya

  • Katz, Zahl. An improved bound on the Hausdorff dimension of Besicovitch sets in ℝ3. 2017. Submitted. [arXiv]
  • Guth, Zahl. Polynomial Wolff axioms and Kakeya-type estimates in ℝ4. 2017. Submitted. [arXiv]
  • Dyatlov, Zahl. Spectral gaps, additive energy, and a fractal uncertainty principle. 2015. GAFA 26(4):1011–1094, 2016. [arXiv] [Journal]
  • Bond, Łaba, Zahl. Quantitative visibility estimates for unrectifiable sets in the plane. 2013. Trans. Amer. Math. Soc. 368, 5475-5513, 2016. [arXiv] [Journal]
  • Zahl. On the Wolff circular maximal function. 2011. Illinois J. Math. 56(4):1281–1295, 2012(2014). [arXiv] [Journal]
  • Zahl. L3 estimates for an algebraic variable coefficient Wolff circular maximal function. 2010. Rev. Mat. Iberoam. 28(4):1061–1090, 2012. [arXiv] [Journal] [errata]

Misc. Combinatorics

  • Hurlbert, Johnson, Zahl. On universal cycles for multisets. 2007. Discrete Math., 309(17):5321–5327, 2009. [arXiv] [Journal]
  • Katz, Zahl. Bounds on degrees of p-adic separating polynomials. 2007. J. Comb. Theory Ser. A, 115(7):1310–1319, 2008. [pdf][Journal]

Talks

Here are some recordings of talks I have given on my research.

  • Cutting curves into segments and incidence geometry. This was a talk given about some results in the papers "Breaking the 3/2 barrier for unit distances in three dimensions" and "Cutting algebraic curves into pseudo-segments and applications." It was given at the Harvard Center of mathematical sciences and applications on November 13, 2017.
  • An improved bound on the Hausdorff dimension of Besicovitch sets in ℝ3 . This was a talk given about some results in the paper by the same title. It was given at MSRI on May 19, 2017.
  • Some questions in discretized additive combinatorics. This was a talk given about some results in the paper "Spectral gaps, additive energy, and a fractal uncertainty principle." It was given at ICERM on February 15, 2016 as part of the workshop "Ergodic, Algebraic and Combinatorial Methods in Dimension Theory."
  • Space Curve Arrangements with Many Incidences. This was a talk given about a preliminary version of the results in the paper "Algebraic curves, rich points, and doubly-ruled surfaces." It was given at IPAM on May 22, 2014 as part of the workshop "Algebraic Techniques for Combinatorial and Computational Geometry: Finding Algebraic Structures in Extremal Combinatorial Configurations"
  • Visibility and discretized projection theorems . This was a talk given about some results in the paper "Quantitative visibility estimates for unrectifiable sets in the plane." It was given at IPAM on May 8, 2014 as part of the workshop "Algebraic Techniques for Combinatorial and Computational Geometry: The Kakeya Problem, Restriction Problem, and Sum-product Theory"
  • Multi-level partitioning theorems. This was a talk given about some results in the paper "A semi-algebraic version of Zarankiewicz's problem." It was given at IPAM on April 7, 2014 as part of the workshop "Algebraic Techniques for Combinatorial and Computational Geometry: Tools from Algebraic Geometry."
  • On the structure of planar point sets that determine few distinct distances. This talk discusses the results in the paper "Few distinct distances implies no heavy lines or circles." It was given at IPAM on March 28, 2014 as part of the workshop "Algebraic Techniques for Combinatorial and Computational Geometry: Combinatorial Geometry Problems at the Algebraic Interface"