Assistant Professor of Mathematics · Johns Hopkins University

Zihui Zhao pronounced Zǐ-huì

I work on regularity problems in geometric measure theory, geometric analysis, elliptic PDEs and harmonic analysis, with a particular focus on:

  • Generalized minimal submanifolds
  • Free boundary problems
  • Quantitative unique continuations
  • Harmonic and elliptic measures

Research for a broad audience: “Brace for anomalies in soap bubbles”

Curriculum vitae arXiv profile

Academic path

Appointments and education

Johns Hopkins University
Assistant Professor of Mathematics (tenure-track), 2023–present.

University of Chicago
NSF Postdoctoral Fellow and L.E. Dickson Instructor, August 2019–2023. Mentor: Prof. Carlos Kenig.

Institute for Advanced Study
Member during the Special Year on Variational Methods in Geometry, September 2018–July 2019. Mentor: Prof. Camillo De Lellis.

University of Washington
Ph.D. in Mathematics, June 2018. Advisor: Prof. Tatiana Toro.

Publications

Publications and preprints

A complete list of preprints is also available on my arXiv page.

  1. On the Asymptotic Properties of Solutions to One-phase Free Boundary Problems With Max Engelstein and Daniel Restrepo. arXiv:2511.08393.
  2. A Note on the Critical Set of Harmonic Functions Near the Boundary With Carlos Kenig. Trans. Amer. Math. Soc. 378 (2025), 3721–3754.
  3. Elliptic Operators in Rough Sets, and the Dirichlet Problem with Boundary Data in Hölder Spaces With Mingming Cao, Pablo Hidalgo-Palencia, José María Martell, and Cruz Prisuelos-Arribas. J. Funct. Anal. 288 no. 5 (2025).
  4. Examples of non-Dini Domains with Large Singular Sets With Carlos Kenig. Advanced Nonlinear Studies 23 no. 1 (2023), Special Issue in honor of David Jerison.
  5. Expansion of Harmonic Functions Near the Boundary of Dini Domains With Carlos Kenig. Rev. Mat. Iberoam. 38 (2022), 2117–2152.
  6. Elliptic Measures for Dahlberg-Kenig-Pipher Operators: Asymptotically Optimal Estimates With Simon Bortz and Tatiana Toro. Math. Ann. 385 (2023), 881–919.
  7. Boundary Unique Continuation on Dini Domains and the Size of the Singular Set With Carlos Kenig. Arch. Ration. Mech. Anal. 245 (2022), 1–88.
  8. Optimal Poisson Kernel Regularity for Elliptic Operators with Hölder-continuous Coefficients in Vanishing Chord-arc Domains With Simon Bortz and Tatiana Toro. J. Funct. Anal. 285 no. 5 (2023).
  9. Uniform Rectifiability and Elliptic Operators Satisfying a Carleson Measure Condition. Part II: The Large Constant Case With Steve Hofmann, José María Martell, Svitlana Mayboroda, and Tatiana Toro. Combined with Part I for publication in Geom. Funct. Anal. 31 (2021), 325–401.
  10. Two Phase Free Boundary Problem for Poisson Kernels With Simon Bortz, Max Engelstein, Max Goering, and Tatiana Toro. Indiana Univ. Math. J. 71 (2022), 251–306.
  11. Dirichlet Energy-minimizers with Analytic Boundary With Camillo De Lellis. Indiana Univ. Math. J. 72 (2023), 1367–1428.
  12. Dirichlet Problem in Domains with Lower Dimensional Boundaries With Joseph Feneuil and Svitlana Mayboroda. Rev. Mat. Iberoam. 37 (2021), 821–910.
  13. Square Function Estimates, BMO Dirichlet Problem, and Absolute Continuity of Harmonic Measure on Lower-dimensional Sets With Svitlana Mayboroda. Anal. PDE 12 (2019), 1843–1890.
  14. Uniform Rectifiability and Elliptic Operators Satisfying a Carleson Measure Condition. Part I: The Small Constant Case With Steve Hofmann, José María Martell, Svitlana Mayboroda, and Tatiana Toro. Submitted on arXiv.
  15. Boundary Rectifiability and Elliptic Operators with W1,1 Coefficients With Tatiana Toro. Adv. Calc. Var. 14 (2021), 37–62.
  16. BMO Solvability and the A-infinity Condition of the Elliptic Measure in Uniform Domains J. Geom. Anal. 28 (2018), 866–908.

Teaching

Selected teaching

Johns Hopkins University · Spring 2026

Graduate Topics Course: The Bernstein Problem for Minimal Graphs

This graduate course uses the classical Bernstein problem as a gateway to modern geometric analysis. Starting from a PDE background, the course builds the necessary tools from geometric measure theory and PDE on manifolds as they arise, with an emphasis on the regularity ideas and examples that shape the theory of minimal graphs.

Lecture notes

University of Chicago · Fall 2022 - Winter 2023 · Fall 2020 - Winter 2021

Honors Calculus (Inquiry-Based Learning)

An inquiry-based learning (IBL) honors calculus course for a highly selected group of first-year undergraduates, with a strong emphasis on student presentations and class discussion.