RESEARCH
Research Interests
Elliptic and Parabolic PDEs, Optimal Transport and Geometric Optics, Free Boundary Problems.
Publications and Preprints
Quantitative convergence of the "bulk'' free boundary in an oscillatory obstacle problem. (with W. M. Feldman) -- Interfaces Free Bound. 26 (2024), no. 1, pp. 31–44.
A perturbative approach to the parabolic optimal transport problem. (with J. Kitagawa) -- SIAM J. Math. Anal. Vol. 55, Issue 6 (2023), 6740-6763.
Regularity for a special case of two-phase Hele-Shaw flow via parabolic integro-differential equations. (with R. Schwab) -- J. Funct. Anal. 285 (2023), no. 8, Paper No. 110066, 83 pp.
Inverse iteration for the Monge-Ampère eigenvalue problem. (with J. Kitagawa) -- Proc. Amer. Math. Soc. 148 (2020), no. 11, 4875–4886.
Exponential convergence of parabolic optimal transport on bounded domains. (with J. Kitagawa) -- Anal. PDE 13 (2020), no. 7, 2183–2204
Harnack inequality for a class of Kolmogorov-Fokker-Planck equations in non-divergence form. (with G. Tralli) -- Arch. Ration. Mech. Anal. 233 (2019), no. 2, 867–900.
Harnack's inequality for a class of non-divergent equations in the Heisenberg group. (with C. E. Gutiérrez and G. Tralli) -- Comm. Partial Differential Equations 42 (2017), no. 10, 1644–1658.
An iterative method for generated Jacobian equations. (with C. E. Gutiérrez) -- Calc. Var. Partial Differential Equations 56 (2017), no. 4, Paper No. 101, 14 pp.
C1,α estimates for the parallel refractor. (with C. E. Gutiérrez and G. Tralli) -- Nonlinear Anal. 142 (2016), 1–25.
PhD Thesis, Temple University, 2018
Papers prior to graduate school:
On the P-scalar curvature. (with J. Corvino) -- J. Geom. Anal. 27 (2017), no. 2, 1589–1623.
On isoperimetric surfaces in general relativity, II. (with J. Corvino, S. Kapita, and H. Wu) -- J. Geom. Phys. 59 (2009), no. 11, 1453–1460.
Funding
NSF DMS-2246611 (2023-2026)
AMS-Simons Travel Grant (2020-2023)
For a true writer each book should be a new beginning where he tries again for something that is beyond attainment. He should always try for something that has never been done or that others have tried and failed. Then sometimes, with great luck, he will succeed.
How simple the writing of literature would be if it were only necessary to write in another way what has been well written. It is because we have had such great writers in the past that a writer is driven far out past where he can go, out to where no one can help him.
Ernest Hemingway, Nobel Banquet Speech
Ultimately, I think that despair is just hubris, because you think you know what's gonna happen. And as Gandalf says, "Not even the wise know all ends."
The unpredictability of what happens next is no curse or taunt. It’s just life, ever maddening, ever mysterious. If you’re frightened, you’re not alone, and a shortfall of confidence is no harbinger of doom. Shoulders back. Chin forward.
You’ll be tripped up by unforeseen obstacles and setbacks. But you’ll also trip across unanticipated bounty and blessings. You’ll quite possibly find yourself someday in a place and role you never expected. You’ll be moved by that.
And you’ll realize that the journey to that point was all the more interesting for its refusal to be scripted, and for its absence of any firm guarantees.