I am a Veblen Instructor in the math department at Princeton and the IAS.
My research interests include geometric analysis and PDE. My email is ochodosh AT math DOT princeton DOT edu.
I previously spent a year at Cambridge as a postdoc. I recieved my Ph.D. from Stanford in 2015, advised by Simon Brendle and co-advised by Michael Eichmair. Before graduate school at Stanford, I completed Part III of Cambridge's Mathematical Tripos, where I was a member of Churchill College. Before that, I was an undergraduate math and physics major at Stanford.
See my past teaching here.
Expository/Lecture Notes are available here.
A splitting theorem for scalar curvature (with Michael Eichmair and Vlad Moraru),
to appear in Comm. Pure Appl. Math. [arXiv:1804.01751]
Characterization of large isoperimetric regions in asymptotically hyperbolic initial data (with Michael Eichmair, Yuguang Shi, and Jintian Zhu),
Minimal surfaces and the Allen-Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates (with Christos Mantoulidis),
Curvature estimates and sheeting theorems for weakly stable CMC hypersurfaces (with Costante Bellettini and Neshan Wickramasekera),
Asymptotically flat three-manifolds contain minimal planes (with Daniel Ketover),
On far-outlying CMC spheres in asymptotically flat Riemannian 3-manifolds (with Michael Eichmair),
Global uniqueness of large stable CMC surfaces in asymptotically flat
3-manifolds (with Michael Eichmair),
Isoperimetry, scalar curvature, and mass in asymptotically flat Riemannian 3-manifolds (with Michael Eichmair, Yuguang Shi, and Haobin Yu),
submitted. [arXiv:1606.04626] [Oberwolfach Report]
Time-periodic Einstein–Klein–Gordon bifurcations of Kerr (with Yakov Shlapentokh-Rothman),
Comm. Math. Phys., vol. 356, no. 3, pp. 1155--1250 (2017). [arXiv:1510.08025] [Journal] [MR 3719550]
Stationary axisymmetric black holes with matter (with Yakov Shlapentokh-Rothman),
to appear in Comm. Geom. Anal. (2018) [arXiv:1510.08024]
Area-minimizing surfaces in asymptotically flat three-manifolds (with Michael Eichmair).
This result was included with this paper for publication. [arXiv:1510.07406]
Minimal hypersurfaces with bounded index (with Daniel Ketover and Davi Maximo),
Invent. Math., vol. 209, no. 3, pp. 617--664 (2017). [arXiv:1509.06724] [Journal] [MR 3681392]
Effective versions of the positive mass theorem (with Alessandro Carlotto and Michael Eichmair),
Invent. Math., vol. 206, no. 3, pp. 975--1016 (2016). [arXiv:1503.05910] [Journal] [MR 3573977]
On the maximum curvature of closed curves in negatively curved manifolds (with Simon Brendle),
Canad. Math. Bull., vol. 58, no. 4, pp. 713--722 (2015). [Local Version] [Journal] [MR 3415662]
On the topology and index of minimal surfaces (with Davi Maximo),
J. Differential Geom., vol. 104, no. 3, pp. 399--418 (2016). [arXiv:1405.7356] [Journal] [Oberwolfach Report][MR 3568626]
Isoperimetric structure of asymptotically conical manifolds (with Michael Eichmair and Alexander Volkmann),
J. Differential Geom., vol. 105, no. 1, pp. 1--19 (2017). [arXiv:1503.05181] [Journal] [MR 3592692]
Large isoperimetric regions in asymptotically hyperbolic manifolds,
Comm. Math. Phys., vol. 343, no. 2, pp. 393--443 (2016). [arXiv:1403.6108] [Journal] [MR 3477343]
Slowly converging Yamabe flows (with Alessandro Carlotto and Yanir Rubinstein),
Geom. Topol., vol. 19, no. 3, pp. 1523--1568 (2015).
[arXiv:1401.3738] [Journal] [MR 3352243] [Errata]
On discontinuity of planar optimal transport maps (with Vishesh Jain, Michael Lindsey, Lyuboslav Panchev and Yanir Rubinstein), J. Topol. Anal., vol. 7, no. 2, pp. 239--260 (2015). This research was carried out by our group mainly during a 2012 SURIM REU. [arXiv:1312.2929] [Journal] [MR 3326301]
A volume comparison theorem for asymptotically hyperbolic manifolds (with Simon Brendle),
Comm. Math. Phys., vol. 332, no. 2, pp. 839--846 (2014). [arXiv:1305.6628] [Journal] [MR 3257665]
Rotational symmetry of conical Kähler-Ricci solitons (with Frederick Tsz-Ho Fong),
Math. Ann., vol. 364, no. 3, pp. 777--792 (2016). [arXiv:1304.0277] [Journal] [MR 3466851]
Expanding Ricci solitons asymptotic to cones,
Calc. Var. Partial Differential Equations, vol. 51, no. 1--2, pp. 1--15 (2014). [arXiv:1303.2983] [Journal] [MR 3247379]
A lack of Ricci bounds for the entropic measure on Wasserstein space over the interval, J. Funct. Anal., vol. 262, no. 10, pp. 4570--4581 (2012). A condensed version of my Part III essay. [arXiv:1111.0058] [Journal] [MR 2900478]
Infinite matrix representations of isotropic pseudodifferential operators,
Methods Appl. Anal., vol. 18, no. 4, pp. 351--372 (2011). A condensed version of my undergraduate thesis (also including an application of the result to prove a commutator characterization for isotropic pseudodifferential operators similar to the one proven by Beals for standard pseudodifferential operators).
[arXiv:1101.4459] [Journal] [MR 2965982]