Reading courses

For mathematics student at Rice:

I am happy to offer independent reading courses when the academic fit is right and time allows. Feel free to get in touch and tell me what you would like to learn, why, which math classes you have already taken, and how you did in them. Please keep in mind that I generally don't offer reading courses on material we cover in regularly taught courses in our department and that I will generally expect that you have completed coursework at the level of our analysis sequence.

Geometry seminar

For mathematics students at Rice:

If you are interested in learning about current research in differential geometry, broadly construed, you are encouraged to attend the Geometry seminar at Rice.

Conferences co-organized

• Geometric moduli spaces - rigidity, genericity, stability. With I. Alonso Lorenzo, B. Hanke, M. Upmeier. ICMS, Scotland, May 19-23, 2025.

• Simons Workshop on Geometric Analysis at Courant. With C. Li, V. Tosatti. Courant Institute, NYU, May 9-11, 2025.

• Geometry of Scalar Curvature 2019. with B. Hanke, P. Piazza, T. Schick, C. Sormani. Il Palazzone Cortona, Italy, July 8-12, 2019.

Support

• NSF Grant DMS-1905165/2050120/2147521.

• NSF Grant DMS-2403728.

• AMS/Simons travel grant.

Publications

• Generic regularity for minimizing hypersurfaces in dimension 11. With O. Chodosh, F. Schulze, Z. Wang. 

• Almgren's three-legged starfish. With J. Marx-Kuo. Submitted.

• Revisiting generic mean curvature flow in R³. With O. Chodosh, K. Choi, F. Schulze. Submitted.

• Mean curvature flow with generic low-entropy initial data II. With O. Chodosh, F. Schulze. Duke Math. J.. To appear.

• Improved generic regularity of codimension-1 minimizing integral currents. With O. Chodosh, F. Schulze. Ars Inven. Anal., May 2024.

• Generic regularity for minimizing hypersurfaces in dimensions 9 and 10. With O. Chodosh, F. Schulze. Submitted.

• Double-well phase transitions are more rigid than minimal hypersurfaces. Proc. Am. Math. Soc., vol. 152 (2024), 1301-1308.

• Decomposing 4-manifolds with positive scalar curvature. With R. H. Bamler, C. Li. Adv. Math., vol. 430 (2023), 109231.

• The p-widths of surfaces. With O. Chodosh. Publ. Math. IHÉS, vol. 137 (2023), 245-342.

• Metrics with λ1(-Δ+kR) >= 0 and flexibility in the Riemannian Penrose Inequality. With C. Li. Commun. Math. Phys., vol. 401 (2023), 1831-1877.

• Mean curvature flow with generic low-entropy initial data. With O. Chodosh, K. Choi, F. Schulze. Duke Math. J., vol. 173 (2024), 1269-1290.

• Variational aspects of phase transitions with prescribed mean curvature. Calc. Var. Partial Differ. Equ., vol. 61 (2022), no. 2, article 43.

• Mean curvature flow with generic initial data. With O. Chodosh, K. Choi, F. Schulze. Invent. Math., vol. 237 (2024), 121-220.

• Ancient gradient flows of elliptic functionals and Morse index. With K. Choi. Am. J. Math., vol. 144 (2022), no. 2, 541-573.

• Minimal hypersurfaces with arbitrarily large area. With O. Chodosh. Int. Math. Res. Not., vol. 2021 (2021), no. 14, 10841-10847.

• Capacity, quasi-local mass, singular fill-ins. With P. Miao, L.-F. Tam. J. Reine Angew. Math., vol. 2020 (2020), no. 768, 55-92.

• Minimal surfaces and the Allen-Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates. With O. Chodosh. Ann. Math., vol. 191 (2020), no. 1, 213-328.

• Positive scalar curvature with skeleton singularities. With C. Li. Math. Ann., vol. 374 (2019), no. 1, 99-131. [Known Misprints]

• Allen-Cahn min-max on surfaces. J. Differ. Geom., vol. 117 (2021), no. 1, 93-135.

• Mean curvature deficit and a quasi-local mass. With P. Miao. Harvard University Center of Mathematical Sciences and Applications (CMSA) Series in Mathematics, vol. 1., Nonlinear analysis in geometry and applied mathematics, 99-107, Int. Press, Somerville, MA, 2017.

• Total mean curvature, scalar curvature, and a variational analog of Brown-York mass. With P. Miao. Commun. Math. Phys., vol. 352 (2017), no. 2, 703-718.

• On the Bartnik mass of apparent horizons. With R. Schoen. Class. Quantum Gravity, vol. 32 (2015), no. 20, 205002.  Appeared on IOPselect 2015, a special edition of journal articles, on the basis of substantial advances, a high degree of novelty, and/or impact on future research.

Reports

• Minimal surfaces and the Allen-Cahn equation on 3-manifolds. Proceedings of the 2024 International Congress of Basic Science, to appear.

• Improved generic regularity of minimizing hypersurfaces. MFO Reports, No. 29 (2023), to appear.

• Decomposing 4-manifolds with positive scalar curvature. MFO Reports, No. 28 (2022), 1593-1595.

• The p-widths of a surface. MFO Reports, No. 35 (2021), 1900-1902.

• Metrics with λ1(-Δ+kR) >= 0 and flexibility in the Riemannian Penrose Inequality. MFO Reports, No. 30 (2021), 1637-1640.

• Ancient gradient flows of elliptic functionals and Morse index. MFO Reports, No. 34 (2019), 2055-2058.

• Minimal surfaces and the Allen-Cahn equation on 3-manifolds. MFO Reports, No. 27 (2018), 1658-1661; also, No. 35 (2018), 2113-2116.

• The curvature on a black hole boundary. CQG+ Insight Piece.

Notes

• T. H. Colding and W. P. Minicozzi's blowup uniqueness and Łojasiewicz inequalities.

• Richard Schoen's lectures on minimal submanifolds. With D. Cheng, C. Li.

• Yi Wang's lectures on harmonic analysis and isoperimetric inequalities. With D. Cheng, O. Chodosh, N. Edelen, C. Henderson, P. Hintz.

• Richard Bamler's lectures on Ricci flow. With O. Chodosh. 

• Brian White's lectures on minimal surfaces. With O. Chodosh.

Theses

• Geometric variational problems in mathematical physics. Advisor: Richard Schoen. Doctoral thesis. [Known Misprints]

• Regularity of hypersurfaces in Rn+1 moving by mean curvature flow.  Advisor: Leon Simon. Undergraduate honors thesis.

The exposition follows Klaus Ecker's Regularity Theory for Mean Curvature Flow.

Academic Background

• Associate Professor, Rice University, now.

• Assistant Professor, Rice University.

• Assistant Professor, Brown University.

• CLE Moore Instructor, Massachusetts Institute of Technology.

• PhD, Stanford University.

• BSc, Stanford University.