Welcome to Keaton Naff's Homepage

About me: 

I completed my Ph.D. in mathematics at Columbia University under the supervision of Simon Brendle. My research interests are in Differential Geometry and PDEs; in particular I am interested in Ricci flow, mean curvature flow, and minimal surfaces. Previously, I was an NSF postdoctoral fellow at MIT under the supervision of Tobias Holck Colding. Currently, I am a postdoctoral research associate at Lehigh University. 

Education: 

• 2016 - 2021, Ph.D. in Mathematics, Columbia University, New York, USA.

• 2012 - 2016, B.A. and M.A. in Mathematics, University of Pennsylvania, Philadelphia, USA.

Publications and preprints:

• Linear stability of the blowdown Ricci shrinker in 4D, preprint 2025 (joint with T. Ozuch). [arXiv:2509.05470]
  ➞ Linear stability of the blowdown Ricci shrinker in 4D: Supplementary Code. [GitHub]

• Planarity and convexity for pinched ancient solutions of mean curvature flow, preprint 2025 (joint with T.-K. Lee, J. Zhu). [arxiv2504.17922]

• Half-space intersection properties for minimal hypersurfaces, J. Geom. Anal. 35 (2025)  (joint with J. J. Zhu). [journal, arXiv:2401.09669]

• Moving monotonicity formulae for minimal submanifolds in constant curvature, preprint 2022 (joint with J. J. Zhu). [arXiv:2210.03263]

• The prescribed point area estimate for minimal submanifolds in constant curvature, to appear Amer. J. Math.  (joint with J. J. Zhu). [arXiv:2206.08302]

• Canonical metrics and ambiKähler structures on 4-manifolds with U(2) symmetry, Cubo 27 (2025), 135-163. (joint with B. Weber). [journal, arxiv:2111.11150]

• A local noncollapsing estimate for mean curvature flow, Amer. J. Math. 146 (2024), 1463-1468. (joint with S. Brendle). [journal, arXiv:2103.15641]

• Uniqueness of compact ancient solutions to the higher dimensional Ricci flow, J. Reine Angew. Math. 2023 (2023), 85-138. (joint with S. Brendle, P. Daskalopoulos, N. Šešum). [journal, arXiv:2102.07180]

• Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions, Geom. Topol. 27 (2023), 153-226. (joint with S. Brendle). [journal, arXiv:2005.05830]

• A canonical neighborhood theorem for the mean curvature flow in higher codimension, Int. Math. Res. Not. IMRN 2022 (2022),  rnac163. [journal, arXiv:2004:08060]

• Singularity models of pinched solutions of mean curvature flow in higher codimension, J. Reine Angew. Math. 2023 (2023), 101-132. [journal, arXiv:1910.03968]

• A planarity estimate for pinched solutions of mean curvature flow, Duke Math. J. 171 (2022), 443-482. [journal, arXiv:1906.08184]

• Ricci shrinking solitons with positive isotropic curvature, preprint 2019. [arXiv:1905.10305]

Teaching: 

Lehigh Univeristy

• Fall 2025: Linear Algebra and ODEs. Instructor. 

• Spring 2025: Linear Algebra and ODEs. Instructor. 

• Fall 2024: Calculus. Instructor. 

Massachusetts Institute of Technology

• Spring 2024: Real Analysis. Instructor. 

• Fall 2023: Geometry of Manifolds (1st-year graduate course). Instructor. 

• Spring 2023: Differential Equations. Teaching assistant. Taught by Prof. Andrew Lawrie. 

• Fall 2022: Curves and Surfaces. Instructor. 

Columbia University

• Spring 2021: Partial Differential Equations. Teaching assistant. Taught by Florian Johne. 

• Fall 2020: Ordinary Differential Equations. Teaching assistant. Taught by Prof. Simon Brendle.

• Summer 2020: Calculus IV. Instructor.

• Spring 2020: Graduate Modern Geometry II. Teaching assistant. Taught by Prof. Simon Brendle.

• Fall 2019: Graduate Modern Geometry I. Teaching assistant. Taught by Prof. Chiu-Chu Melissa Liu.

• Summer 2019: Calculus IV. Instructor.

• Fall 2018: Undergraduate Differential Geometry. Teaching assistant. Taught by Prof. Richard Hamilton.

• Summer 2018: Undergraduate Ordinary Differential Equations. Teaching assistant. Taught by Karsten Gimre.

• Spring 2018: Undergraduate Partial Differential Equations. Teaching assistant. Taught by Prof. Ovidiu Savin.

• Fall 2017: Undergraduate Differential Geometry. Teaching assistant. Taught by Prof. Richard Hamilton.