Biography:Tom Ilmanen
Tom Ilmanen | |
|---|---|
| Born | 1961 |
| Nationality | American |
| Education | Ph.D. in Mathematics |
| Alma mater | University of California, Berkeley |
| Occupation | Mathematician |
| Known for | Research in differential geometry, proof of Riemannian Penrose conjecture |
Tom Ilmanen (born 1961) is an American mathematician specializing in differential geometry and the calculus of variations. He is a professor at ETH Zurich.[1] He obtained his PhD in 1991 at the University of California, Berkeley with Lawrence Craig Evans as supervisor.[2] Ilmanen and Gerhard Huisken used inverse mean curvature flow to prove[3] the Riemannian Penrose conjecture, which is the fifteenth problem in Yau's list of open problems,[4] and was resolved at the same time in greater generality by Hubert Bray using alternative methods.[5]
In their 2001 paper,[3] Huisken and Ilmanen made a conjecture on the mathematics of general relativity, about the curvature in spaces with very little mass: as the mass of the space shrinks to zero, the curvature of the space also shrinks to zero. This was proved in 2023 by Conghan Dong and Antoine Song.[6][7]
In an influential preprint (Singularities of mean curvature flow of surfaces - 1995), Ilmanen conjectured:
For a smooth one-parameter family of closed embedded surfaces in Euclidean 3-space flowing by mean curvature, every tangent flow at the first singular time has multiplicity one. [8]
This has become known as the "multiplicity-one" conjecture. Richard Bamler and Bruce Kleiner proved the multiplicity-one conjecture in a 2023 preprint.[9][10]
Ilmanen received a Sloan Fellowship in 1996.[11]
He wrote the research monograph Elliptic Regularization and Partial Regularity for Motion by Mean Curvature.[12]
Selected publications
- Huisken, Gerhard, and Tom Ilmanen. "The inverse mean curvature flow and the Riemannian Penrose inequality." Journal of Differential Geometry 59.3 (2001): 353–437. DOI: 10.4310/jdg/1090349447
- Ilmanen, Tom. "Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature." Journal of Differential Geometry 38.2 (1993): 417–461.
- Feldman, Mikhail, Tom Ilmanen, and Dan Knopf. "Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons." Journal of Differential Geometry 65.2 (2003): 169–209.
References
- ↑ "Prof. Dr. Tom Ilmanen". 2020-05-11. https://math.ethz.ch/research/geometric-analysis-pde/tom-ilmanen.html.
- ↑ Tom Ilmanen at the Mathematics Genealogy Project
- ↑ 3.0 3.1 Huisken, Gerhard; Ilmanen, Tom (2001-11-01). "The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality". Journal of Differential Geometry 59 (3): 353–437. doi:10.4310/jdg/1090349447. ISSN 0022-040X. https://projecteuclid.org/journals/journal-of-differential-geometry/volume-59/issue-3/The-Inverse-Mean-Curvature-Flow-and-the-Riemannian-Penrose-Inequality/10.4310/jdg/1090349447.pdf. Retrieved 2025-03-31.
- ↑ Greene, Robert Everist; Yau, Shing-Tung (1993). Differential Geometry: Partial Differential Equations on Manifolds. Proceedings of Symposia in Pure Mathematics. 54.1. doi:10.1090/pspum/054.1. ISBN 978-0-8218-1494-9.
- ↑ Mars, Marc (2009). "Present status of the Penrose inequality". Classical and Quantum Gravity 26 (19). doi:10.1088/0264-9381/26/19/193001.
- ↑ Nadis, Steve (30 November 2023), "A Century Later, New Math Smooths Out General Relativity", Quanta Magazine, https://www.quantamagazine.org/a-century-later-new-math-smooths-out-general-relativity-20231130/
- ↑ Dong, Conghan; Song, Antoine (2025). "Stability of Euclidean 3-space for the positive mass theorem". Inventiones Mathematicae 239 (1): 287–319. doi:10.1007/s00222-024-01302-z. ISSN 0020-9910. Bibcode: 2025InMat.239..287D.
- ↑ Colding, Tobias; Minicozzi, William (2012-03-01). "Generic mean curvature flow I; generic singularities". Annals of Mathematics 175 (2): 755–833. doi:10.4007/annals.2012.175.2.7. ISSN 0003-486X. http://annals.math.princeton.edu/wp-content/uploads/annals-v175-n2-p07-p.pdf. Retrieved 2025-04-01.
- ↑ Nadis, Steve (2025-03-31). "A New Proof Smooths Out the Math of Melting". https://www.quantamagazine.org/a-new-proof-smooths-out-the-math-of-melting-20250331/.
- ↑ Bamler, Richard; Kleiner, Bruce (2023). "On the Multiplicity-One Conjecture for Mean Curvature Flows of Surfaces". arXiv:2312.02106 [math.DG].
- ↑ "Fellows Database | Alfred P. Sloan Foundation". https://sloan.org/fellows-database.
- ↑ Ilmanen, Tom (1994). Elliptic Regularization and Partial Regularity for Motion by Mean Curvature. Providence, R.I: American Mathematical Soc.. ISBN 978-0-8218-2582-2.
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