Tom Ilmanen

{{Short description|American mathematician}}

{{Infobox person

| name = Tom Ilmanen

| birth_date = 1961

| nationality = American

| occupation = Mathematician

| known_for = Research in differential geometry, proof of Riemannian Penrose conjecture

| education = Ph.D. in Mathematics

| alma_mater = University of California, Berkeley

}}

Tom Ilmanen (born 1961) is an American mathematician specializing in differential geometry and the calculus of variations. He is a professor at ETH Zurich.{{cite web | title=Prof. Dr. Tom Ilmanen| website=ETH Zurich - Department of Mathematics | date=2020-05-11 | url=https://math.ethz.ch/research/geometric-analysis-pde/tom-ilmanen.html | access-date=2025-03-31}} He obtained his PhD in 1991 at the University of California, Berkeley with Lawrence Craig Evans as supervisor.{{MathGenealogy|id=31671}} Ilmanen and Gerhard Huisken used inverse mean curvature flow to prove{{cite journal | last=Huisken | first=Gerhard | last2=Ilmanen | first2=Tom | title=The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality | journal=Journal of Differential Geometry | volume=59 | issue=3 | date=2001-11-01 | issn=0022-040X | doi=10.4310/jdg/1090349447 | doi-access=free | url=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 | access-date=2025-03-31 | pages=353-437}} the Riemannian Penrose conjecture, which is the fifteenth problem in Yau's list of open problems,Differential Geometry: Partial Differential Equations on Manifolds. (1993). In R. Greene & S.-T. Yau (Eds.), Proceedings of Symposia in Pure Mathematics. American Mathematical Society. https://doi.org/10.1090/pspum/054.1 https://doi.org/10.1090/pspum/054.1 and was resolved at the same time in greater generality by Hubert Bray using alternative methods.Mars, M. (2009). "[https://doi.org/10.1088/0264-9381/26/19/193001 Present status of the Penrose inequality]". Classical and Quantum Gravity (Vol. 26, Issue 19, p. 193). IOP Publishing.

In their 2001 paper, 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.{{citation|url=https://www.quantamagazine.org/a-century-later-new-math-smooths-out-general-relativity-20231130/|title=A Century Later, New Math Smooths Out General Relativity|magazine=Quanta Magazine|date=30 November 2023|first=Steve|last=Nadis}}{{cite journal | last=Dong | first=Conghan | last2=Song | first2=Antoine | title=Stability of Euclidean 3-space for the positive mass theorem | journal=Inventiones mathematicae | volume=239 | issue=1 | date=2025 | issn=0020-9910 | doi=10.1007/s00222-024-01302-z | pages=287–319| arxiv=2302.07414 }}

In an influential preprint (Singularities of mean curvature flow of surfaces - 1995), Ilmanen conjectured:

{{Blockquote

|text=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. {{cite journal | last=Colding | first=Tobias | last2=Minicozzi | first2=William | title=Generic mean curvature flow I; generic singularities | journal=Annals of Mathematics | volume=175 | issue=2 | date=2012-03-01 | issn=0003-486X | doi=10.4007/annals.2012.175.2.7 | doi-access=free | pages=755–833 | url=http://annals.math.princeton.edu/wp-content/uploads/annals-v175-n2-p07-p.pdf | access-date=2025-04-01}}

}}

This has become known as the "multiplicity-one" conjecture. Richard Bamler and Bruce Kleiner proved the multiplicity-one conjecture in a 2023 preprint.{{cite web | last=Nadis | first=Steve | title=A New Proof Smooths Out the Math of Melting | website=Quanta Magazine | date=2025-03-31 | url=https://www.quantamagazine.org/a-new-proof-smooths-out-the-math-of-melting-20250331/ | access-date=2025-03-31}}{{cite web | title=On the Multiplicity-One Conjecture for Mean Curvature Flows of Surfaces | last=Bamler | first=Richard | last2=Kleiner | first2=Bruce | url=https://arxiv.org/pdf/2312.02106 | access-date=2025-03-31}}

Ilmanen received a Sloan Fellowship in 1996.{{Cite web|url=https://sloan.org/fellows-database|title=Fellows Database | Alfred P. Sloan Foundation|website=sloan.org}}

He wrote the research monograph Elliptic Regularization and Partial Regularity for Motion by Mean Curvature.{{cite book | last=Ilmanen | first=Tom | title=Elliptic Regularization and Partial Regularity for Motion by Mean Curvature | publisher=American Mathematical Soc. | publication-place=Providence, R.I | date=1994 | isbn=978-0-8218-2582-2 | page=}}