# Antoine Song

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**Antoine Y. Song** (born 18 July 1992 in Paris) is a French[1] mathematician whose research concerns [differential geometry](/source/Differential_geometry) and [geometric analysis](/source/Geometric_analysis). He is a professor at [Caltech](/source/Caltech).[2] In 2018, he proved [Yau's conjecture](/source/Yau's_conjecture).

## Education

Antoine Song was a student at the [École Normale Supérieure de Paris](/source/%C3%89cole_normale_sup%C3%A9rieure_(Paris)) from 2012 to 2015. He obtained a bachelor's and a master's degree in mathematics from the [Université Pierre et Marie Curie](/source/Pierre_and_Marie_Curie_University) (Paris 6).[1] He obtained his Ph.D. from [Princeton University](/source/Princeton_University) in 2019 under the supervision of [Fernando Codá Marques](/source/Fernando_Cod%C3%A1_Marques).[3]

## Mathematical work

It is known that any closed surface possesses infinitely many [closed geodesics](/source/Closed_geodesic). The first problem in the minimal submanifolds section of [Yau](/source/Shing-Tung_Yau)'s list ([Yau's conjecture](/source/Yau's_conjecture)) asks whether any closed [three-manifold](/source/Three-manifold) has infinitely many closed smooth [immersed](/source/Immersion_(mathematics)#Immersed_plane_curves) minimal surfaces. At the time it was known from [Almgren–Pitts min-max theory](/source/Almgren%E2%80%93Pitts_min-max_theory) the existence of at least one minimal surface. Kei Irie, [Fernando Codá Marques](/source/Fernando_Cod%C3%A1_Marques), and [André Neves](/source/Andr%C3%A9_Neves) solved this problem in the [generic](/source/Generic_property) case [4] and later Song proved it in full generality.[5]

Together with Conghan Dong,[6] he proved a conjecture from 2001 by [G. Huisken](/source/Gerhard_Huisken) and [T. Ilmanen](/source/Tom_Ilmanen) on the mathematics of [general relativity](/source/General_relativity), about the [curvature](/source/Curvature) in spaces with very little [mass](/source/Mass).[7]

## Honours and awards

He was a [Clay Research Fellow](/source/Clay_Mathematics_Institute) (2019–2024).[8]

He is a [Sloan Fellow](/source/Sloan_Research_Fellowship).[9][10]

He delivered the 2021–2022 [Peccot Lectures](/source/Peccot_Lectures) (in 2022, due to the coronavirus pandemic).[11]

In 2024, he received the *Frontiers of Science Award*.[12]

## Selected publications

- Song, Antoine (2023). "Existence of infinitely many minimal hypersurfaces in closed manifolds". *[Annals of Mathematics](/source/Annals_of_Mathematics)*. **197** (3). Second Series. [arXiv:1806.08816](https://arxiv.org/abs/1806.08816). [doi:10.4007/annals.2023.197.3.1](https://doi.org/10.4007/annals.2023.197.3.1). MR 4564260. Zbl 1526.53013.
- Marques, Fernando C.; Neves, André; Song, Antoine (2019). "Equidistribution of minimal hypersurfaces for generic metrics". *[Inventiones Mathematicae](/source/Inventiones_Mathematicae)*. **216** (2): 421–443. [arXiv:1712.06238](https://arxiv.org/abs/1712.06238). [Bibcode:2019InMat.216..421M](https://ui.adsabs.harvard.edu/abs/2019InMat.216..421M). [doi:10.1007/s00222-018-00850-5](https://doi.org/10.1007/s00222-018-00850-5)[13]
- Dong, Conghan & Song, Antoine (2025). "Stability of Euclidean 3-space for the positive mass theorem". *Inventiones Mathematicae*. **239** (1): 287. [arXiv:2302.07414](https://arxiv.org/abs/2302.07414). [Bibcode:2025InMat.239..287D](https://ui.adsabs.harvard.edu/abs/2025InMat.239..287D). [doi:10.1007/s00222-024-01302-z](https://doi.org/10.1007/s00222-024-01302-z)[14]

## References

1. ["Song's CV"](https://web.math.princeton.edu/~aysong/moncv.pdf)

1. ["Antoine Song | the Division of Physics, Mathematics and Astronomy"](https://pma.caltech.edu/people/antoine-song)

1. Irie, Kei; Marques, Fernando; Neves, André (2018). "Density of minimal hypersurfaces for generic metrics". *[Annals of Mathematics](/source/Annals_of_Mathematics)*. **187** (3). [doi:10.4007/annals.2018.187.3.8](https://doi.org/10.4007/annals.2018.187.3.8)

1. Song 2023.

1. Dong & Song 2025.

1. Nadis, Steve (30 November 2023), ["A Century Later, New Math Smooths Out General Relativity"](https://www.quantamagazine.org/a-century-later-new-math-smooths-out-general-relativity-20231130/), *[Quanta Magazine](/source/Quanta_Magazine)*

1. ["Antoine Song | Clay Mathematics Institute"](https://www.claymath.org/people/antoine-song). *www.claymath.org*

1. ["Caltech Professors Win 2024 Sloan Fellowships"](https://www.caltech.edu/about/news/caltech-professors-win-2024-sloan-fellowships). 20 February 2024.

1. ["2024 Fellows | Alfred P. Sloan Foundation"](https://sloan.org/fellowships/2024-Fellows)

1. ["Cours Peccot 2021-2022 | CNRS Mathématiques"](https://www.insmi.cnrs.fr/fr/cnrsinfo/cours-peccot-2021-2022). 27 September 2021.

1. ["Twenty French mathematicians win Frontiers of Science 2024 prize (Beijing) | CNRS Mathématiques"](https://www.insmi.cnrs.fr/en/cnrsinfo/prix-frontiers-of-science-2024). *www.insmi.cnrs.fr*. July 14, 2024.

1. ["Math Duo Maps the Infinite Terrain of Minimal Surfaces"](https://www.quantamagazine.org/math-duo-maps-the-infinite-terrain-of-minimal-surfaces-20190312/). 12 March 2019.

1. ["A Century Later, New Math Smooths Out General Relativity"](https://www.quantamagazine.org/a-century-later-new-math-smooths-out-general-relativity-20231130/). 30 November 2023.

## External links

- [The Geometry of Minimal Surfaces: An Interview with Antoine Song (October 13, 2023)](https://www.caltech.edu/about/news/Geometry_of_Minimal_Surfaces)

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