# Helmut Hofer

**Helmut Hofer** is a German and American mathematician who works on symplectic geometry, dynamical systems, and partial differential equations, and who is described by the [Institute for Advanced Study](https://www.edgepedia.org/institute-for-advanced-study) as one of the founders of the area of symplectic topology.<sup>[1](https://www.ias.edu/scholars/hofer)</sup> His fundamental contributions to the field have led to a new area of mathematics known as Hofer geometry.<sup>[1](https://www.ias.edu/scholars/hofer)</sup> The American Academy of Arts and Sciences credits him with fundamental contributions to [Hamiltonian mechanics](https://www.edgepedia.org/hamiltonian-mechanics), symplectic geometry, and related analysis and geometry.<sup>[2](https://www.amacad.org/person/helmut-h-hofer)</sup> According to his Academia Europaea member record, he holds German and US citizenship,<sup>[3](https://www.ae-info.org/ae/User/Hofer_Helmut?skin=raw)</sup> has been a member of the National Academy of Sciences since 2008,<sup>[4](https://www.ias.edu/sites/default/files/2025-04-17_hofer_CV%20Publicationlist.pdf)</sup> and, per the Institute for Advanced Study, is the Hermann Weyl Professor, Emeritus, at the Institute for Advanced Study (IAS) in Princeton.<sup>[1](https://www.ias.edu/scholars/hofer)</sup> 

| Key facts | |
|---|---|
| Field | Symplectic geometry, symplectic topology, Hamiltonian dynamics, partial differential equations<sup>[1](https://www.ias.edu/scholars/hofer)</sup> |
| Signature work | The monograph *Symplectic Invariants and Hamiltonian Dynamics* (Birkhäuser) and the 1990 construction of a bi-invariant metric on symplectic diffeomorphism groups<sup>[5](https://link.springer.com/book/10.1007/978-3-0348-0104-1)</sup><sup> • </sup><sup>[6](https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics/article/abs/on-the-topological-properties-of-symplectic-maps/8247940DE12DBF9E582C426924E608EE)</sup> |
| Career | Zürich and Bath (1979–85), Rutgers (1985–89), Bochum (1989–93), ETH Zürich (1993–97), Courant Institute (1997–2009), Institute for Advanced Study (2009–26, emeritus 7/2026)<sup>[4](https://www.ias.edu/sites/default/files/2025-04-17_hofer_CV%20Publicationlist.pdf)</sup><sup> • </sup><sup>[1](https://www.ias.edu/scholars/hofer)</sup> |
| Training | Diplom 1979 and Ph.D. 1981, Universität Zürich; dissertation on a variational approach to resonance problems, written under Peter Hess<sup>[4](https://www.ias.edu/sites/default/files/2025-04-17_hofer_CV%20Publicationlist.pdf)</sup><sup> • </sup><sup>[7](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=33913)</sup> |
| Honors | Ostrowski Prize 1999; NAS member 2008; Heinz Hopf Prize 2013; American Academy of Arts and Sciences 2020<sup>[4](https://www.ias.edu/sites/default/files/2025-04-17_hofer_CV%20Publicationlist.pdf)</sup> |
| Editorial role | Managing editor of *Inventiones Mathematicae* since 2008, editorial board from 2007<sup>[1](https://www.ias.edu/scholars/hofer)</sup> |

## Career

Hofer attended the Fichte-Gymnasium Krefeld from 1966 to 1974 and earned a Diplom in [Mathematics](https://www.edgepedia.org/mathematics) at the University of Zürich in 1979, followed by a Ph.D. there in 1981.<sup>[4](https://www.ias.edu/sites/default/files/2025-04-17_hofer_CV%20Publicationlist.pdf)</sup> His dissertation, *A Variational Approach to a Class of Resonance Problems with Application to a Wave Equation Problem*, was written under Peter Hess.<sup>[7](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=33913)</sup>

His early appointments were Assistant in Mathematics at Zürich (1979–81), Oberassistent there (1981–82; Academia Europaea records the post as 1981–1983<sup>[3](https://www.ae-info.org/ae/User/Hofer_Helmut?skin=raw)</sup>), and Lecturer in Pure Mathematics at the [University of Bath](https://www.edgepedia.org/university-of-bath) (1983–85).<sup>[4](https://www.ias.edu/sites/default/files/2025-04-17_hofer_CV%20Publicationlist.pdf)</sup> He then moved to [Rutgers University](https://www.edgepedia.org/rutgers-university) as Assistant Professor (1985–87), Associate Professor (1987–88), and Professor (1988–89).<sup>[4](https://www.ias.edu/sites/default/files/2025-04-17_hofer_CV%20Publicationlist.pdf)</sup>

In 1989 he took a C4-Professorship at Ruhr-Universität Bochum, serving as Dekan in 1992–93, and in 1993 became Professor at ETH Zürich.<sup>[4](https://www.ias.edu/sites/default/files/2025-04-17_hofer_CV%20Publicationlist.pdf)</sup> From 1997 to 2006 he was Professor at the Courant Institute of New York University, and Silver Professor there from 2006 to 2009.<sup>[4](https://www.ias.edu/sites/default/files/2025-04-17_hofer_CV%20Publicationlist.pdf)</sup> He joined the IAS School of Mathematics as Professor on July 1, 2009, and became Hermann Weyl Professor there in 2019.<sup>[4](https://www.ias.edu/sites/default/files/2025-04-17_hofer_CV%20Publicationlist.pdf)</sup> 

## Representative work

**Symplectic capacities.** The Hofer–Zehnder capacity of an open subset of a symplectic manifold measures the size of the set in terms of the periodic orbits of the autonomous Hamiltonian flows it supports; when such a capacity can be shown to be finite, the almost existence theorem follows, a result with broad consequences for Hamiltonian flows.<sup>[8](https://msp.org/gt/2005/9-4/gt-v9-n4-p01-p.pdf)</sup> These invariants connect rigidity phenomena in symplectic topology with global periodic phenomena in Hamiltonian systems, and they are the main theme of his monograph *Symplectic Invariants and Hamiltonian Dynamics* (Birkhäuser; reissued by Springer), which grew out of lecture courses given in 1991 and at the 1992 Borel Seminar in Bern and whose later chapters treat the Arnold conjecture, [Floer homology](https://www.edgepedia.org/floer-homology), and symplectic homology.<sup>[5](https://link.springer.com/book/10.1007/978-3-0348-0104-1)</sup><sup> • </sup><sup>[9](https://unina2.on-line.it/sebina/repository/catalogazione/documenti/Symplectic%20Invariants%20and%20hamiltonian%20dynamics.pdf)</sup>

**The Hofer metric.** In a 1990 paper in the *Proceedings of the Royal Society of Edinburgh* Hofer constructed a metric for symplectic diffeomorphism groups and proved generalized symplectic fixed point theorems.<sup>[6](https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics/article/abs/on-the-topological-properties-of-symplectic-maps/8247940DE12DBF9E582C426924E608EE)</sup> The metric is bi-invariant on the group of compactly supported symplectic diffeomorphisms of \(\mathbb{R}^{2n}\) and is defined through the oscillations of generating Hamiltonians; the associated displacement energy measures the distance between the identity map and the diffeomorphisms that displace a given set from itself.<sup>[10](https://doi.org/10.1007/978-3-0348-8540-9_5)</sup> This bi-invariant metric on the group of Hamiltonian diffeomorphisms gave rise to the field known as Hofer geometry.<sup>[2](https://www.amacad.org/person/helmut-h-hofer)</sup>

**Floer homology and the Weinstein conjecture.** Together with Andreas Floer he published joint papers, one treating coherent orientations for periodic orbit problems (Mathematische Zeitschrift, 1993) and another dealing with symplectic homology of open sets in \(C^n\) (Mathematische Zeitschrift, 1994), according to Hofer's Curriculum Vitae and Publication List.<sup>[4](https://www.ias.edu/sites/default/files/2025-04-17_hofer_CV%20Publicationlist.pdf)</sup> Since then, Floer homology has been applied to the study of refined Hofer–Zehnder capacities near closed symplectic submanifolds, which partially recovers existence results of Arnold for charged-particle Hamiltonian flows on a torus.<sup>[8](https://msp.org/gt/2005/9-4/gt-v9-n4-p01-p.pdf)</sup> Hofer's introduction of holomorphic curves into contact geometry led to the solution of fundamental cases of the Weinstein conjecture on periodic orbits; a 2024 survey records his proof of the conjecture for the three-sphere and for overtwisted contact three-manifolds as the breakthrough establishing a relation between holomorphic curves and closed Reeb orbits.<sup>[2](https://www.amacad.org/person/helmut-h-hofer)</sup><sup> • </sup><sup>[11](https://arxiv.org/html/2410.19936)</sup>

**Symplectic Field Theory, finite energy foliations, polyfolds, feral curves.** The 2000 paper *Introduction to Symplectic Field Theory* sketched a theory providing an approach to [Gromov–Witten invariants](https://www.edgepedia.org/gromov-witten-invariants) of symplectic manifolds and, at the same time, a rich source of new invariants of contact manifolds and their Legendrian submanifolds.<sup>[12](https://ar5iv.labs.arxiv.org/html/math/0010059)</sup> A 2024 survey describes SFT as a highly ambitious project that first appeared in crystallized form around 2000, with its foundational compactness paper a five-author work.<sup>[11](https://arxiv.org/html/2410.19936)</sup> With Wysocki and Zehnder he introduced the theory of finite energy foliations and, later, Polyfold Theory, intended to give rigorous foundations for constructing Floer-like theories; with Fish he introduced the theory of feral curves.<sup>[2](https://www.amacad.org/person/helmut-h-hofer)</sup> His papers with Wysocki and Zehnder include the [Annals of Mathematics](https://www.edgepedia.org/annals-of-mathematics) study of finite energy foliations of tight three-spheres and Hamiltonian dynamics (2003), the 2017 Memoirs of the AMS volume on applications of polyfold theory to Gromov–Witten theory, and the book *Polyfold and Fredholm Theory* (Springer, 2021), which runs to over 1,000 pages.<sup>[4](https://www.ias.edu/sites/default/files/2025-04-17_hofer_CV%20Publicationlist.pdf)</sup> 

## Honors and recognition

Hofer received an Alfred P. Sloan Fellowship (1987–89), the 1999 Ostrowski Prize, and election to the National Academy of Sciences in 2008.<sup>[4](https://www.ias.edu/sites/default/files/2025-04-17_hofer_CV%20Publicationlist.pdf)</sup> According to his curriculum vitae, ETH Zürich awarded him the Heinz Hopf Prize in 2013.<sup>[4](https://www.ias.edu/sites/default/files/2025-04-17_hofer_CV%20Publicationlist.pdf)</sup> He was an invited speaker at the International Congress of Mathematicians in Kyoto in 1990.<sup>[4](https://www.ias.edu/sites/default/files/2025-04-17_hofer_CV%20Publicationlist.pdf)</sup>

## At the Institute for Advanced Study

Hofer first came to the IAS as a Member in 1987, with later visits in 2001–02 and 2005, before joining the faculty in July 2009.<sup>[1](https://www.ias.edu/scholars/hofer)</sup> He served as faculty from 7/2009 to 6/2026 and has been listed as Emeritus in the School of Mathematics since 7/2026.<sup>[1](https://www.ias.edu/scholars/hofer)</sup> The Mathematics Genealogy Project records 22 doctoral students and 76 descendants, including Kai Cieliebak and Matthias Schwarz (ETH Zürich, 1996), Chris Wendl (NYU, 2005), and Umberto Hryniewicz (NYU, 2008).<sup>[7](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=33913)</sup>

## Recent developments

In March 2023 the *Annals of Mathematics* published the 206-page paper *Feral pseudoholomorphic curves and minimal sets*, following the 2022 paper *Almost Existence from the Feral Curve Perspective and Some Questions* in *Ergodic Theory and Dynamical Systems*.<sup>[4](https://www.ias.edu/sites/default/files/2025-04-17_hofer_CV%20Publicationlist.pdf)</sup> His CV was updated in April 2025,<sup>[4](https://www.ias.edu/sites/default/files/2025-04-17_hofer_CV%20Publicationlist.pdf)</sup> he gave the Gauss Lecture of the German Mathematical Society in October 2025,<sup>[4](https://www.ias.edu/sites/default/files/2025-04-17_hofer_CV%20Publicationlist.pdf)</sup> and he moved to emeritus status at the IAS in July 2026.<sup>[1](https://www.ias.edu/scholars/hofer)</sup>

## References

1. [Helmut Hofer | Scholars | Institute for Advanced Study](https://www.ias.edu/scholars/hofer)
2. [Helmut H. Hofer | American Academy of Arts and Sciences](https://www.amacad.org/person/helmut-h-hofer)
3. [Helmut Hofer – Academia Europaea member record](https://www.ae-info.org/ae/User/Hofer_Helmut?skin=raw)
4. [Curriculum Vitae and Publication List, Helmut Hofer (IAS, updated April 2025)](https://www.ias.edu/sites/default/files/2025-04-17_hofer_CV%20Publicationlist.pdf)
5. [Symplectic Invariants and Hamiltonian Dynamics (Springer book page)](https://link.springer.com/book/10.1007/978-3-0348-0104-1)
6. [On the topological properties of symplectic maps (Proc. Royal Soc. Edinburgh A, 1990)](https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics/article/abs/on-the-topological-properties-of-symplectic-maps/8247940DE12DBF9E582C426924E608EE)
7. [Helmut Hofer - The Mathematics Genealogy Project](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=33913)
8. [Squeezing in Floer theory and refined Hofer–Zehnder capacities (Geometry & Topology)](https://msp.org/gt/2005/9-4/gt-v9-n4-p01-p.pdf)
9. [Symplectic Invariants and Hamiltonian Dynamics (full text)](https://unina2.on-line.it/sebina/repository/catalogazione/documenti/Symplectic%20Invariants%20and%20hamiltonian%20dynamics.pdf)
10. [Geometry of compactly supported symplectic mappings in R^2n (Springer chapter)](https://doi.org/10.1007/978-3-0348-8540-9_5)
11. [Symplectic field theory: an overview (arXiv survey, 2024)](https://arxiv.org/html/2410.19936)
12. [Introduction to Symplectic Field Theory (arXiv mirror)](https://ar5iv.labs.arxiv.org/html/math/0010059)

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*Topic: Encyclopedia › Physical world and mathematics › Physical and mathematical scientists › Mathematicians and statisticians › Topologists and geometers*

*Initially written Sep 21, 2026 · Reviewed: Sep 25, 2026 · Edited: Sep 25, 2026; Oct 11, 2026; Oct 15, 2026 · Last review: Sep 25, 2026*

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