mathematics

Helmut Hofer

Helmut Hofer is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Helmut Hofer rather than just read about it. In short: Helmut Hermann W. Hofer (born February 28, 1956) is a German-American mathematician, one of the founders of the area of symplectic topology.

Key takeaways

Helmut Hofer belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Helmut Hofer to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Helmut Hofer from memory before moving on to harder problems.

Reference excerpt

Helmut Hermann W. Hofer (born February 28, 1956) is a German-American mathematician, one of the founders of the area of symplectic topology. He is a member of the National Academy of Sciences, and the recipient of the 1999 Ostrowski Prize and the 2013 Heinz Hopf Prize. Since 2009, he has been a faculty member at the Institute for Advanced Study in Princeton, New Jersey. He currently works on symplectic geometry, dynamical systems, and partial differential equations. His contributions to the field include Hofer geometry. Hofer was elected to the American Academy of Arts and Sciences in 2020. He was an invited speaker at the International Congress of Mathematicians (ICM) in 1990 in Kyoto and a plenary speaker at the ICM in 1998 in Berlin. He is currently an editor of Annals of Mathematics.

Selected publications Ekeland, Ivar; Hofer, Helmut (1985). "Periodic solutions with prescribed minimal period for convex autonomous Hamiltonian systems". Inventiones Mathematicae. 81 (1): 155–188. Bibcode:1985InMat..81..155E. doi:10.1007/BF01388776. MR 0796195. S2CID 120891372. Hofer, Helmut; Zehnder, Eduard (1987). "Periodic solutions on hypersurfaces and a result by C. Viterbo". Inventiones Mathematicae. 90 (1): 1–9. Bibcode:1987InMat..90....1H. doi:10.1007/BF01389030. MR 0906578. S2CID 121157240. Ekeland, Ivar; Hofer, Helmut (1989). "Symplectic topology and Hamiltonian dynamics". Mathematische Zeitschrift. 200 (3): 355–378. doi:10.1007/BF01215653. MR 0978597. S2CID 123600347. Hofer, Helmut (1990). "On the topological properties of symplectic maps". Proceedings of the Royal Society of Edinburgh, Section A. 115 (1–2): 25–38. doi:10.1017/S0308210500024549. MR 1059642. S2CID 121426158. Hofer, Helmut (1993). "Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three". Inventiones Mathematicae. 114 (3): 515–563. Bibcode:1993InMat.114..515H. doi:10.1007/BF01232679. MR 1244912. S2CID 123618375. Hofer, Helmut; Zehnder, Eduard (2011). Symplectic invariants and Hamiltonian dynamics. Modern Birkhäuser Classics. Basel: Birkhäuser Verlag. doi:10.1007/978-3-0348-0104-1. ISBN 978-3-0348-0103-4. MR 2797558. Hofer, Helmut; Wysocki, Krzysztof; Zehnder, Eduard (1998). "The dynamics on three-dimensional strictly convex energy surfaces". Annals of Mathematics. 148 (1): 197–289. doi:10.2307/120994. JSTOR 120994. MR 1652928. Hofer, Helmut; Wysocki, Krzysztof; Zehnder, Eduard (2003). "Finite energy foliations of tight three-spheres and Hamiltonian dynamics". Annals of Mathematics. 157 (1): 125–255. doi:10.4007/annals.2003.157.125. MR 1954266.

Notes

External links Helmut Hofer at the Mathematics Genealogy Project Oberwolfach photos of Helmut Hofer

Illustrations

Worked examples

Example 1 — a first encounter with Helmut Hofer

Start with the simplest possible case. Write down what Helmut Hofer claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Helmut Hofer before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Helmut Hofer ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Helmut Hofer

In research: Helmut Hofer appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Helmut Hofer in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Helmut Hofer is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1956 births, 20th-century German mathematicians, 21st-century German mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Helmut Hofer outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Helmut Hofer in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Helmut Hofer means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Helmut Hofer out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Helmut Hofer in simple terms?

Helmut Hermann W. Hofer (born February 28, 1956) is a German-American mathematician, one of the founders of the area of symplectic topology.

Why does Helmut Hofer matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Helmut Hofer?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Helmut Hofer.