mathematics

Joel Hass

Joel Hass 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 Joel Hass rather than just read about it. In short: Joel Hass (born January 3, 1956) is an American mathematician and professor of mathematics at the University of California, Davis. His research focuses on low-dimensional geometry and topology, including 3-manifolds, minimal surfaces, knot theory, and computational complexity questions in topology.

Key takeaways

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

Reference excerpt

Joel Hass (born January 3, 1956) is an American mathematician and professor of mathematics at the University of California, Davis. His research focuses on low-dimensional geometry and topology, including 3-manifolds, minimal surfaces, knot theory, and computational complexity questions in topology.

Education and career Hass was born in Tel Aviv on January 3, 1956, to Julian and Aliza Hass, and attended public schools in Queens, New York, after his family emigrated to the United States. He spent parts of his childhood in England, and later entered Hunter College before transferring to Columbia University. Hass received a B.A. from Columbia University in 1976, an M.A. from the University of California, Berkeley in 1978, and a Ph.D. from Berkeley in 1981. His doctoral thesis, Minimal surfaces in low dimensional manifolds, was supervised by Robion Kirby. After completing his doctorate, Hass held postdoctoral and visiting positions at the Hebrew University of Jerusalem, the University of Michigan, the Mathematical Sciences Research Institute, and the University of California, Berkeley. He joined the faculty of the University of California, Davis in 1988, became professor in 1994, and served as chair of the UC Davis Department of Mathematics from 2010 to 2014. Hass was a member of the Institute for Advanced Study in 1990-1991, 2000-2001, and 2015-2016, and a research professor at MSRI in 2006. He also held visiting positions at the Technion, the University of Melbourne, and the Hebrew University of Jerusalem. From 2017 to 2020 he directed the UC Davis branch of COSMOS, a California summer program for high school students in mathematics and science. In 2021 he was one of the founders of the Association for Mathematical Research and served as its first president.

Research contributions Hass's work centers on low-dimensional topology and geometry. With Michael Freedman and Peter Scott, he proved foundational results on least-area incompressible surfaces in 3-manifolds. Hass also constructed examples of bounded 3-manifolds admitting negatively curved metrics with concave boundary, a result highlighted in Robion Kirby's biography as an unexpected example in 3-dimensional geometry. With Roger Schlafly, Hass proved the equal-volume case of the double bubble conjecture, showing that the standard double bubble minimizes surface area among enclosures of two equal volumes in three-dimensional Euclidean space. Hass has also worked on algorithmic problems in knot theory and 3-manifold topology. With Jeffrey Lagarias and Nicholas Pippenger, he proved that the unknotting problem is in NP. With Lagarias, he gave an exponential upper bound on the number of Reidemeister moves needed to transform a diagram of the unknot into a standard circle. With Ian Agol and William Thurston, he proved that knot genus in 3-manifolds is NP-complete. His later work includes applications of geometry and topology to biological shape analysis, including work with Patrice Koehl on conformal methods for comparing genus-zero surfaces and shapes arising from bones, proteins, and teeth.

Awards and honors National Science Foundation Postdoctoral Fellow, 1984-1986 Rothschild Fellow, 1987-1988 Alfred P. Sloan Research Fellow, 1989-1991 Fellow of the American Mathematical Society, class of 2013

Selected publications

Research papers Freedman, Michael; Hass, Joel; Scott, Peter (1983). "Least area incompressible surfaces in 3-manifolds". Inventiones Mathematicae. 71 (3): 609–642. doi:10.1007/BF02095997. hdl:2027.42/46610. Hass, Joel (1994). "Bounded 3-manifolds admit negatively curved metrics with concave boundary". Journal of Differential Geometry. 40 (3): 449–459. Hass, Joel; Lagarias, Jeffrey C.; Pippenger, Nicholas (1999). "The computational complexity of knot and link problems". Journal of the ACM. 46 (2): 185–211. arXiv:math/9807016. doi:10.1145/301970.301971. Hass, Joel; Schlafly, Roger (2000). "Double bubbles minimize". Annals of Mathematics. Second Series. 151 (2): 459–515. arXiv:math/0003157. doi:10.2307/121042. JSTOR 121042. Hass, Joel; Lagarias, Jeffrey C. (2001). "The number of Reidemeister moves needed for unknotting". Journal of the American Mathematical Society. 14 (2): 399–428. arXiv:math/9807012. doi:10.1090/S0894-0347-01-00358-7. Agol, Ian; Hass, Joel; Thurston, William P. (2006). "The computational complexity of knot genus and spanning area". Transactions of the American Mathematical Society. 358 (9): 3821–3850. arXiv:math/0205057. doi:10.1090/S0002-9947-05-03919-X. Hass, Joel; Koehl, Patrice (2017). "Comparing shapes of genus-zero surfaces". Journal of Applied and Computational Topology. 1 (1): 57–87. doi:10.1007/s41468-017-0004-y.

Books Adams, Colin; Hass, Joel; Thompson, Abigail (1998). How to Ace Calculus: The Streetwise Guide. New York: W. H. Freeman. ISBN 978-0716731603. Adams, Colin; Hass, Joel; Thompson, Abigail (2001). How to Ace the Rest of Calculus: The Streetwise Guide. New York: W. H. Freeman. ISBN 978-0716741749. Thomas, George B.; Weir, Maurice D.; Hass, Joel (2014). Thomas' Calculus (13th ed.). Pearson.

References

External links Official website Joel Hass at the Mathematics Genealogy Project Google Scholar profile

Illustrations

Worked examples

Example 1 — a first encounter with Joel Hass

Start with the simplest possible case. Write down what Joel Hass 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 Joel Hass 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 Joel Hass 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 Joel Hass

In research: Joel Hass 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 Joel Hass 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: Joel Hass is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1956 births, 20th-century American mathematicians, 21st-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Joel Hass 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.

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How to study Joel Hass in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Joel Hass means in your own words.
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Frequently asked questions

What is Joel Hass in simple terms?

Why does Joel Hass 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 Joel Hass?

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 Joel Hass.