Joel Spruck

Key takeaways

Joel Spruck 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 Spruck to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Joel Spruck from memory before moving on to harder problems.

Reference excerpt

Joel Spruck (born 1946) is a mathematician, J. J. Sylvester Professor of Mathematics at Johns Hopkins University, whose research concerns geometric analysis and elliptic partial differential equations. He obtained his PhD from Stanford University with the supervision of Robert S. Finn in 1971.

Mathematical contributions Spruck is well known in the field of elliptic partial differential equations for his series of papers "The Dirichlet problem for nonlinear second-order elliptic equations," written in collaboration with Luis Caffarelli, Joseph J. Kohn, and Louis Nirenberg. These papers were among the first to develop a general theory of second-order elliptic differential equations which are fully nonlinear, with a regularity theory that extends to the boundary. Caffarelli, Nirenberg & Spruck (1985) has been particularly influential in the field of geometric analysis since many geometric partial differential equations are amenable to its methods. With Basilis Gidas, Spruck studied positive solutions of subcritical second-order elliptic partial differential equations of Yamabe type. With Caffarelli, they studied the Yamabe equation on Euclidean space, proving a positive mass-style theorem on the asymptotic behavior of isolated singularities. In 1974, Spruck and David Hoffman extended a mean curvature-based Sobolev inequality of James H. Michael and Leon Simon to the setting of submanifolds of Riemannian manifolds. This has been useful for the study of many analytic problems in geometric settings, such as for Gerhard Huisken's study of mean curvature flow in Riemannian manifolds and for Richard Schoen and Shing-Tung Yau's study of the Jang equation in their resolution of the positive energy theorem in general relativity. In the late 80s, Stanley Osher and James Sethian developed the level-set method as a computational tool in numerical analysis. In collaboration with Lawrence Evans, Spruck pioneered the rigorous study of the level-set flow, as adapted to the mean curvature flow. The level-set approach to mean curvature flow is important in the technical ease with topological change that can occur along the flow. The same approach was independently developed by Yun Gang Chen, Yoshikazu Giga, and Shun'ichi Goto. The works of Evans–Spruck and Chen–Giga–Goto found significant application in Gerhard Huisken and Tom Ilmanen's solution of the Riemannian Penrose inequality of general relativity and differential geometry, where they adopted the level-set approach to the inverse mean curvature flow. In 1994 Spruck was an invited speaker at the International Congress of Mathematicians in Zurich.

Major publications Hoffman, David; Spruck, Joel. Sobolev and isoperimetric inequalities for Riemannian submanifolds. Comm. Pure Appl. Math. 27 (1974), 715–727. Gidas, B.; Spruck, J. A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differential Equations 6 (1981), no. 8, 883–901. Gidas, B.; Spruck, J. Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598. Caffarelli, L.; Nirenberg, L.; Spruck, J. The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation. Comm. Pure Appl. Math. 37 (1984), no. 3, 369–402. Caffarelli, L.; Kohn, J.J.; Nirenberg, L.; Spruck, J. The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations. Comm. Pure Appl. Math. 38 (1985), no. 2, 209–252. Caffarelli, L.; Nirenberg, L.; Spruck, J. The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian. Acta Math. 155 (1985), no. 3–4, 261–301. Caffarelli, Luis A.; Gidas, Basilis; Spruck, Joel. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297. Evans, L.C.; Spruck, J. Motion of level sets by mean curvature. I. J. Differential Geom. 33 (1991), no. 3, 635–681. Spruck, Joel; Yang, Yi Song. Topological solutions in the self-dual Chern-Simons theory: existence and approximation. Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), no. 1, 75–97.

Prizes Simons Fellowship (2012–2013) Fellow of the American Mathematical Society (2013 inauguration) Guggenheim Fellowship (1999–2000)

References

External links Joel Spruck publications indexed by Google Scholar

Worked examples

Example 1 — a first encounter with Joel Spruck

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

In research: Joel Spruck 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 Spruck 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 Spruck is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1946 births, American mathematicians, Fellows of the American Mathematical Society, so understanding it makes those chapters shorter.
In everyday life: Look for Joel Spruck 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.

How to study Joel Spruck in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Joel Spruck 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 Joel Spruck out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Joel Spruck in simple terms?

Joel Spruck (born 1946) is a mathematician, J. J.

Why does Joel Spruck 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 Spruck?

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 Spruck.