mathematics

Sheldon Katz

Sheldon Katz 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 Sheldon Katz rather than just read about it. In short: Sheldon H. Katz (19 December 1956, Brooklyn) is an American mathematician, specializing in algebraic geometry and its applications to string theory.

Key takeaways

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

Reference excerpt

Sheldon H. Katz (19 December 1956, Brooklyn) is an American mathematician, specializing in algebraic geometry and its applications to string theory.

Background and career In 1973 Katz won first prize in the U.S.A. Mathematical Olympiad. He received in 1976 his bachelor's degree from MIT and in 1980 his Ph.D. from Princeton University under Robert C. Gunning with thesis Deformations of Linear Systems, Divisors and Weierstrass Points on Curves. At the University of Utah, he was an instructor from 1980 to 1984. At the University of Oklahoma he was an assistant professor from 1984 to 1987. At Oklahoma State University, he became in 1987 an assistant professor, in 1989 an associate professor, in 1994 a full professor, in 1997 Southwestern Bell Professor, and in 1999 Regents Professor. Since 2001 he has been a professor at the University of Illinois, Urbana-Champaign, where he was chair of the department in 2006–2011. For the academic year 1982/83 he was a visiting scholar at the Institute for Advanced Study. He was a visiting professor at the Mittag-Leffler Institute (1997), at Duke University (1991/92) and at the University of Bayreuth (1989). His research on algebraic geometry and its applications to string theory (including mirror symmetry) and supersymmetry has been published in prestigious journals in mathematics and physics. In 2013 he was elected a Fellow of the American Mathematical Society.

Selected publications

Articles with Bruce Crauder: Crauder, Bruce; Katz, Sheldon (1989). "Cremona transformations with smooth irreducible fundamental locus". American Journal of Mathematics. 111 (2): 289–307. doi:10.2307/2374511. JSTOR 2374511. with Alberto Albano: Albano, Alberto; Katz, Sheldon (1991). "Lines on the Fermat quintic threefold and the infinitesimal generalized Hodge conjecture". Trans. Amer. Math. Soc. 324 (1): 353–368. doi:10.1090/s0002-9947-1991-1024767-6. MR 1024767. with David R. Morrison and M. Ronen Plesser: Katz, Sheldon; Morrison, David R.; Plesser, M.Ronen (1996). "Enhanced gauge symmetry in type 11 string theory". Nuclear Physics B. 477 (1): 105–140. arXiv:hep-th/9601108. Bibcode:1996NuPhB.477..105K. doi:10.1016/0550-3213(96)00331-8. S2CID 14482596. with Eric Sharpe: Katz, Sheldon; Sharpe, Eric (2003). "D-branes, open string vertex operators, and Ext groups". Adv. Theor. Math. Phys. 6 (6): 979–1030. arXiv:hep-th/0208104. Bibcode:2002hep.th....8104K. doi:10.4310/ATMP.2002.v6.n6.a1. S2CID 14199444. with Tony Pantev and E. Sharpe: Katz, Sheldon; Pantev, Tony; Sharpe, Eric (2003). "D-branes, orbifolds, and Ext groups". Nuclear Physics B. 673 (1): 263–300. arXiv:hep-th/0212218. Bibcode:2003NuPhB.673..263K. doi:10.1016/j.nuclphysb.2003.09.022. S2CID 17710799. with Andrei Caldararu and E. Sharpe: Caldararu, Andrei; Katz, Sheldon; Sharpe, Eric (2004). "D-branes, B fields, and Ext groups". Advances in Theoretical and Mathematical Physics. 7 (3): 381–404. arXiv:hep-th/0302099. doi:10.4310/atmp.2003.v7.n3.a1. S2CID 7443345. with Ron Donagi and E. Sharpe: Donagi, Ron; Katz, Sheldon; Sharpe, Eric (2005). "Spectra of D-branes with Higgs vevs". Advances in Theoretical and Mathematical Physics. 8 (5): 813–859. arXiv:hep-th/0309270. doi:10.4310/atmp.2004.v8.n5.a3. S2CID 229996. with D. Morrison, Sakura Schäfer-Nameki, and James Sully: Katz, Sheldon; Morrison, David R.; Schäfer-Nameki, Sakura; Sully, James (2011). "Tate's algorithm and F-theory". JHEP. 94 (8): 1108. arXiv:1106.3854. Bibcode:2011JHEP...08..094K. doi:10.1007/JHEP08(2011)094. S2CID 119184488. with Jinwon Choi and Albrecht Klemm: Choi, Jinwon; Katz, Sheldon; Klemm, Albrecht (2014). "The refined BPS index from stable pair invariants". Communications in Mathematical Physics. 328 (3): 903–954. arXiv:1210.4403. Bibcode:2014CMaPh.328..903C. doi:10.1007/s00220-014-1978-0. S2CID 119708881.

Books with Rahul Pandharipande, Cumrun Vafa, Ravi Vakil, Eric Zaslow, Kentaro Hori, Albrecht Klemm, Richard Thomas: Mirror Symmetry, Clay Mathematics Monographs, vol. 1, 2003 with David A. Cox: Mirror Symmetry and Algebraic Geometry. AMS. 1999. ISBN 9780821821275. Enumerative Geometry and String Theory. Student Mathematical Library. AMS. 2006. ISBN 9780821836873.

References

Worked examples

Example 1 — a first encounter with Sheldon Katz

Start with the simplest possible case. Write down what Sheldon Katz 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 Sheldon Katz 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 Sheldon Katz 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 Sheldon Katz

In research: Sheldon Katz 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 Sheldon Katz 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: Sheldon Katz 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 Sheldon Katz 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 Sheldon Katz in 20 minutes

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

Frequently asked questions

What is Sheldon Katz in simple terms?

Sheldon H. Katz (19 December 1956, Brooklyn) is an American mathematician, specializing in algebraic geometry and its applications to string theory.

Why does Sheldon Katz 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 Sheldon Katz?

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 Sheldon Katz.