mathematics

Lawrence Ein

Lawrence Ein 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 Lawrence Ein rather than just read about it. In short: Lawrence Man Hou Ein (born 18 November 1955) is a mathematician who works in algebraic geometry. Education and career Lawrence Ein received in 1976 his bachelor's degree from UCLA and in 1981 his Ph.D. from the University of California, Berkeley under the direction of Robin Hartshorne with thesis Stable vector bundles on projective spaces in char p > 0 (which was published in Mathematische Annalen).

Key takeaways

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

Reference excerpt

Lawrence Man Hou Ein (born 18 November 1955) is a mathematician who works in algebraic geometry.

Education and career Lawrence Ein received in 1976 his bachelor's degree from UCLA and in 1981 his Ph.D. from the University of California, Berkeley under the direction of Robin Hartshorne with thesis Stable vector bundles on projective spaces in char p > 0 (which was published in Mathematische Annalen). For the academic year 1981–1982 he was an American Mathematical Society Postdoctoral Research Fellow at the Institute for Advanced Study. From 1982 to 1984 he was a C.L.E. Moore Instructor at MIT. He became from 1984 to 1987 an assistant professor, from 1987 to 1989 an associate professor, and from 1989 to the present a full professor at the University of Illinois at Chicago. He has been a visiting professor at the University of Hong Kong, UCLA, the University of Michigan, Harvard University, the University of Nancy, and the MSRI in Berkeley. Ein has served on the editorial boards of the Transactions and Memoirs of the American Mathematical Society from 1995 to 2003, Communications in Algebra from 1995 to the present, Geometriae Dedicata from 2004 to 2013, and Serdica Mathematical Journal from 2004 to the present. Ein was a Sloan Fellow from 1986 to 1989. He was an Invited Speaker of the ICM in 2006 in Madrid. He was elected a Fellow of the American Mathematical Society in 2012.

Selected publications Ein, Lawrence (1986). "Hilbert schemes of smooth space curves". Annales Scientifiques de l'École Normale Supérieure. 19 (4): 469–478. doi:10.24033/asens.1513. with Aaron Bertram and Robert Lazarsfeld: Bertram, A.; Ein, Lawrence; Lazarsfeld, Robert (1991). "Surjectivity of Gaussian maps for line bundles of large degree on curves". Algebraic geometry. Lecture Notes in Mathematics book series (LNM, volume 1479). Vol. 1479. Berlin; Heidelberg: Springer. pp. 15–25. doi:10.1007/BFb0086260. ISBN 978-3-540-54456-2. with Aaron Bertram and R. Lazarsfeld: Bertram, Aaron; Ein, Lawrence; Lazarsfeld, Robert (1991). "Vanishing theorems, a theorem of Severi, and the equations defining projective varieties". J. Amer. Math. Soc. 4 (3): 587–602. doi:10.1090/S0894-0347-1991-1092845-5. with R. Lazarsfeld: Ein, Lawrence; Lazarsfeld, Robert (1993). "Global generation of pluricanonical and adjoint linear series on smooth projective threefolds". J. Amer. Math. Soc. 6 (4): 875–903. doi:10.1090/S0894-0347-1993-1207013-5. with R. Lazarsfeld: Ein, Lawrence; Lazarsfeld, Robert (1997). "Singularities of theta divisors and the birational geometry of irregular varieties". J. Amer. Math. Soc. 10: 243–258. doi:10.1090/S0894-0347-97-00223-3. with Jean-Pierre Demailly and R. Lazarsfeld: Demailly, Jean-Pierre; Ein, Lawrence; Lazarsfeld, Robert (2000). "A subadditivity property of multiplier ideals". arXiv:math/0002035. with R. Lazarsfeld and Karen E. Smith: Ein, Lawrence; Lazarsfeld, Robert; Smith, Karen E (2001). "Uniform bounds and symbolic powers on smooth varieties". Inventiones Mathematicae. 144 (2): 241–262. arXiv:math/0005098. Bibcode:2001InMat.144..241E. doi:10.1007/s002220100121. S2CID 14857907. with Tommaso de Fernex: Beltrametti, Mauro C.; Catanese, Fabrizio; Ciliberto, Ciro; Lanteri, Antonio; Pedrini, Claudio (2002). "Resolution of indeterminacy of pairs". Algebraic geometry. Walter de Gruyter. pp. 165–177. ISBN 9783110198072. with R. Lazarsfeld: Ein, Lawrence; Lazarsfeld, Robert (2012). "Asymptotic syzygies of algebraic varieties". Inventiones Mathematicae. 190 (3): 603–646. arXiv:1103.0483. Bibcode:2012InMat.190..603E. doi:10.1007/s00222-012-0384-5. S2CID 119329785. with R. Lazarsfeld: Ein, Lawrence; Lazarsfeld, Robert (2015). "The gonality conjecture on syzygies of algebraic curves of large degree". Publications Mathématiques de l'IHÉS. 122 (1): 301–313. arXiv:1407.4445. doi:10.1007/s10240-015-0072-2. S2CID 119324291.

References

Illustrations

Worked examples

Example 1 — a first encounter with Lawrence Ein

Start with the simplest possible case. Write down what Lawrence Ein 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 Lawrence Ein 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 Lawrence Ein 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 Lawrence Ein

In research: Lawrence Ein 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 Lawrence Ein 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: Lawrence Ein is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1955 births, 20th-century American mathematicians, 21st-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Lawrence Ein 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 Lawrence Ein in 20 minutes

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

Frequently asked questions

What is Lawrence Ein in simple terms?

Lawrence Man Hou Ein (born 18 November 1955) is a mathematician who works in algebraic geometry. Education and career Lawrence Ein received in 1976 his bachelor's degree from UCLA and in 1981 his Ph.D. from the University of California, Berkeley under the direction of Robin Hartshorne with thesis S…

Why does Lawrence Ein 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 Lawrence Ein?

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 Lawrence Ein.