Alan Weinstein Explained

Alan Weinstein
Birth Place:	New York, United States
Fields:	Mathematics
Alma Mater:	University of California, Berkeley
Thesis Title:	The Cut Locus and Conjugate Locus of a Riemannian Manifold
Thesis Year:	1967
Doctoral Advisor:	Shiing-Shen Chern
Doctoral Students:	Theodore Courant Viktor Ginzburg Steve Omohundro Steven Zelditch Yong-Geun Oh
Known For:	Marsden-Weinstein quotient Weinstein conjecture Symplectic groupoid Symplectic category
Awards:	Sloan Research Fellowship, 1971 Guggenheim Fellowship, 1985

Alan David Weinstein (born 17 June 1943) is a professor of mathematics at the University of California, Berkeley, working in the field of differential geometry, and especially in Poisson geometry.

Early life and education

Weinstein was born in New York City.^[1] After attending Roslyn High School,^[2] Weinstein obtained a bachelor's degree at the Massachusetts Institute of Technology in 1964. His teachers included, among others, James Munkres, Gian-Carlo Rota, Irving Segal, and, for the first senior course of differential geometry, Sigurður Helgason. He received a PhD at University of California, Berkeley in 1967 under the direction of Shiing-Shen Chern. His dissertation was entitled "The cut locus and conjugate locus of a Riemannian manifold".^[3]

Career

Weinstein worked then at MIT on 1967 (as Moore instructor) and at Bonn University in 1968/69. In 1969 he returned to Berkeley as assistant professor and from 1976 he is full professor. During 1975/76 he visited IHES in Paris and during 1978/79 he was visiting professor at Rice University. Weinstein was awarded in 1971 a Sloan Research Fellowship^[4] and in 1985 a Guggenheim Fellowship.^[5] In 1978 he was invited speaker at the International Congress of Mathematicians in Helsinki.^[6] In 1992 he was elected Fellow of the American Academy of Arts and Sciences^[7] and in 2012 Fellow of the American Mathematical Society.^[8] In 2003 he was awarded a honorary doctorate from Universiteit Utrecht.^{[9] [10]}

Research

Weinstein's works cover many areas in differential geometry and mathematical physics, including Riemannian geometry, symplectic geometry, Lie groupoids, geometric mechanics and deformation quantization.^[11]

Among his most important contributions, in 1971 he proved a tubular neighbourhood theorem for Lagrangians in symplectic manifolds.^[12]

In 1974 he worked with Jerrold Marsden on the theory of reduction for mechanical systems with symmetries, introducing the famous Marsden–Weinstein quotient.^[13]

In 1978 he formulated a celebrated conjecture on the existence of periodic orbits,^[14] which has been later proved in several particular cases and has led to many new developments in symplectic and contact geometry.^[15]

In 1981 he formulated a general principle, called symplectic creed, stating that "everything is a Lagrangian submanifold".^[16] Such insight has been constantly quoted as the source of inspiration for many results in symplectic geometry.

Building on the work of André Lichnerowicz, in a 1983 foundational paper^[17] Weinstein proved many results which laid the ground for the development of modern Poisson geometry. A further influential idea in this field was its introduction of symplectic groupoids.^{[18] [19]}

He is author of more than 50 research papers in peer-reviewed journals and he has supervised 34 PhD students.

Books

• Geometric Models for Noncommutative Algebras (with A. Cannas da Silva), Berkeley Mathematics Lecture Notes series, American Mathematical Society (1999)^[20]
• Lectures on the Geometry of Quantization (with S. Bates), Berkeley Mathematics Lecture Notes series, American Mathematical Society (1997)^[21]
• Basic Multivariable Calculus (with J.E. Marsden and A.J. Tromba), W.A. Freeman and Company, Springer-Verlag (1993),
• Calculus, I, II, III (with J.E. Marsden), 2nd ed., Springer-Verlag (1985), now out of print and free at CaltechAUTHORS.^{[22] [23] [24]}
• Calculus Unlimited (with J.E. Marsden), Benjamin/Cummings (1981), now out of print and free at CaltechAUTHORS.^[25]

External links

• Weinstein's home page.
• Alain Weinstein at University of California, Berkeley

Further reading

• The breadth of symplectic and Poisson geometry: festschrift in honor of Alan Weinstein

Notes and References

1. American Men and Women of Science, Thomson Gale, 2005
2. Bursztyn . Henrique . Fernandes . Rui Loja . Rui Loja Fernandes . 2023-01-01 . A Conversation with Alan Weinstein . . 70 . 1 . 1 . 10.1090/noti2595 . 254776861 . 0002-9920. free .
3. Web site: Alan Weinstein - The Mathematics Genealogy Project. 2021-07-17. www.mathgenealogy.org.
4. Web site: Past Fellows Alfred P. Sloan Foundation. 2021-07-17. sloan.org. en. 2018-03-14. https://web.archive.org/web/20180314000756/https://sloan.org/past-fellows. dead.
5. Web site: John Simon Guggenheim Foundation Alan David Weinstein. 2021-07-17. en-US.
6. Book: Proceedings of the International Congress of Mathematician 1978. 1980. Lehto. Olii. 2. Helsinki. 803.
7. Web site: Alan David Weinstein. 2021-07-17. American Academy of Arts & Sciences. en.
8. http://www.ams.org/profession/fellows-list List of Fellows of the American Mathematical Society
9. Web site: Archive Honorary Doctorates . 2023-01-28 . . en.
10. Fall 2003 . Honors and Awards . Berkeley Mathematics Newsletter . X . 1 . 10.
11. Book: The Breadth of Symplectic and Poisson Geometry - Festschrift in Honor of Alan Weinstein . . 2005 . 978-0-8176-3565-7 . Marsden . Jerrold . Jerrold E. Marsden . ix - xii . en . Preface . Progress in Mathematics . 232 . 10.1007/b138687 . Ratiu . Tudor . Tudor Ratiu.
12. 1971-06-01. Symplectic manifolds and their lagrangian submanifolds. Advances in Mathematics. en. 6. 3. 329–346. 10.1016/0001-8708(71)90020-X. free. 0001-8708. Weinstein. Alan.
13. Marsden. Jerrold. Weinstein. Alan. 1974-02-01. Reduction of symplectic manifolds with symmetry. Reports on Mathematical Physics. en. 5. 1. 121–130. 10.1016/0034-4877(74)90021-4. 1974RpMP....5..121M . 0034-4877.
14. 1979-09-01. On the hypotheses of Rabinowitz' periodic orbit theorems. Journal of Differential Equations. en. 33. 3. 353–358. 10.1016/0022-0396(79)90070-6. free. 0022-0396. Weinstein. Alan. 1979JDE....33..353W .
15. Pasquotto. Federica. 2012-09-01. A Short History of the Weinstein Conjecture. Jahresbericht der Deutschen Mathematiker-Vereinigung. en. 114. 3. 119–130. 10.1365/s13291-012-0051-1. 120567013. 1869-7135.
16. Weinstein . Alan . July 1981 . Symplectic geometry . . 5 . 1 . 1–13 . 10.1090/S0273-0979-1981-14911-9 . Project Euclid. free .
17. Weinstein. Alan. 1983-01-01. The local structure of Poisson manifolds. Journal of Differential Geometry. 18. 3. 10.4310/jdg/1214437787. free. 0022-040X.
18. Weinstein. Alan. 1987. Symplectic groupoids and Poisson manifolds. Bulletin of the American Mathematical Society. en. 16. 1. 101–104. 10.1090/S0273-0979-1987-15473-5. free. 0273-0979.
19. Coste. A.. Dazord. P.. Weinstein. A.. 1987. Groupoïdes symplectiques. Publications du Département de mathématiques (Lyon). fr. 2A. 1–62.
20. Web site: Geometric Models for Noncommutative Algebras. 2021-07-17. bookstore.ams.org.
21. Web site: Lectures on the Geometry of Quantization. 2021-07-17. bookstore.ams.org.
22. Book: Marsden. Jerrold E.. Calculus I. Weinstein. Alan J.. 1985. Springer . 9780387909745 .
23. Book: Marsden. Jerrold E.. Calculus II. Weinstein. Alan J.. 1985. Springer . 9780387909752 .
24. Book: Marsden. Jerrold E.. Calculus III. Weinstein. Alan J.. 1985. Springer . 9780387909851 .
25. Book: Marsden. Jerrold. Weinstein. Alan J.. Calculus Unlimited. 1981. Benjamin/Cummings Publishing Company . 9780805369328 .