Alan David Weinstein (born 17 June 1943) is a professor of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

at the

University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California), is a Public university, public Land-grant university, land-grant research university in Berkeley, California, United States. Founded in 1868 and named after t ...

, working in the field of

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, and especially in

Poisson geometry In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hami ...

Early life and education

Weinstein was born in New York City. After attending Roslyn High School, Weinstein obtained a bachelor's degree at the

Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a Private university, private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of moder ...

in 1964. His teachers included, among others,

James Munkres James Raymond Munkres (born ) is a Professor Emeritus of mathematics at MIT and the author of several texts in the area of topology, including ''Topology'' (an undergraduate-level text), ''Analysis on Manifolds'', ''Elements of Algebraic Topology' ...

Gian-Carlo Rota Gian-Carlo Rota (April 27, 1932 – April 18, 1999) was an Italian-American mathematician and philosopher. He spent most of his career at the Massachusetts Institute of Technology, where he worked in combinatorics, functional analysis, proba ...

Irving Segal Irving Ezra Segal (1918–1998) was an American mathematician known for work on theoretical quantum mechanics. He shares credit for what is often referred to as the Segal–Shale–Weil representation. Early in his career Segal became known for h ...

, and, for the first senior course of differential geometry, Sigurður Helgason. He received a PhD at

in 1967 under the direction of

Shiing-Shen Chern Shiing-Shen Chern (; , ; October 26, 1911 – December 3, 2004) was a Chinese American mathematician and poet. He made fundamental contributions to differential geometry and topology. He has been called the "father of modern differential geome ...

. His dissertation was entitled "''The

cut locus In differential geometry, the cut locus of a point on a manifold is the closure of the set of all other points on the manifold that are connected to by two or more distinct shortest geodesics. More generally, the cut locus of a closed set on ...

and conjugate locus of a

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

''".

Career

Weinstein worked then at MIT on 1967 (as Moore instructor) and at

Bonn University The University of Bonn, officially the Rhenish Friedrich Wilhelm University of Bonn (), is a public research university in Bonn, North Rhine-Westphalia, Germany. It was founded in its present form as the () on 18 October 1818 by Frederick Will ...

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 William Marsh Rice University, commonly referred to as Rice University, is a Private university, private research university in Houston, Houston, Texas, United States. Established in 1912, the university spans 300 acres. Rice University comp ...

. Weinstein was awarded in 1971 a

Sloan Research Fellowship The Sloan Research Fellowships are awarded annually by the Alfred P. Sloan Foundation since 1955 to "provide support and recognition to early-career scientists and scholars". This program is one of the oldest of its kind in the United States. ...

and in

1985 The year 1985 was designated as the International Youth Year by the United Nations. Events January * January 1 ** The Internet's Domain Name System is created. ** Greenland withdraws from the European Economic Community as a result of a n ...

Guggenheim Fellowship Guggenheim Fellowships are Grant (money), grants that have been awarded annually since by the John Simon Guggenheim Memorial Foundation, endowed by the late Simon Guggenheim, Simon and Olga Hirsh Guggenheim. These awards are bestowed upon indiv ...

. In 1978 he was invited speaker at the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...

in Helsinki. In

1992 1992 was designated as International Space Year by the United Nations. Events January * January 1 – Boutros Boutros-Ghali of Egypt replaces Javier Pérez de Cuéllar of Peru as United Nations Secretary-General. * January 6 ** The Republ ...

he was elected Fellow of the

American Academy of Arts and Sciences The American Academy of Arts and Sciences (The Academy) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and other ...

and in 2012 Fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

. In 2003 he was awarded a

honorary doctorate An honorary degree is an academic degree for which a university (or other degree-awarding institution) has waived all of the usual requirements. It is also known by the Latin phrases ''honoris causa'' ("for the sake of the honour") or '' ad hon ...

from

Universiteit Utrecht Utrecht University (UU; , formerly ''Rijksuniversiteit Utrecht'') is a public research university in Utrecht, Netherlands. Established , it is one of the oldest universities in the Netherlands. In 2023, it had an enrollment of 39,769 students, an ...

Research

Weinstein's works cover many areas in

and

mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...

, including

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...

symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...

Lie groupoid In mathematics, a Lie groupoid is a groupoid where the set \operatorname of objects and the set \operatorname of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are sm ...

geometric mechanics Geometric mechanics is a branch of mathematics applying particular geometric methods to many areas of mechanics, from mechanics of particles and rigid bodies to fluid mechanics and control theory. Geometric mechanics applies principally to system ...

and

deformation quantization In mathematics and physics, deformation quantization roughly amounts to finding a (quantum) algebra whose classical limit is a given (classical) algebra such as a Lie algebra or a Poisson algebra. In physics Intuitively, a deformation of a math ...

. Among his most important contributions, in 1971 he proved a tubular neighbourhood theorem for Lagrangians in

symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sy ...

s. In 1974 he worked with

Jerrold Marsden Jerrold Eldon Marsden (August 17, 1942 – September 21, 2010) was a Canadian mathematician. He was the Carl F. Braun Professor of Engineering and Control & Dynamical Systems at the California Institute of Technology. Marsden is listed as an ISI ...

on the theory of reduction for mechanical systems with

symmetries Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

, introducing the famous Marsden–Weinstein quotient. In 1978 he formulated a celebrated

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

on the existence of

periodic orbit In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Iterated functions Given ...

s, which has been later proved in several particular cases and has led to many new developments in symplectic and

contact geometry In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution ...

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

André Lichnerowicz André Lichnerowicz (; January 21, 1915, Bourbon-l'Archambault – December 11, 1998, Paris) was a French differential geometer and mathematical physicist. He is considered the founder of modern Poisson geometry. Biography His grandfather Jan f ...

, in a 1983 foundational paper Weinstein proved many results which laid the ground for the development of modern

. A further influential idea in this field was its introduction of symplectic groupoids. 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,

(1999)
''Lectures on the Geometry of Quantization''
(with S. Bates), Berkeley Mathematics Lecture Notes series,

(1997) *
Basic Multivariable Calculus
' (with J.E. Marsden and A.J. Tromba), W.A. Freeman and Company,

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

(1993), *
Calculus, I, II, III
' (with J.E. Marsden), 2nd ed.,

(1985), now out of print and free at CaltechAUTHORS. *
Calculus Unlimited
' (with J.E. Marsden), Benjamin/Cummings (1981), now out of print and free at CaltechAUTHORS.

Notes

External links

Weinstein's home page

Alain Weinstein
at