Alan Weinstein

Alan Weinstein is an American mathematician renowned for his foundational contributions to symplectic geometry and related areas of mathematical physics. A professor emeritus at the University of California, Berkeley, his research has profoundly influenced the modern understanding of Hamiltonian mechanics and geometric quantization. His work includes the development of key concepts like the Weinstein conjecture and the Weinstein tubular neighborhood theorem.

Biography

Alan Weinstein was born in New York City and completed his undergraduate studies at the Massachusetts Institute of Technology. He earned his Ph.D. in 1967 from the University of California, Berkeley under the supervision of Shlomo Sternberg. Following his doctorate, he held positions at University of California, San Diego and University of Oxford before returning to University of California, Berkeley as a faculty member, where he spent the majority of his career. He has been a visiting scholar at numerous prestigious institutions, including the Institut des Hautes Études Scientifiques and the Mathematical Sciences Research Institute.

Mathematical contributions

Weinstein's research has been central to the development of modern symplectic geometry and its applications to classical mechanics. He introduced the influential Weinstein conjecture, which concerns the existence of closed characteristics on contact manifolds and has connections to Hamiltonian dynamics. His proof of the Weinstein tubular neighborhood theorem provided a fundamental tool for studying Lagrangian submanifolds. He made pioneering advances in Poisson geometry, including the study of Poisson-Lie groups and the development of symplectic groupoids as objects for deformation quantization. His work with Albert S. Schwarz on the BFV formalism provided a rigorous geometric framework for BRST quantization in theoretical physics.

Awards and honors

Throughout his career, Weinstein has received significant recognition for his mathematical achievements. He was awarded a Sloan Fellowship in 1970 and a Guggenheim Fellowship in 1980. In 1999, he was elected a fellow of the American Academy of Arts and Sciences. He was an invited speaker at the International Congress of Mathematicians in Warsaw in 1983. His influential textbook, "Lectures on Symplectic Manifolds," has been a standard reference in the field for decades.

Selected publications

Weinstein's extensive publication record includes several landmark papers and books. Key works include "Symplectic Manifolds and their Lagrangian Submanifolds" in *Advances in Mathematics*, and "The Local Structure of Poisson Manifolds" in the *Journal of Differential Geometry*. His monograph *Lectures on Symplectic Manifolds*, published by the American Mathematical Society, is widely cited. Other significant papers involve collaborations with figures like Victor Guillemin on convexity properties of moment maps and with Mikhail Gromov on aspects of symplectic topology.

Academic service and leadership

Weinstein has played a major role in the governance and direction of mathematical organizations. He served as the Director of the Mathematical Sciences Research Institute in Berkeley from 2001 to 2004. He has been an active member of the American Mathematical Society, serving on numerous committees and editorial boards, including for the *Journal of Symplectic Geometry*. At University of California, Berkeley, he chaired the Department of Mathematics and mentored a generation of doctoral students who have become leading researchers in geometric analysis and mathematical physics.

Category:American mathematicians Category:University of California, Berkeley faculty Category:Symplectic geometers Category:1943 births Category:Living people