Principal Investigator: Alan D. Weinstein
This grant supports travel to Brazil for US participants (primarily graduate students and postdocs) at the conference "Poisson 2010," to be held in IMPA in Rio de Janeiro, July 26-30, preceded by a preparatory school of three days in length. With roots in classical mechanics about 200 years ago and work of Sophus Lie about a century ago, the subject of Poisson geometry crystallized through work of Lichnerowicz and Kirillov in the 1970's. Its influence now extends to a wide variety of areas, including symplectic geometry and topology, deformation theory, representation theory, algebraic geometry, integrable hamiltonian systems, and field theory. In recent years, Poisson geometry has found new applications in string topology, the geometric Langlands program, and the geometry of complex surfaces. Developments in these and other timely subjets will be addressed in the school and conference.
The meeting, whose full title is "Poisson Geometry in Mathematics and Physics," is the seventh in a biannual series which brings together mathematicians and mathematical physicists with common interests in Poisson geometry and its applications. This geometry provides a very important bridge between mathematics and physics and between classical and quantum physics. Speakers at the school and conference have been chosen not only for the importance of their results but also for their ability to communicate them to a broad audience of mathematicians and physicists. Proceedings will be published in a manner which makes them accessible at low (or no) cost to a wide readership, in order to stimulate further study and research in the rapidly growing area of Poisson geometry. Detailed information about the conference, and about others in the biannual series, may be found at poissongeometry.org.
The Poisson bracket operation, introduced in the early 19th century by Simeon Poisson as an algebraic tool in the study of classical dynamics, was found in the 20th century to be an essential structure governing the transition between classical and quantum mechanics. In the late 20th century, it was found that so-called Poisson structures carried a deep geometric meaning, and their study since then has had ramifications not only in mechanics, but throughout mathematics and physics, in such as representation theory, symplectic geometry, partial differential equations, and quantum field theory. Beginning in 1998, a series of biennial international meetings has been held with the aim of bringing together mathematiciansand mathematical physicists from diverse areas (both scientific and geographic), having common interests in Poisson geometry. Since 2006, the event has consisted of a "school" several days long designed to prepare young participants, followed by a five-day research conference. The scope of Poisson geometry and its applications has continued to expand, so that invited lectures at "Poisson 2010" covered such new topics as string topology, the geometric Langlands program, vertex algebras,and the geometry of complex surfaces. Other areas represented were generalized complexgeometry, higher-order algebraic structures, deformation quantization, continuum mechanics, stacks in symplectic geometry, classification of singular Poisson structures, integrable systems, noncommutative Poisson geometry, microlocal analysis, and Poisson sigma models. Proceedings of the conference were published in the Bulletin of the Brazilian Mathematical society, and video of all the lectures is available at http://video.impa.br/index.php?page=poisson-geometry-in-mathematics-and-physics. The present grant supported the travel expenses of nine young US mathematicians (graduate students and postdocs) to attend Poisson 2010 in Rio de Janeiro, as well as part of the travel expenses of the principal investigator, who was an organizer of the meeting. Despite the availability of these funds, the participation by young US mathematicians was disappointingly low, and so the small amount of funds remaining in the grant were used to support travel to a weekend conference, Gone Fishing, held in November 2011 at The Washington Unversity in St. Louis with the aim of stimulating interest in Poisson geometry among North American mathematicians. 18 half-hour talks were given there, several of them by graduate students and recent PhD's. At Poisson 2012, held in Utrecht (Netherlands), a similar grant supported travel for 18 US participants (either citizens or foreign nationals based in the US). Poisson 2014 will be held at the University of Illinois at Champaign-Urbana and will be the first of the series to take place in the US. Links to information about all the Poisson 20xx and Gone Fishing conferences may be found at poissongeometry.org.
