Generalized complex and holomorphic Poisson geometry (10w5072)


Henrique Bursztyn (Instituto Nacional de Matemática Pura e Aplicada)

(Utrecht University)

(University of Toronto)

Nigel Hitchin (Oxford University)

Jacques Hurtubise (McGill University)

Ruxandra Moraru (University of Waterloo)


The Banff International Research Station will host the "Generalized complex and holomorphic Poisson geometry" workshop from April 11 to 16, 2010.

Symplectic and algebraic geometry are fundamental fields of
mathematical research, which have been brought together in the last two decades by new developments in string theory and the theory of integrable systems.

The origins of symplectic geometry can be traced back to Lagrange's work in celestial mechanics. Lagrange first introduced the notion of Poisson bracket to describe the motions of planets and other celestial bodies, and subsequently used it as a fundamental ingredient in the mathematical formulation of other mechanical systems. Lagrange's formalism later evolved (most notably through the work of Poisson, Lie, and Hamilton) into what is known today as symplectic geometry. The language of symplectic geometry is now used to describe everything from the solar system, to elementary particles such as quarks and photons, to dynamical systems appearing in oceanography and atmospheric science.

Mathematicians who work in the area of modern algebraic geometry study systems of algebraic equations, but from a geometric point of view. Instead of finding explicit solutions to the equations, it is often more important to describe the geometric structure of the space of all solutions. For example, even if we are not able to compute all its solutions, we know that any complex polynomial equation of degree d in one variable always has d of them, up to multiplicity.

In recent years, two new developments have strongly linked
symplectic geometry to algebraic geometry. The first is the
Homological Mirror Symmetry Conjecture of Kontsevich. Mirror
symmetry was first discovered by physicists in string theory as a
duality between certain Calabi-Yau manifolds. Kontsevich then
proposed in 1994 that this statement could be interpreted as a
specific equivalence between the algebraic geometry of one and the symplectic geometry of the other. The second development came in mathematics with the introduction of generalized complex geometry by Hitchin in 2002 as a hybrid of complex and symplectic geometry. The purpose of this conference is to understand the connections between these two developments.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the US National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnologí­a (CONACYT).