Advances in hyperkähler and holomorphic symplectic geometry (12w5126)


Marco Gualtieri (University of Toronto)

Jacques Hurtubise (McGill University)

Daniel Huybrechts (University of Bonn)

(University of Massachusetts Amherst)

Ruxandra Moraru (University of Waterloo)

(University of North Carolina)


The Banff International Research Station will host the "Advances in hyperkahler and holomorphic symplectic geometry" workshop from March 11th to March 16th, 2012.

Hyperk"ahler manifolds were defined by Calabi in 1978, and have since played a prominent role in many areas of mathematics and physics.

In mathematics, the quaternions are a number system that extends the complex numbers. They were introduced by Hamilton in 1843 to describe mechanical systems in three-dimensional space; in fact, several key results of theoretical physics, such as Maxwell's equations of electromagnetism, were first expressed in the language of quaternions. They are still widely used today in computer graphics, robotics, orbital dynamics, and signal processing to represent rotations and orientations in three dimensions.

A hyperk"ahler manifold is a space which admits a metric that is compatible with an action of the quaternions on its tangent vectors. For instance, K3 surfaces and complex tori are the only compact four-dimensional hyperk"ahler manifolds that exist. Interestingly, many other examples arise as spaces of solutions to the anti-self-dual Yang-Mills equations, which are a generalisation of Maxwell's equations; moreover, hyperk"ahler manifolds are Ricci-flat, and therefore correspond to vacuum solutions of Einstein's equations.

Yau's solution to the Calabi conjecture allows us to deduce that hyperk"ahler metrics exist on certain spaces, called compact holomorphic symplectic manifolds, coming from algebraic geometry. Though they exist, an explicit description of these metrics is not known. Recently, however, physicists were (2009) able to explicitly compute hyperk"ahler metrics on certain noncompact varieties using enumerative invariants on Calabi-Yau manifolds, opening a completely new direction in the field.

The goal of the workshop is to bring together the main researchers in Riemannian geometry, algebraic geometry, and physics who focus on hyperk"ahler and holomorphic symplectic geometry, for the purpose of attacking the open questions in the field, absorbing the recent progress coming from the different sub-disciplines, and for exposing graduate students to the most pressing questions which demand further investigation.

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 U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).