Will explore Weinstein manifolds

Assistant Professor Thomas Kragh will receive funding from Knut and Alice Wallenberg Foundation to recruit an international researcher for a postdoctoral position at the Department of Mathematics, Uppsala University.

The planned project focuses on answering questions in symplectic geometry using methods from algebraic topology. Symplectic geometry in its most modern form is closely related to developments in modern physics, such as string theory and quantum field theory, but it also has implications for the understanding of dynamic systems in classical physics, the original area of symplectic geometry from the 19th century. At that time, Newton's equations were given geometric clothing that made it easier to find solutions and describe them. 

Symplectic geometry, for example, describes the geometry of a space in a mechanical system, called phase space. For a moving object, at each moment its trajectory is determined by its location and speed. Together, these determine a surface element which provides basic structure of symplectic geometry. This geometry describes directions in which the system can evolve – it describes movement.

Geometry was also the tool used to develop Albert Einstein’s general theory of relativity about gravity. Using Riemann's geometry from the mid-19th century, Einstein described how the mass of stars, galaxies and even black holes make space curve which, in his theory, has four dimensions (three space dimensions and one time dimension). As Riemann's geometry is related to the classical mechanical description of gravity, symplectic geometry is related to its quantum mechanical description through the theory of super-strings. These describe our world in eleven dimensions (although we can only perceive four of them).

The as yet unexplored Weinstein manifolds are a special class of symplectic spaces. The idea of the project is to apply the methods from algebraic K-theory, which have already been successful in the study of common topological manifolds, to reveal the properties of the Weinstein manifolds.