Geometry and physics in an intimate union

Lukas Nakamura received his doctoral degree in mathematics from Uppsala University in 2024. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor Maxim Kontsevich, Institut des Hautes Études Scientifiques (IHES), Bures-sur-Yvette, France.

When symplectic geometry, the topic of this project was created in the nineteenth century, neither quantum mechanics nor Einstein’s general theory of relativity, let alone the string theory of modern physics, existed. Geometry grew from the need to replace Newton’s complicated differential equations with simpler methods of describing motion. At each moment, the trajectory of a moving object is determined by its position and velocity – by a pair of quantities that together form a two-dimensional surface: the basic structure of symplectic geometry. 

Since the emergence of modern physics, symplectic geometry has experienced a renaissance. The twentieth century saw the development of two new pillars of physics: quantum mechanics, which describes the smallest parts of the universe, and Albert Einstein’s general theory of relativity, which describes the whole universe. Efforts to reconcile the two continue, and one promising theory for this reconciliation is called string theory. This builds our world in eleven dimensions rather than Einstein’s four, and its smallest components are tiny strings instead of quantum particles. 

Modern symplectic geometry is further developed in interaction with string theory and quantum field theory. Nowadays it not only tackles two-dimensional spaces, but all abstract spaces with an even number of dimensions. Spaces with odd dimensions are treated by contact geometry, which is closely related to symplectic geometry. The interaction between these two geometries and mathematical operations inside and in between different geometric spaces are the subjects of the current project.