The number of knots is infinite, and one of the challenges is to find new calculable methods of differentiating them. “Mathematical knots are formed by interlacing and looping a piece of string and then joining its ends.”
Modern symplectic geometry has links to a field of mathematics called topology, and particularly the study of knots. Uppsala is suddenly on the world map, and we are becoming part of the global research infrastructure.” Knotted problemsīut symplectic geometry can also be used to study more dimensions – four, six, and so on, which has rekindled interest in the field. Among other things, we can now attract pre-eminent postdocs from universities such as Stanford, Princeton and Berkeley. “The Wallenberg Scholar grant is helping to build a strong mathematical environment at an advanced level. A symplectic surface became a measure of the two interlaced manifolds: position and velocity, a two-dimensional geometry.
Before Hamilton’s work, complicated calculations was needed to describe these motions, but Hamilton gave the formulas a geometrical language, making the equations easier to solve. The term was coined in the 1800s, when William Hamilton, an Irish mathematician, reformulated classical Newtonian mechanics, which for example explains planetary orbits or how an apple falls. It is a “wilder” form of geometry in which certain two-dimensional areas must remain unchanged during manipulation. Symplectic geometry is some distance away from traditional schoolbook geometry. One interdisciplinary field is symplectic geometry, a flourishing branch of mathematics that Tobias came across while working as a postdoc at Stanford around the turn of the millennium.