Algebraic Geometry and Complex Systems

Algebraic geometry has revealed complex structures among fundamental natural phenomena, such as solitary waves. The Korteweg-de Vries (KdV) equation is a prominent nonlinear PDE modelling these waves, with solutions encoded by algebro-geometric objects such as -maximal real- algebraic curves and -totally positive- Grassmannians. The KdV equation transforms into the modified KdV (mKdV) equation, linking the dynamics of the underlying algebraic curve with its curvature. The mKdV equation also finds applications in DNA modelling. We will explore these relationships, along with the mathematical questions they raise, and showcase the computational machinery we have developed to advance this research. Our tools include computer algebra algorithms designed to study geometric data.