Solutions - At the Intersection of Algebraic-Geometry and Mathematical Physics

Algebraic-geometry grew out of the study of geometric objects defined by polynomial equations, like elliptic curves and Grassmannian manifolds. In the 20th century, it developed a reputation for being an especially arcane area of “pure” mathematics. That is why it may be surprising to learn that it has applications in mathematical physics, the subject that can accurately predict the behavior of waves and particles. In this talk, I will explain how it was discovered in the late 20th century that certain nonlinear partial differential equations used by physicists are actually just equations of algebraic-geometry “in disguise”. This discovery has far-reaching implications. The interplay between the two fields has allowed us to answer open questions in physics using algebraic-geometry and vice versa. And, because these equations have “soliton” solutions which are waves that behave like particles, this active area of research may also help resolve some of the enduring mysteries of quantum physics.