|Todor Eliseev Milanov
Last Update 2022/11/21
The Korteweg-de Vries (KdV) equation is a mathematical model of the motion of a wave in shallow waters. It has been studied extensively from a number of different perspectives. In particular, it was discovered that KdV is a reduction of a more universal equation, known as the Kadomtsev- Petviashvili (KP) equation. It turns out that the solutions of KP can be parameterized by the points of an infinite Grassmanian. The latter is a central object in both geometry and representation theory. I am deeply impressed by the unity of seemingly different areas of mathematics on one side and nature on the other.
At the end of the 20th century it was discovered that the KdV equation governs the amplitudes of string motions in a vacuum. I have been interested in finding other equations, similar to KdV, which characterize the string amplitudes in more interesting spaces that have non-trivial topologies. More precisely, I am using complex geometry and representation theory to obtain a characterization of the string amplitudes. It seems that there are some new geometrical objects, as well as some new representation theories, that are still awaiting discovery.
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