An amazing phenomenon called solitons 鈥� are isolated waves interacting in a collision in a special way. As wave functions, solitons are described by partial differential equations, but the study of the internal symmetries of these unusual equations leads to other areas of modern mathematics: integrable systems, representation theory, algebraic geometry, computational mathematics, modeling, applied mathematics.
In this course, we will describe how the theory of representations of infinite-dimensional algebraic structures and the combinatorics of symmetric functions help to construct families of solutions of the famous Kadomtsev-Petviashvili soliton hierarchy. Surprisingly, the functions that define these decisions independently play an important role in representation theory itself.
We plan to cover the following topics:
- Examples of soliton equations and their solutions. Properties of solitons.
- Derivatives of Hirota.
- KdV equation and KP equation in terms of Hirota derivatives.
- Introduction to the theory of symmetric functions.
- Semi-infinite exterior algebra.
- Formal distributions.
- The action of infinite-dimensional algebraic structures on a semi-infinite exterior algebra and on the algebra of symmetric functions.
- Boson-fermion correspondence.
- Bilinear form of the KP hierarchy.
- Solutions to the bilinear KP identity.
- Schur functions as solutions to a bilinear KP identity.
- Representation of symmetric functions through the action of vertex operators on a vacuum vector.
