**Yuly Billig
**(U of New Brunswich)

Abstract: In 1971 Ryogo Hirota invented a method to produce exact N-soliton solutions for many important non-linear partial differential equations. > Ten years later it was discovered by the Kyoto school that there is a hidden Kac-Moody group action on the space of solitons in the Hirota's method. Another important ingredient of this theory is the construction of the vertex operators discovered by physicists in the string theory. One obtains soliton solutions of the hierarchy of non-linear PDEs from the representation of Kac-Moody algebras by vertex operators.