Title: 
Hidden quantum group structure on solution spaces of certain PDE Systems

Abstract:

I describe a systematic method for solving PDEs of certain type, 
which arise in conformal field theory, and in the theory of Schramm-Loewner 
evolutions, with boundary conditions given by specified asymptotic 
behavior of the solutions. Our method is a correspondence associating 
vectors in a tensor product representation of a quantum group to Coulomb gas 
type integral functions, in which properties of the functions are encoded 
in natural, representation theoretical properties of the vectors. 
In particular, this hidden quantum group structure on the solution space 
of such PDEs enables explicit calculation of the asymptotics and monodromy 
properties of the solutions. This also leads us to a generalization of 
the Temperley-Lieb algebra, defined in terms of a diagrammatic 
representation. This generalized diagram algebra is the commutant 
algebra of the quantum group, generalizing the quantum Schur-Weyl duality.