Tau functions, convolution symmetries and applications
Лекция- Математика
The first part of this talk will review the theory of Tau functions for the KP hierarchy of integrable flows, as developed by Sato and by Segal and Wilson, with extensions to the MKP and 2-Toda lattice hierarchies. The universal geometrical setting consists of abelian group actions on infinite Grassmann manifolds and Flag manifolds, with the dynamics determined by Hirota bilinear relations. This gives rise, through the Plucker embedding, to the operator formulation on the fermionic Fock space, with the Hirota relations seen as equivalent to the Plucker relations.
Other applications of tau functions include: matrix model partition functions and correlators, partition functions for integrable random processes, such as random 3-D partitions, and crystal growth, and generating functions for invariants on moduli spaces. These are best understood in terms of a new class of abelian group actions, distinct from the usual ''shift flows'' that appear in the KP type hierarchies, which may be viewed as convolution symmetries acting on the Hilbert space Grassmannian.