Professor Gieseker will investigate deformations of infinite dimensional integrable systems such as the KdV hierarchy. The KdV hierarchy is a sequence of mutually commuting differential operators in one variable. The object of this grant is to produce deformations of this and similar hierarchies which preserve complete integrability. Professor Gieseker has produced such a deformation of the KdV hierarchy and will seek to produce many more deformations of KdV and also to find deformations of other infinite dimensional completely integrable systems.
Certain important equations of nonlinear science are completely integrable, which roughly means that there are a lot of conserved quantities of these equations. These equations are applicable in a wide variety of practical situations, for instance in understanding laser optics. Professor Gieseker proposes to apply the methods of algebraic geometry to find discretizations of this type of equation so that the discretizations are integrable. This means finding algorithms to solve these equations on the computer so that the solutions computed on the computer have the same type of conserved quantities as the original equations. Professor Gieseker will continue to investigate his discretization of KdV, as well as develop new discretizations of other infinite dimensional integrable systems.
