| This dissertation contains a study of a two dimensional generalization of Toda equations, introduced by Gieseker in [Gie1]. These are completely integrable differential-difference equations, and are constructed using algebraic geometry. They provide integrable discretizations of the KP equation. We put these systems into Hamiltonian framework by introducing appropriate Poisson brackets. These brackets are non-local, and possess interesting arithmetic and combinatorial properties. One feature that makes them ubiquotious is, they give birth to new brackets for other systems, the examples being periodic band matrix systems that are discussed in [vM-M]. One gets a bi-Hamiltonian interpretation for the band matrix systems by combining the new brackets with the old.; The first chapter of the dissertation contains exposition to the material and a collection of relatively well-known results. The second chapter relates discrete KP equations to algebraic geometry. The correspondence between the relevant classes of objects is stated and proven in detail.; The third chapter constitutes the core of the thesis. The Poisson brackets that we mentioned are introduced here. The involutivity of the conserved quantities is proven. A bi-Hamiltonian structure for the band matrix systems is gotten. Surprisingly, there seems to be no bi-Hamiltonian interpretation of the discrete KP systems in general.; In the end, there is a combinatorial result related to the brackets. The brackets naturally define antisymmetric products on a certain set of objects on the toroidal grid. We call these objects "toroidal pipe diagrams" because of their shape. It seems that each of these toroidal pipe diagrams can be interpreted as a 1-cycle on the grid which itself is a discrete torus. Then the product defines an intersection pairing of cycles. This is reminiscent of Goldman and Turaev's intersection pairing on (continuous) surfaces [T], so we guess that there is a link, and this may be an instance of "discrete" string topology [C-S]. |
