Geometry of the KP Hierarchy and String Equations

In part I, we review the basic results of the KP and Toda lattice theory, Hamiltonian and orthogonal polynomials theories. Also we will give a brief introduction to the constrained KP hierarchy. In Part II, we use three sections to deal with three different topics. In S1, we show that a family of compatible symplectic structures for an isospectral pseudo -differential operator L (of the KP hierarchy) is obtained by taking the Lie derivative of the first symplectic structure with respect to symmetry Virasoro vector fields V _{n}, which satisfy [ V_{n},V_{m} ]=(m-n)V_{m+n}.In S2, we will study constrained KP hierarchies, a special case being the AKNS hierarchy. We will present an approach to find the constraints induced on the tau function of the KP hierarchy. In particular, we give a set of nice formula in the case of the AKNS hierarchy. We also consider the same problem for more general constrained KP hierarchies. In S3, we will set up the connections between orthogonal polynomial systems and the Toda lattice, string equations, and Virasoro constraints. We first define the (1, q) problem for orthogonal polynomials and give all possible solutions for the (1, 1) problem. We will prove two theorems which give the connections between the (1, q) problem and the weight of orthogonal polynomials. After we set up the connections we can get the string equations and Virasoro constraints of the Toda lattice. At the end of that section we give a few examples.