The discrete Dirac operator and the discrete generalized Weierstrass representation in pseudo-Euclidean spaces

In this thesis we consider the problem of finding a integrable discretization of the Dirac operator. We show that an appropriate deformation of the spectral properties of the eigen-function of the smooth Dirac operator leads to a discrete integrable Dirac operator. We use this discrete Dirac operator to construct a discrete analogue of the modified Novikov--Veselov hierarchy and a discrete analogue of the generalized Weierstrass representation of isotropically embedded surfaces in pseudo-Euclidean spaces.