Development of analysis and discretization in differential geometry

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This is research on discrete curves and surfaces established by the use of transformation theory, a theory that can be viewed as a discrete construction within the smooth category that allows for natural notions of discretization. Previously understood discrete isothermic surfaces have been extended here to the wider class of discrete Omega surfaces, together with a transformation theory for this wider class that preserves underlying structures in the corresponding smooth case. In particular, Darboux transformations for the full class have been developed. Additionally, discretization of the potential mKdV equation has been seen from the perspective of Darboux transformations of curves. Also, in a related topic, singularities and signature changes of smooth surfaces in de Sitter 3-space have been explored, with particular application to various catenoids in that space.