Development of analysis and discretization in differential geometry

这是对使用变换理论建立的离散曲线和曲面的研究，该理论可以被视为光滑类别中的离散结构，允许自然的离散化概念。以前理解的离散等温曲面在这里被扩展到更广泛的离散欧米茄曲面，以及在相应光滑情况下保留底层结构的这一更广泛类别的变换理论。特别是，已经开发了整个类的达布变换。另外，从曲线的达布变换的角度，对势能mKdV方程进行了离散化。此外，在一个相关的主题中，研究了de Sitter三维空间中光滑表面的奇异性和特征变化，并特别应用于该空间中的各种链状体。

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.