Development of analysis and discretization in differential geometry

这是利用变换理论建立的关于离散曲线和曲面的研究，该理论可以被视为光滑范畴内的离散构造，允许自然的离散化概念。以前所理解的离散等温面已经在这里扩展到更广泛的离散Omega曲面，以及针对这一更广泛的类的变换理论，该变换理论在相应的光滑情况下保留了底层结构。特别是，针对整个类的Darboux变换已经被开发出来。此外，还从曲线的达布变换的角度对势mKdV方程进行了离散化处理。另外，在一个相关的主题中，研究了De Sitter三维空间中光滑曲面的奇点和符号变化，并特别地应用于该空间中的各种链状曲面。

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.