Development of analysis and discretization in differential geometry

这是对离散曲线和曲面的研究，通过使用变换理论建立，该理论可以被视为光滑范畴内的离散构造，允许离散化的自然概念。 以前理解的离散等温表面已被扩展到这里的更广泛的类离散欧米茄表面，连同变换理论，为这个更广泛的类，保留底层结构在相应的光滑的情况下。 特别是，达布变换的全类已经开发。 此外，从曲线的达布变换的角度来看，已经看到了潜在的mKdV方程的离散化。 此外，在一个相关的主题，奇异性和签名的变化，在德西特3-空间的光滑表面进行了探讨，特别适用于各种悬链在该空间。

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.