This chapter surveys conventional and new mathematics involved in the development of GUT. The most important conventional mathematics includes the theories of generalized curves and surfaces and the integrated version of Pontrjagin maximum principle developed by L. C. Young. We summarize their original development here because they are quantitative models of many important physical concepts. The boundary year for the new mathematics is 1998, the year of publication of the counterexamples to Fermat’s last theorem that proves the false conjecture and catalyzed the development of new mathematics such as (a) the new real number system, (b) generalized integral, derivative and fractal and (c) the complex vector plane (fully developed here except (c)). The introduction to the complex vector plane that rectifies the complex number system is presented but its full development requires re-writing of the rectified complex number system as the basis for rectification of complex analysis. The introduction of qualitative mathematics paved the way for qualitative modeling, the crucial factor for the discovery of the superstring and the 11 initial laws of nature required for the solution of the gravitational n-body problem by serving as foundations of GUT. The d-sequence of a dark number qualitatively models the superstring, the generalized curve quantitatively models its path and that of an elementary particle and the new real number system quantitatively models time and distance of ordinary space. The generalized surface quantitatively models the expanding Cosmic Sphere before its burst at t = 1.5 billion years from the start of the Big Bang.