Surface Ricci flow deforms the Riemannian metric proportional to the curvature, such that the curvature evolves according to a non-linear heat diffusion process and eventually becomes constant. Surface Ricci flow is a powerful tool to construct Riemannian metrics by prescribing curvatures. The smooth surface Ricci flow has been generalized to the discrete setting. In this talk, we introduce the main concepts, theorems of discrete surface Ricci flow and show its existence, uniqueness and convergence. Optimal mass transportation has intrinsic relation to the Alexandrov theory in convex geometry, which is formulated as the Monge-Ampere equation. In this talk, we introduce a variational approach to solve the Alexandrov problem, which also gives the optimal transportation map. Furthermore, we will briefly introduce the applications of both methods in engineering and medicine fields, especially in machine learning.

Brief bio: Dr. Xiangfeng David Gu got his B.S. from Tsinghua University in 1994, PhD from Harvard University in 2002, supervised by a Fields medalist: Prof. Shing-Tung Yau. Dr. Gu is a tenured professor in Computer Science Department and Applied Mathematics Department of State University of New York at Stony brook. Dr. Gu won US NSF Career award, Morningside Gold medal in applied mathematics in the 6th International Congress of Chinese Mathematician 2013. Dr. Gu and Prof.Yau are the major founders of an emerging interdisciplinary field: Computational Conformal Geometry, which applies modern geometry in engineering and medicine fields. He has published more than 270 articles in top level journals and conferences in graphics, vision, visualization, medical imaging and networking fields; three monographs in mathematics and computer science. He has obtained several international patents, some of them have been licensed to Siemens and GE.