This two-volume set presents forty-eight papers on applied mathematics by Fields Medal winner Shing-Tung Yau. The papers are organized topically into thirteen sections: circuit theory, MRI data analysis, matrix eigenvalues, scientific computing, evolutionary dynamics (in mathematical biology), quantum information (mathematical physics), nonlinear filtering, graph theory, medical imaging, computer vision, computer graphics and geometric modeling, artificial intelligence, and computational geometry and numerical analysis. Also included are commentaries on Yau's work in the various topics: S.T. Chui on Yau's work in circuit theory; Xudong Huang on his work in MRI Data analysis; Mei-Heng Yueh on his work in scientific computing; Gábor Lippner on his work in quantum state transfer; Xiuqiong Chen on his work in nonlinear filtering; Yong Lin on his works on graphs; and Xianfeng David Gu on his work in computer graphics, computer vision, digital geometry processing, medical imaging and artificial intelligence. The collection begins with prefaces by Bjorn Engquist (University of Texas) and Tony Chang (president, KAUST). Shing-Tung Yau is a Professor Emeritus of Harvard University, where he was the William Caspar Graustein Professor of Mathematics and Professor of Physics at Harvard University. He is now Chair Professor of Mathematics at Tsinghua University in Beijing. He is the winner of the Fields Medal, the National Medal of Science, the Crafoord Prize, the Veblen Prize, the Wolf Prize, and a MacArthur Fellowship. A member of the National Academy of Sciences, Yau has received honorary degrees from ten universities. He is also the director of several mathematical institutes in China. Yau has written and edited more than twenty books and is the editor of numerous mathematics journals. This is the second of two volumes.

This book introduces differential geometry and cutting-edge findings from the discipline by incorporating both classical approaches and modern discrete differential geometry across all facets and applications, including graphics and imaging, physics and networks. With curvature as the centerpiece, the authors present the development of differential geometry, from curves to surfaces, thence to higher dimensional manifolds; and from smooth structures to metric spaces, weighted manifolds and complexes, and to images, meshes and networks. The first part of the book is a differential geometric study of curves and surfaces in the Euclidean space, enhanced while the second part deals with higher dimensional manifolds centering on curvature by exploring the various ways of extending it to higher dimensional objects and more general structures and how to return to lower dimensional constructs. The third part focuses on computational algorithms in algebraic topology and conformal geometry, applicable for surface parameterization, shape registration and structured mesh generation. The volume will be a useful reference for students of mathematics and computer science, as well as researchers and engineering professionals who are interested in graphics and imaging, complex networks, differential geometry and curvature.