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This invaluable book is based on the notes of a graduate course on differential geometry which the author gave at the Nankai Institute of Mathematics. It consists of two parts: the first part contains an introduction to the geometric theory of characteristic classes due to Shiing-shen Chern and André Weil, as well as a proof of the Gauss-Bonnet-Chern theorem based on the Mathai-Quillen construction of Thom forms; the second part presents analytic proofs of the Poincaré-Hopf index formula, as well as the Morse inequalities based on deformations introduced by Edward Witten.
Contents:

Chern classes. --- Index theorems. --- Complexes. --- Linear complexes --- Algebras, Linear --- Coordinates --- Geometry --- Line geometry --- Transformations (Mathematics) --- Differential operators --- Global analysis (Mathematics) --- Index theory (Mathematics) --- Manifolds (Mathematics) --- Chern characteristic classes --- Chern's characteristic classes --- Chern's classes --- Classes, Chern --- Characteristic classes

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This book highlights quantum optics technologies that can revolutionize the way we encode, store, transmit, and handle information. These technologies can help us overcome bottlenecks in classical physics-based information technology in information transmission capacity, computing speed, and information security. The book provides readers with new perspectives on potential applications of the quantum theory. Besides, the book summaries the research advances in quantum optics and atom optics, including manipulation and construction of the quantum states of photons and even atoms, molecules, and matter at the quantum level, and new phenomena and technologies brought about by the interactions between photons and the quantum states of matter. The book provides extensive and thoroughly exhaustive coverage of quantum optics. It is suitable for researchers and graduate students of optical physics and quantum optics.

Quantum optics. --- Quantum computers. --- Quantum communication. --- Atoms. --- Molecules. --- Quantum Optics. --- Quantum Computing. --- Quantum Communications and Cryptography. --- Atomic, Molecular and Chemical Physics.

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Differential geometry. Global analysis

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This unique volume, resulting from a conference at the Chern Institute of Mathematics dedicated to the memory of Xiao-Song Lin, presents a broad connection between topology and physics as exemplified by the relationship between low-dimensional topology and quantum field theory.The volume includes works on picture (2+1)-TQFTs and their applications to quantum computing, Berry phase and Yang-Baxterization of the braid relation, finite type invariant of knots, categorification and Khovanov homology, Gromov-Witten type invariants, twisted Alexander polynomials, Faddeev knots, generalized Ricci flo

Low-dimensional topology --- Quantum field theory --- Algebraic topology --- Field theory (Physics) --- Topology, Low-dimensional --- Manifolds (Mathematics)