Spin/Pin-structures on vector bundles have long featured prominently in differential geometry, in particular providing part of the foundation for the original proof of the renowned Atiyah–Singer Index Theory. More recently, they have underpinned the symplectic topology foundations of the so-called real sector of the mirror symmetry of string theory.This semi-expository three-part monograph provides an accessible introduction to Spin- and Pin-structures in general, demonstrates their role in the orientability considerations in symplectic topology, and presents their applications in enumerative geometry.Part I contains a systematic treatment of Spin/Pin-structures from different topological perspectives and may be suitable for an advanced undergraduate reading seminar. This leads to Part II, which systematically studies orientability problems for the determinants of real Cauchy-Riemann operators on vector bundles. Part III introduces enumerative geometry of curves in complex projective varieties and in symplectic manifolds, demonstrating some applications of the first two parts in the process. Two appendices review the Cech cohomology perspective on fiber bundles and Lie group covering spaces.Key Features: oThis topic is very important in topology, geometry, and high energy physicsoComprehensive treatment from different perspectives (classic, modern standard, and new), and describes explicitly the correspondences between these different perspectivesoUnique in applying properties of relative Spin- and Pin-structures to obtain properties of orientations of real Cauchy-Riemann operators on vectorsoBesides the rigorous theories, also contains many interconnected examples, and presents in detail the applications to real enumerative in the final partoSelf-contained and comes with background information; part of the book is accessible to advanced undergraduatesoOne of the author Aleksey Zinger is a renowned expert in this field. The other author Xujia Chen is an emerging star

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