Enumerative geometry and string theory

Perhaps the most famous example of how ideas from modern physics have revolutionized mathematics is the way string theory has led to an overhaul of enumerative geometry, an area of mathematics that started in the eighteen hundreds. Century-old problems of enumerating geometric configurations have now been solved using new and deep mathematical techniques inspired by physics! The book begins with an insightful introduction to enumerative geometry. From there, the goal becomes explaining the more advanced elements of enumerative algebraic geometry. Along the way, there are some crash courses on intermediate topics which are essential tools for the student of modern mathematics, such as cohomology and other topics in geometry. The physics content assumes nothing beyond a first undergraduate course. The focus is on explaining the action principle in physics, the idea of string theory, and how these directly lead to questions in geometry. Once these topics are in place, the connection between physics and enumerative geometry is made with the introduction of topological quantum field theory and quantum cohomology. This volume was copublished with the Institute for Advanced Study/Park City Mathematics Institute. Read more... IAS/Park City Mathematics Institute -- Preface -- Acknowledgements -- ch. 1. Warming up to enumerative geometry -- Exercises -- ch. 2. Enumerative geometry in the projective plane -- Exercises -- ch. 3. Stable maps and enumerative geometry -- Exercises -- ch. 4. Crash course in topology and manifolds -- Exercises -- ch. 5. Crash course in C[infinity] manifolds and cohomology -- Exercises -- ch. 6. Cellular decompositions and line bundles -- Exercises -- ch. 7. Enumerative geometry of lines -- Exercises -- ch. 8. Excess intersection -- Exercises -- ch. 9. Rational curves on the quintic threefold -- Exercises -- ch. 10. Mechanics -- Exercises -- ch. 11. Introduction to supersymmetry -- Exercises -- ch. 12. Introduction to string theory -- Exercises -- ch. 13. Topological quantum field theory -- Exercises -- ch. 14. Quantum cohomology and enumerative geometry -- Exercises -- Bibliography -- Index Title......Page 1 Contents......Page 5 Preface......Page 9 1. Warming up to Enumerative Geometry......Page 15 2. Enumerative Geometry in the Projective Plane......Page 27 3. Stable Maps and Enumerative Geometry......Page 41 4. Crash Course in Topology and Manifolds......Page 57 5. Crash Course in C^∞ Manifolds and Cohomology......Page 71 6. Cellular Decompositions and Line Bundles......Page 91 7. Enumerative Geometry of Lines......Page 109 8. Excess Intersection......Page 125 9. Rational Curves on the Quintic Threefold......Page 139 10. Mechanics......Page 149 11. Introduction to Supersymmetry......Page 159 12. Introduction to String Theory......Page 171 13. Topological Quantum Field Theory......Page 187 14. Quantum Cohomology and Enumerative Geometry......Page 199 Bibliography......Page 211 Index......Page 215