فیزیک و ریاضیات همولوژی پیوند: سمینار ریاضیات عالی، ۲۴ ژوئن تا ۵ ژوئیه ۲۰۱۳، مرکز تحقیقات ریاضی، دانشگاه مونترال، کبک، کانادا

Throughout recent history, the theory of knot invariants has been a fascinating melting pot of ideas and scientific cultures, blending mathematics and physics, geometry, topology and algebra, gauge theory, and quantum gravity. The 2013 Séminaire de Mathématiques Supérieures in Montréal presented an opportunity for the next generation of scientists to learn in one place about the various perspectives on knot homology, from the mathematical background to the most recent developments, and provided an access point to the relevant parts of theoretical physics as well. This volume presents a cross-section of topics covered at that summer school and will be a valuable resource for graduate students and researchers wishing to learn about this rapidly growing field. Cover 1 Title page 4 Contents 6 Preface 8 Chern-Simons theory and knot invariants 10 1. Topological Field Theories 10 2. Salient Features of Knots and Links 10 3. Braids and Braid Group 13 4. Three-manifolds and their invariants 15 5. Abelian Chern-Simons Theory 16 6. Non-abelian Chern-Simons Theory 18 7. Conclusions 27 8. Appendix 27 References 29 Tensor product algebras, Grassmannians and Khovanov homology 32 1. Introduction 32 2. Tensor product algebras of sl(2) 34 3. The geometry of Grassmannians 47 4. Khovanov homology 51 References 65 Lectures on knot homology and quantum curves 68 Foreword 68 1. Why knot homology? 70 2. The classical A-polynomial 78 3. Quantization 83 4. Categorification 94 5. Epilogue: super-A-polynomial 100 References 103 An introduction to knot Floer homology 108 1. Introduction 108 2. Properties and applications 110 3. The original definition 115 4. Grid diagrams 125 5. The cube of resolutions 128 6. Surgery formulas 132 References 140 Lectures on knot homology 146 1. Introduction 146 2. Quadruply-graded homology 152 3. Knot homology and planar curves 161 4. Torus knot homology and DAHA 165 5. Torus knot homology, Hilbert schemes and rational DAHA 167 6. Matrix factorizations and Koszul models 173 7. Other directions 178 References 181 Back Cover 188 "Throughout recent history, the theory of knot invariants has been a fascinating melting pot of ideas and scientific cultures, blending mathematics and physics, geometry, topology and algebra, gauge theory, and quantum gravity. The 2013 Séminaire de Mathématiques Supérieures in Montréal presented an opportunity for the next generation of scientists to learn in one place about the various perspectives on knot homology, from the mathematical background to the most recent developments, and provided an access point to the relevant parts of theoretical physics as well. This volume presents a cross-section of topics covered at that summer school and will be a valuable resource for graduate students and researchers wishing to learn about this rapidly growing field."--Page 4 de la couverture The 2013 Seminaire de Mathematiques Superieures in Montreal presented an opportunity for the next generation of scientists to learn in one place about the various perspectives on knot homology. This volume presents a cross-section of topics covered at that summer school and will be a valuable resource for graduate students and researchers wishing to learn about this rapidly growing field.