هندسه گرمسیری و سیستم‌های قابل یکپارچه‌سازی: کنفرانس هندسه گرمسیری و سیستم‌های قابل یکپارچه‌سازی 3-8 ژوئیه 2011 مدرسه ریاضیات و آمار ... بریتانیا (ریاضیات معاصر)

This volume contains the proceedings of the conference on tropical geometry and integrable systems, held July 3–8, 2011, at the University of Glasgow, United Kingdom. One of the aims of this conference was to bring together researchers in the field of tropical geometry and its applications, from apparently disparate ends of the spectrum, to foster a mutual understanding and establish a common language which will encourage further developments of the area. This aim is reflected in these articles, which cover areas from automata, through cluster algebras, to enumerative geometry. In addition, two survey articles are included which introduce ideas from researchers on one end of this spectrum to researchers on the other. This book is intended for graduate students and researchers interested in tropical geometry and integrable systems and the developing links between these two areas. Introduction 8 Introduction to tropical algebraic geometry 12 1. Introduction 12 2. Tropical varieties 14 3. Drawing curves in the plane 17 4. The Fundamental and Structure theorems 19 5. The computational approach to tropical varieties 22 6. Connections to subvarieties of toric varieties 24 References 27 Tropical curves and integrable piecewise linear maps 32 1. Introduction 32 Acknowledgments 34 2. Tropical curve theory 34 3. Tropical periodic Toda lattice 38 4. Periodic BBS 41 References 50 Counting algebraic curves with tropical geometry 52 1. Enumerative Algebraic Geometry 52 2. Tropical Geometry 54 3. Counting Algebraic Curves tropically 57 4. Applications 60 References 63 Hurwitz numbers, ribbon graphs, and tropicalization 66 1. Introduction 66 2. The classical viewpoints: covers and permutations 68 3. Double Hurwitz numbers and labeled ribbon graphs 71 4. From ribbon graphs to permutations 76 5. The tropical definition 79 References 83 Sperner property, matroids and finite-dimensional Gorenstein algebras 84 Introduction 84 1. Posets and the Sperner property 85 2. Matroids 86 3. Finite-dimensional Gorenstein algebras and Lefschetz property 88 4. Gorenstein algebras associated to matroids 89 5. Main result 90 6. Gröbner fan of J_{M} 92 References 94 Block triangular bilinear forms and braid group action 96 1. A quadratic Poisson algebra and its reductions 96 2. The Dubrovin–Ugaglia bracket and braid group action 98 3. General block size 100 4. Quantisation 102 Acknowledgements 104 References 105 Tropicalization method in cluster algebras 106 1. Introduction 106 2. Cluster algebra with coefficients 107 3. Tropicalization in cluster algebras 111 4. Application I: Periodicities of Y-systems 116 5. Application II: Dilogarithm identities 120 References 124 An application of the max-plus spectral theory to an ultradiscrete analogue of the Lax pair 128 1. Introduction 128 2. Simplified Lax pair 131 3. Adding constraints 140 4. Conclusions and projects 144 5. Acknowledgement 144 References 144 A KdV cellular automaton without integers 146 1. Introduction 146 2. A brief recall of the continuous and the discrete situation 149 3. A linear problem for the ultradiscrete KdV equation 151 4. Bound states for the ultradiscrete KdV equation 153 5. Tau functions and the dressing procedure 156 6. The case of degenerate spectrum 160 7. Concluding remarks 164 References 165 Two of the nine papers in this collection introduce tropical geometry and integrable systems to specialists in the other, emphasizing the max-plus algebra and other features common to both areas of mathematics. Other topics include counting algebraic curves with tropical geometry, block triangular bilinear forms and braid group action, the topicalization method in cluster algebras, and an application of the max-plus spectral theory to an ultradiscrete analogue of the Lax pair. There is no index. Annotation ©2012 Book News, Inc., Portland, OR (booknews.com) Chris Athorne, Diane Maclagan, Ian Strachan, Editors. Includes Bibliographical References.