Holdings: Geometric combinatorics

Geometric combinatorics:

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Format:	Book
Language:	English
Published:	Providence, RI American Math. Soc. 2007 [Princeton, NJ] Inst. for Advanced Study
Series:	IAS/Park City mathematics series 13
Subjects:	Analyse combinatoire Géométrie combinatoire Combinatorial geometry Combinatorial analysis Kombinatorische Geometrie Konferenzschrift
Online Access:	Inhaltsverzeichnis Klappentext
Physical Description:	XII, 691 S. Ill., graph. Darst.
ISBN:	0821837362 9780821837368

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adam_text	Contents Preface xi Ezra Miller and Victor Reiner What is Geometric Combinatorics? An Overview of the Graduate Summer School 1 Bibliography 17 Alexander Barvinok Lattice Points, Polyhedra, and Complexity 19 Introduction 21 Lecture 1. Inspirational Examples. Valuations 23 Valuations 26 Problems 27 Lecture 2. Identities in the Algebra of Polyhedra 29 Problems 34 Lecture 3. Generating Functions and Cones. Continued Fractions 39 Generating Functions and Cones 39 Continued Fractions 42 Computing f(K, x) for 2-dimensional Cones 43 The Computational Complexity 44 Problems 45 Lecture 4. Rational Polyhedra and Rational Functions 47 Problems 51 Lecture 5. Computing Generating Functions Fast 53 Why Do We Need Generating Functions? 53 What "Fast" and "Short" Means 54 Problems 57 Concluding Remarks 58 Bibliography 61 iv CONTENTS Sergey Fomin and Nathan Reading Root Systems and Generalized Associahedra 63 Lecture 1. Reflections and Roots 67 1.1. The Pentagon Recurrence 67 1.2. Reflection Groups 68 1.3. Symmetries of Regular Polytopes 70 1.4. Root Systems 73 1.5. Root Systems of Types A, B, C, and D 75 Lecture 2. Dynkin Diagrams and Coxeter Groups 77 2.1. Finite Type Classification 77 2.2. Coxeter Groups 79 2.3. Other "Finite Type" Classifications 80 2.4. Reduced Words and Permutohedra 82 2.5. Coxeter Element and Coxeter Number 84 Lecture 3. Associahedra and Mutations 87 3.1. Associahedron 87 3.2. Cyclohedron 93 3.3. Matrix Mutations 95 3.4. Exchange Relations 96 Lecture 4. Cluster Algebras 101 4.1. Seeds and Clusters 101 4.2. Finite Type Classification 104 4.3. Cluster Complexes and Generalized Associahedra 106 4.4. Polytopal Realizations of Generalized Associahedra 108 4.5. Double Wiring Diagrams and Double Bruhat Cells 111 Lecture 5. Enumerative Problems 115 5.1. Catalan Combinatorics of Arbitrary Type 115 5.2. Generalized Narayana Numbers 120 5.3. Non-crystallographic Types 124 5.4. Lattice Congruences and the Weak Order 125 Bibliography 129 Robin Forman Topics in Combinatorial Differential Topology and Geometry 133 Lecture 1. Discrete Morse Theory 137 1. Introduction I37 2. Cell Complexes and CW Complexes 138 3. The Morse Theory 143 4. A More Combinatorial Language 146 5. Our First Example: The Real Projective Plane 147 6. Sphere Theorems 148 7. Our Second Example 148 8. Exercises for Lecture 1 I5I CONTENTS v Lecture 2. Discrete Morse Theory, continued 153 1. Suspensions and Discrete Morse Theory 153 2. Monotone Graph Properties 155 3. The Morse Complex 161 4. Canceling Critical Points 165 5. Exercises for Lecture 2 166 Lecture 3. Discrete Morse Theory and Evasiveness 169 1. The Main Results 169 2. Betti Numbers for General Sets of Faces 175 3. Exercises for Lecture 3 180 Lecture 4. The Charney-Davis Conjectures 181 1. Introduction 181 2. Exercises for Lecture 4 187 Lecture 5. From Analysis to Combinatorics 189 1. Hodge Theory and the Hopf-Charney-Davis Conjectures 189 2. The Charney-Davis Conjecture and the /i-vector 194 3. Exercises for Lecture 5 198 Bibliography 201 Mark Hairnan and Alexander Woo Geometry of q and q, í- Analogs in Combinatorial Enumeration 207 Introduction 209 Lecture 1. Kostka Numbers and q-Analogs 211 1.1. Definition of Kostka Numbers 211 1.2. Κχμ in Symmetric Functions 212 1.3. Sn Representations 212 1.4. GLn Representations 214 1.5. The g-Analog Κχμ(α) 215 1.6. Exercises 216 Lecture 2. Catalan Numbers, Trees, Lagrange Inversion, and their g-Analogs 217 2.1. Catalan Numbers 217 2.2. Rooted Trees 218 2.3. The Lagrange Inversion Formula 219 2.4. g-Analogs 220 2.5. g-Lagrange Inversion 222 2.6. Exercises 226 Lecture 3. Macdonald Polynomials 227 3.1. Symmetric Function Bases and the Involution ω 227 3.2. Plethystic Substitution 228 3.3. The Cauchy Kernel and Hall Inner Product 228 3.4. Dominance Ordering 229 3.5. Definition of Macdonald Polynomials 229 3.6. More Properties of Macdonald Polynomials 232 3.7. Exercises 234 vi CONTENTS Lecture 4. Connecting Macdonald Polynomials and g-Lagrange Inversion; (q, ŕ)- Analogs 235 4.1. The Operator V and a (q, ^-Analog of kn{q) 235 4.2. Proof of Theorem 7 236 4.3. First Remarks on Positivity 239 4.4. Exercises 240 Lecture 5. Positivity and Combinatorics? 241 5.1. Representation Theory of #μ(χ; q, t) 241 5.2. Representation Theory of Ven 243 5.3. Combinatorics of Ven 243 5.4. Combinatorics of Ημ(χ;ς, t) 245 5.5. Exercises 245 Bibliography 247 Dmitry N. Kozlov Chromatic Numbers, Morphism Complexes, and Stiefel- Whitney Characteristic Classes 249 Preamble 251 Lecture 1. Introduction 253 1.1. The Chromatic Number of a Graph 253 1.2. The Category of Graphs 256 Lecture 2. The Functor Hom (-, -) 261 2.1. Complexes of Graph Homomorphisms 261 2.2. Morphism Complexes 264 2.3. Historic Detour 267 2.4. More about the Hom -Complexes 269 2.5. Folds 273 Lecture 3. Stiefel- Whitney Classes and First Applications 277 3.1. Elements of the Principal Bundle Theory 277 3.2. Properties of Stiefel- Whitney Classes 279 3.3. First Applications of Stiefel- Whitney Classes to Lower Bounds of Chromatic Numbers of Graphs 281 Lecture 4. The Spectral Sequence Approach 285 4.1. Hom-i.-construction 285 4.2. Spectral Sequence Generalities 288 4.3. The Standard Spectral Sequence Converging to #(Нопц_(Т, G)) 293 Lecture 5. The Proof of the Lovász Conjecture 295 5.1. Formulation of the Conjecture and Sketch of the Proof 295 5.2. Completing the Sketch for the Case к is Odd 297 5.3. Completing the Sketch for the Case к is Even 301 CONTENTS vii Lecture 6. Summary and Outlook 305 6.1. Homotopy Tests, Z2-Tests, and Families of Test Graphs 305 6.2. Conclusion and Open Problems 308 Bibliography 311 Robert MacPherson Equivariant Invariants and Linear Geometry 317 Introduction 319 0.1. Spaces with a Torus Action 320 0.2. Linear Graphs 322 0.3. Rings and Modules 323 Lecture 1. Equivariant Homology and Intersection Homology 327 1.1. Introduction 327 1.2. Simplicial Complexes 328 1.3. Pseudomanifolds 329 1.4. Ordinary Homology Theory 330 1.5. Basic Definitions of Equivariant Topology 332 1.6. Equivariant Homology 333 1.7. Formal Properties of Equivariant Homology 335 1.8. Torus Equivariant Cohomology of a Point 337 1.9. The Equivariant Cohomology of a 2-Sphere 338 1.10. Equivariant Intersection Cohomology 340 Lecture 2. Moment Graphs 343 2.1. Assumptions on the Action of Τ on X 343 2.2. The Moment Graph 344 2.3. Complex Projective Line and the Line Segment 346 2.4. Projective (η — l^Space and the Simplex 347 2.5. Quadric Hypersurfaces and the Cross-Polytope 348 2.6. Grassmannians and Hypersimplices 351 2.7. The Flag Manifold and the Permutahedron 353 2.8. Toric Varieties and Convex Polyhedra 354 2.9. Moment Maps 357 Lecture 3. The Cohomology of a Linear Graph 359 3.1. The Definition of the Cohomology of a Linear Graph 359 3.2. Interpreting W(Q) for Small і 360 3.3. Piecewise Polynomial Functions 361 3.4. Morse Theory 362 3.5. Perfect Morse Functions 364 3.6. Determining U{G) as a O(T) Module 366 3.7. Poincaré Duality 366 3.8. The Main Theorems 367 Lecture 4. Computing Intersection Homology 371 4.1. Graphs Arising from Reflection Groups 371 4.2. Upward Saturated Subgraphs 372 viii CONTENTS 4.3. Sheaves on Graphs 373 4.4. A Criterion for Perfection 374 4.5. Definition of the Sheaf M 375 4.6. The Main Results 377 4.7. Flag Varieties and Generalized Schubert Varieties 377 Lecture 5. Cohomology as Functions on a Variety 379 5.1. The Fixed Point Arrangement 379 5.2. How to Compute the Fixed Point Arrangement 380 5.3. The Main Result 381 5.4. Springer Varieties 382 5.5. Relation with Lecture 3 385 Bibliography 387 Richard P. Stanley An Introduction to Hyperplane Arrangements 389 Lecture 1. Basic Definitions, the Intersection Poset and the Characteristic Polynomial 391 1.1. Basic Definitions 391 1.2. The Intersection Poset 397 1.3. The Characteristic Polynomial 398 Exercises 400 Lecture 2. Properties of the Intersection Poset and Graphical Arrangements 403 2.1. Properties of the Intersection Poset 403 2.2. The Number of Regions 409 2.3. Graphical Arrangements 414 Exercises 419 Lecture 3. Matroids and Geometric Lattices 421 3.1. Matroids 421 3.2. The Lattice of Flats and Geometric Lattices 423 Exercises 428 Lecture 4. Broken Circuits, Modular Elements, and Supersolvability 431 4.1. Broken Circuits 431 4.2. Modular Elements 437 4.3. Supersolvable Lattices 442 Exercises 446 449 449 452 454 456 459 466 467 468 Lecture 5. Finite Fields 5.1. The Finite Field Method 5.2. The Shi Arrangement 5.3. Exponential Sequences of Arrangements 5.4. The Catalan Arrangement 5.5. Interval Orders 5.6. Intervals with Generic Lengths 5.7. Other Examples Exercises CONTENTS ix Lecture 6. Separating Hyperplanes 475 6.1. The Distance Enumerator 475 6.2. Parking Functions and Tree Inversions 478 6.3. The Distance Enumerator of the Shi Arrangement 483 6.4. The Distance Enumerator of a Supersolvable Arrangement 487 6.5. The Varchenko Matrix 490 Exercises 491 Bibliography 495 Michelle L. Wachs, Poset Topology: Tools and Applications 497 Introduction 499 Lecture 1. Basic Definitions, Results, and Examples 501 1.1. Order Complexes and Face Posets 501 1.2. The Möbius Function 505 1.3. Hyperplane and Subspace Arrangements 507 1.4. Some Connections with Graphs, Groups and Lattices 512 1.5. Poset Homology and Cohomology 513 1.6. Top Cohomology of the Partition Lattice 515 Lecture 2. Group Actions on Posets 519 2.1. Group Representations 519 2.2. Representations of the Symmetric Group 521 2.3. Group Actions on Poset (Co)homology 526 2.4. Symmetric Functions, Plethysm, and Wreath Product Modules 528 Lecture 3. Shellability and Edge Labelings 537 3.1. Shellable Simplicial Complexes 537 3.2. Lexicographic Shellability 541 3.3. CL-shellability and Coxeter Groups 553 3.4. Rank Selection 558 Lecture 4. Recursive Techniques 563 4.1. Cohen-Macaulay Complexes 563 4.2. Recursive Atom Orderings 567 4.3. More Examples 569 4.4. The Whitney Homology Technique 573 4.5. Bases for the Restricted Block Size Partition Posets 579 4.6. Fixed Point Möbius Invariant 586 Lecture 5. Poset Operations and Maps 587 5.1. Operations: Alexander Duality and Direct Product 587 5.2. Quillen Fiber Lemma 591 5.3. General Poset Fiber Theorems 596 5.4. Fiber Theorems and Subspace Arrangements 599 5.5. Inflations of Simplicial Complexes 601 Bibliography 605 x CONTENTS Günter M. Ziegler Convex Polytopes: Extremal Constructions and /-Vector Shapes 617 Introduction 619 Lecture 1. Constructing 3-Dimensional Polytopes 621 1.1. The Cone of /-vectors 623 1.2. The Steinitz Theorem 625 1.3. Steinitz' Theorem via Circle Packings 628 Lecture 2. Shapes of /-Vectors 643 2.1. Unimodality Conjectures 644 2.2. Basic Examples 644 2.3. Global Constructions 647 2.4. Local Constructions 649 Lecture 3. 2-Simple 2-Simplicial 4-Polytopes 653 3.1. Examples 654 3.2. 2-Simple 2-Simplicial 4-Polytopes 657 3.3. Deep Vertex Truncation 659 3.4. Constructing DVT(Stack(n, 4)) 661 Lecture 4. /-Vectors of 4-Polytopes 665 4.1. The /-Vector Cone 666 4.2. Fatness and the Upper Bound Problem 669 4.3. The Lower Bound Problem 671 Lecture 5. Projected Products of Polygons 673 5.1. Products and Deformed Products 673 5.2. Computing the /-Vector 674 5.3. Deformed Products 674 5.4. Surviving a Generic Projection 678 5.5. Construction 678 Appendix: A Short Introduction to polymake (by Thilo Schröder and Nikolaus Witte) 681 A.I. Getting Started 681 A.2. The polymake System 684 Bibliography 687 Geometrie combinatórios describes a wide area of mathematics that is primarily the study of geometric objects and their combinatorial structure. Perhaps the most familiar examples are polytopes and simplicial complexes, but the subject is much broader. This volume is a compilation of expository articles at the interface between combinatorics and geometry, based on a three-week program of lectures at the Institute for Advanced Study/Park City Mathematics Institute (IAS/PCMI) summer program on Geometric Combinatorics. The topics covered include posets, graphs, hyperplane arrangements, discrete Morse theory, and more. These objects are considered from multiple perspectives, such as in enumerative or topological contexts, or in the presence of discrete or continuous group actions. Most of the exposition is aimed at graduate students or researchers learning the material for the first time. Many of the articles include substantial numbers of exercises, and all include numerous examples. The reader is led quickly to the state of the art and current active research by worldwide authorities on their respective subjects.
adam_txt	Contents Preface xi Ezra Miller and Victor Reiner What is Geometric Combinatorics? An Overview of the Graduate Summer School 1 Bibliography 17 Alexander Barvinok Lattice Points, Polyhedra, and Complexity 19 Introduction 21 Lecture 1. Inspirational Examples. Valuations 23 Valuations 26 Problems 27 Lecture 2. Identities in the Algebra of Polyhedra 29 Problems 34 Lecture 3. Generating Functions and Cones. Continued Fractions 39 Generating Functions and Cones 39 Continued Fractions 42 Computing f(K, x) for 2-dimensional Cones 43 The Computational Complexity 44 Problems 45 Lecture 4. Rational Polyhedra and Rational Functions 47 Problems 51 Lecture 5. Computing Generating Functions Fast 53 Why Do We Need Generating Functions? 53 What "Fast" and "Short" Means 54 Problems 57 Concluding Remarks 58 Bibliography 61 iv CONTENTS Sergey Fomin and Nathan Reading Root Systems and Generalized Associahedra 63 Lecture 1. Reflections and Roots 67 1.1. The Pentagon Recurrence 67 1.2. Reflection Groups 68 1.3. Symmetries of Regular Polytopes 70 1.4. Root Systems 73 1.5. Root Systems of Types A, B, C, and D 75 Lecture 2. Dynkin Diagrams and Coxeter Groups 77 2.1. Finite Type Classification 77 2.2. Coxeter Groups 79 2.3. Other "Finite Type" Classifications 80 2.4. Reduced Words and Permutohedra 82 2.5. Coxeter Element and Coxeter Number 84 Lecture 3. Associahedra and Mutations 87 3.1. Associahedron 87 3.2. Cyclohedron 93 3.3. Matrix Mutations 95 3.4. Exchange Relations 96 Lecture 4. Cluster Algebras 101 4.1. Seeds and Clusters 101 4.2. Finite Type Classification 104 4.3. Cluster Complexes and Generalized Associahedra 106 4.4. Polytopal Realizations of Generalized Associahedra 108 4.5. Double Wiring Diagrams and Double Bruhat Cells 111 Lecture 5. Enumerative Problems 115 5.1. Catalan Combinatorics of Arbitrary Type 115 5.2. Generalized Narayana Numbers 120 5.3. Non-crystallographic Types 124 5.4. Lattice Congruences and the Weak Order 125 Bibliography 129 Robin Forman Topics in Combinatorial Differential Topology and Geometry 133 Lecture 1. Discrete Morse Theory 137 1. Introduction I37 2. Cell Complexes and CW Complexes 138 3. The Morse Theory 143 4. A More Combinatorial Language 146 5. Our First Example: The Real Projective Plane 147 6. Sphere Theorems 148 7. Our Second Example 148 8. Exercises for Lecture 1 I5I CONTENTS v Lecture 2. Discrete Morse Theory, continued 153 1. Suspensions and Discrete Morse Theory 153 2. Monotone Graph Properties 155 3. The Morse Complex 161 4. Canceling Critical Points 165 5. Exercises for Lecture 2 166 Lecture 3. Discrete Morse Theory and Evasiveness 169 1. The Main Results 169 2. Betti Numbers for General Sets of Faces 175 3. Exercises for Lecture 3 180 Lecture 4. The Charney-Davis Conjectures 181 1. Introduction 181 2. Exercises for Lecture 4 187 Lecture 5. From Analysis to Combinatorics 189 1. Hodge Theory and the Hopf-Charney-Davis Conjectures 189 2. The Charney-Davis Conjecture and the /i-vector 194 3. Exercises for Lecture 5 198 Bibliography 201 Mark Hairnan and Alexander Woo Geometry of q and q, í- Analogs in Combinatorial Enumeration 207 Introduction 209 Lecture 1. Kostka Numbers and q-Analogs 211 1.1. Definition of Kostka Numbers 211 1.2. Κχμ in Symmetric Functions 212 1.3. Sn Representations 212 1.4. GLn Representations 214 1.5. The g-Analog Κχμ(α) 215 1.6. Exercises 216 Lecture 2. Catalan Numbers, Trees, Lagrange Inversion, and their g-Analogs 217 2.1. Catalan Numbers 217 2.2. Rooted Trees 218 2.3. The Lagrange Inversion Formula 219 2.4. g-Analogs 220 2.5. g-Lagrange Inversion 222 2.6. Exercises 226 Lecture 3. Macdonald Polynomials 227 3.1. Symmetric Function Bases and the Involution ω 227 3.2. Plethystic Substitution 228 3.3. The Cauchy Kernel and Hall Inner Product 228 3.4. Dominance Ordering 229 3.5. Definition of Macdonald Polynomials 229 3.6. More Properties of Macdonald Polynomials 232 3.7. Exercises 234 vi CONTENTS Lecture 4. Connecting Macdonald Polynomials and g-Lagrange Inversion; (q, ŕ)- Analogs 235 4.1. The Operator V and a (q, ^-Analog of kn{q) 235 4.2. Proof of Theorem 7 236 4.3. First Remarks on Positivity 239 4.4. Exercises 240 Lecture 5. Positivity and Combinatorics? 241 5.1. Representation Theory of #μ(χ; q, t) 241 5.2. Representation Theory of Ven 243 5.3. Combinatorics of Ven 243 5.4. Combinatorics of Ημ(χ;ς, t) 245 5.5. Exercises 245 Bibliography 247 Dmitry N. Kozlov Chromatic Numbers, Morphism Complexes, and Stiefel- Whitney Characteristic Classes 249 Preamble 251 Lecture 1. Introduction 253 1.1. The Chromatic Number of a Graph 253 1.2. The Category of Graphs 256 Lecture 2. The Functor Hom (-, -) 261 2.1. Complexes of Graph Homomorphisms 261 2.2. Morphism Complexes 264 2.3. Historic Detour 267 2.4. More about the Hom -Complexes 269 2.5. Folds 273 Lecture 3. Stiefel- Whitney Classes and First Applications 277 3.1. Elements of the Principal Bundle Theory 277 3.2. Properties of Stiefel- Whitney Classes 279 3.3. First Applications of Stiefel- Whitney Classes to Lower Bounds of Chromatic Numbers of Graphs 281 Lecture 4. The Spectral Sequence Approach 285 4.1. Hom-i.-construction 285 4.2. Spectral Sequence Generalities 288 4.3. The Standard Spectral Sequence Converging to #(Нопц_(Т, G)) 293 Lecture 5. The Proof of the Lovász Conjecture 295 5.1. Formulation of the Conjecture and Sketch of the Proof 295 5.2. Completing the Sketch for the Case к is Odd 297 5.3. Completing the Sketch for the Case к is Even 301 CONTENTS vii Lecture 6. Summary and Outlook 305 6.1. Homotopy Tests, Z2-Tests, and Families of Test Graphs 305 6.2. Conclusion and Open Problems 308 Bibliography 311 Robert MacPherson Equivariant Invariants and Linear Geometry 317 Introduction 319 0.1. Spaces with a Torus Action 320 0.2. Linear Graphs 322 0.3. Rings and Modules 323 Lecture 1. Equivariant Homology and Intersection Homology 327 1.1. Introduction 327 1.2. Simplicial Complexes 328 1.3. Pseudomanifolds 329 1.4. Ordinary Homology Theory 330 1.5. Basic Definitions of Equivariant Topology 332 1.6. Equivariant Homology 333 1.7. Formal Properties of Equivariant Homology 335 1.8. Torus Equivariant Cohomology of a Point 337 1.9. The Equivariant Cohomology of a 2-Sphere 338 1.10. Equivariant Intersection Cohomology 340 Lecture 2. Moment Graphs 343 2.1. Assumptions on the Action of Τ on X 343 2.2. The Moment Graph 344 2.3. Complex Projective Line and the Line Segment 346 2.4. Projective (η — l^Space and the Simplex 347 2.5. Quadric Hypersurfaces and the Cross-Polytope 348 2.6. Grassmannians and Hypersimplices 351 2.7. The Flag Manifold and the Permutahedron 353 2.8. Toric Varieties and Convex Polyhedra 354 2.9. Moment Maps 357 Lecture 3. The Cohomology of a Linear Graph 359 3.1. The Definition of the Cohomology of a Linear Graph 359 3.2. Interpreting W(Q) for Small і 360 3.3. Piecewise Polynomial Functions 361 3.4. Morse Theory 362 3.5. Perfect Morse Functions 364 3.6. Determining U{G) as a O(T) Module 366 3.7. Poincaré Duality 366 3.8. The Main Theorems 367 Lecture 4. Computing Intersection Homology 371 4.1. Graphs Arising from Reflection Groups 371 4.2. Upward Saturated Subgraphs 372 viii CONTENTS 4.3. Sheaves on Graphs 373 4.4. A Criterion for Perfection 374 4.5. Definition of the Sheaf M 375 4.6. The Main Results 377 4.7. Flag Varieties and Generalized Schubert Varieties 377 Lecture 5. Cohomology as Functions on a Variety 379 5.1. The Fixed Point Arrangement 379 5.2. How to Compute the Fixed Point Arrangement 380 5.3. The Main Result 381 5.4. Springer Varieties 382 5.5. Relation with Lecture 3 385 Bibliography 387 Richard P. Stanley An Introduction to Hyperplane Arrangements 389 Lecture 1. Basic Definitions, the Intersection Poset and the Characteristic Polynomial 391 1.1. Basic Definitions 391 1.2. The Intersection Poset 397 1.3. The Characteristic Polynomial 398 Exercises 400 Lecture 2. Properties of the Intersection Poset and Graphical Arrangements 403 2.1. Properties of the Intersection Poset 403 2.2. The Number of Regions 409 2.3. Graphical Arrangements 414 Exercises 419 Lecture 3. Matroids and Geometric Lattices 421 3.1. Matroids 421 3.2. The Lattice of Flats and Geometric Lattices 423 Exercises 428 Lecture 4. Broken Circuits, Modular Elements, and Supersolvability 431 4.1. Broken Circuits 431 4.2. Modular Elements 437 4.3. Supersolvable Lattices 442 Exercises 446 449 449 452 454 456 459 466 467 468 Lecture 5. Finite Fields 5.1. The Finite Field Method 5.2. The Shi Arrangement 5.3. Exponential Sequences of Arrangements 5.4. The Catalan Arrangement 5.5. Interval Orders 5.6. Intervals with Generic Lengths 5.7. Other Examples Exercises CONTENTS ix Lecture 6. Separating Hyperplanes 475 6.1. The Distance Enumerator 475 6.2. Parking Functions and Tree Inversions 478 6.3. The Distance Enumerator of the Shi Arrangement 483 6.4. The Distance Enumerator of a Supersolvable Arrangement 487 6.5. The Varchenko Matrix 490 Exercises 491 Bibliography 495 Michelle L. Wachs, Poset Topology: Tools and Applications 497 Introduction 499 Lecture 1. Basic Definitions, Results, and Examples 501 1.1. Order Complexes and Face Posets 501 1.2. The Möbius Function 505 1.3. Hyperplane and Subspace Arrangements 507 1.4. Some Connections with Graphs, Groups and Lattices 512 1.5. Poset Homology and Cohomology 513 1.6. Top Cohomology of the Partition Lattice 515 Lecture 2. Group Actions on Posets 519 2.1. Group Representations 519 2.2. Representations of the Symmetric Group 521 2.3. Group Actions on Poset (Co)homology 526 2.4. Symmetric Functions, Plethysm, and Wreath Product Modules 528 Lecture 3. Shellability and Edge Labelings 537 3.1. Shellable Simplicial Complexes 537 3.2. Lexicographic Shellability 541 3.3. CL-shellability and Coxeter Groups 553 3.4. Rank Selection 558 Lecture 4. Recursive Techniques 563 4.1. Cohen-Macaulay Complexes 563 4.2. Recursive Atom Orderings 567 4.3. More Examples 569 4.4. The Whitney Homology Technique 573 4.5. Bases for the Restricted Block Size Partition Posets 579 4.6. Fixed Point Möbius Invariant 586 Lecture 5. Poset Operations and Maps 587 5.1. Operations: Alexander Duality and Direct Product 587 5.2. Quillen Fiber Lemma 591 5.3. General Poset Fiber Theorems 596 5.4. Fiber Theorems and Subspace Arrangements 599 5.5. Inflations of Simplicial Complexes 601 Bibliography 605 x CONTENTS Günter M. Ziegler Convex Polytopes: Extremal Constructions and /-Vector Shapes 617 Introduction 619 Lecture 1. Constructing 3-Dimensional Polytopes 621 1.1. The Cone of /-vectors 623 1.2. The Steinitz Theorem 625 1.3. Steinitz' Theorem via Circle Packings 628 Lecture 2. Shapes of /-Vectors 643 2.1. Unimodality Conjectures 644 2.2. Basic Examples 644 2.3. Global Constructions 647 2.4. Local Constructions 649 Lecture 3. 2-Simple 2-Simplicial 4-Polytopes 653 3.1. Examples 654 3.2. 2-Simple 2-Simplicial 4-Polytopes 657 3.3. Deep Vertex Truncation 659 3.4. Constructing DVT(Stack(n, 4)) 661 Lecture 4. /-Vectors of 4-Polytopes 665 4.1. The /-Vector Cone 666 4.2. Fatness and the Upper Bound Problem 669 4.3. The Lower Bound Problem 671 Lecture 5. Projected Products of Polygons 673 5.1. Products and Deformed Products 673 5.2. Computing the /-Vector 674 5.3. Deformed Products 674 5.4. Surviving a Generic Projection 678 5.5. Construction 678 Appendix: A Short Introduction to polymake (by Thilo Schröder and Nikolaus Witte) 681 A.I. Getting Started 681 A.2. The polymake System 684 Bibliography 687 Geometrie combinatórios describes a wide area of mathematics that is primarily the study of geometric objects and their combinatorial structure. Perhaps the most familiar examples are polytopes and simplicial complexes, but the subject is much broader. This volume is a compilation of expository articles at the interface between combinatorics and geometry, based on a three-week program of lectures at the Institute for Advanced Study/Park City Mathematics Institute (IAS/PCMI) summer program on Geometric Combinatorics. The topics covered include posets, graphs, hyperplane arrangements, discrete Morse theory, and more. These objects are considered from multiple perspectives, such as in enumerative or topological contexts, or in the presence of discrete or continuous group actions. Most of the exposition is aimed at graduate students or researchers learning the material for the first time. Many of the articles include substantial numbers of exercises, and all include numerous examples. The reader is led quickly to the state of the art and current active research by worldwide authorities on their respective subjects.
any_adam_object	1
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spelling	Geometric combinatorics Ezra Miller ... ed. Providence, RI American Math. Soc. 2007 [Princeton, NJ] Inst. for Advanced Study XII, 691 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier IAS/Park City mathematics series 13 Analyse combinatoire Géométrie combinatoire Combinatorial geometry Combinatorial analysis Kombinatorische Geometrie (DE-588)4140733-7 gnd rswk-swf (DE-588)1071861417 Konferenzschrift gnd-content Kombinatorische Geometrie (DE-588)4140733-7 s DE-604 Miller, Ezra Sonstige oth Erscheint auch als Online-Ausgabe 978-1-4704-3912-5 IAS/Park City mathematics series 13 (DE-604)BV010402400 13 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016276835&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016276835&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext
spellingShingle	Geometric combinatorics IAS/Park City mathematics series Analyse combinatoire Géométrie combinatoire Combinatorial geometry Combinatorial analysis Kombinatorische Geometrie (DE-588)4140733-7 gnd
subject_GND	(DE-588)4140733-7 (DE-588)1071861417
title	Geometric combinatorics
title_auth	Geometric combinatorics
title_exact_search	Geometric combinatorics
title_exact_search_txtP	Geometric combinatorics
title_full	Geometric combinatorics Ezra Miller ... ed.
title_fullStr	Geometric combinatorics Ezra Miller ... ed.
title_full_unstemmed	Geometric combinatorics Ezra Miller ... ed.
title_short	Geometric combinatorics
title_sort	geometric combinatorics
topic	Analyse combinatoire Géométrie combinatoire Combinatorial geometry Combinatorial analysis Kombinatorische Geometrie (DE-588)4140733-7 gnd
topic_facet	Analyse combinatoire Géométrie combinatoire Combinatorial geometry Combinatorial analysis Kombinatorische Geometrie Konferenzschrift
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016276835&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016276835&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV010402400
work_keys_str_mv	AT millerezra geometriccombinatorics

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