Geometric combinatorics:
Gespeichert in:
| Format: | Buch |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
Providence, RI
American Math. Soc.
2007
[Princeton, NJ] Inst. for Advanced Study |
| Schriftenreihe: | IAS/Park City mathematics series
13 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | XII, 691 S. Ill., graph. Darst. |
| ISBN: | 0821837362 9780821837368 |
Internformat
MARC
| LEADER | 00000nam a2200000zcb4500 | ||
|---|---|---|---|
| 001 | BV023073713 | ||
| 003 | DE-604 | ||
| 005 | 20240527 | ||
| 007 | t| | ||
| 008 | 080110s2007 xxuad|| |||| 00||| eng d | ||
| 010 | |a 2007060782 | ||
| 020 | |a 0821837362 |c alk. paper |9 0-8218-3736-2 | ||
| 020 | |a 9780821837368 |9 978-0-8218-3736-8 | ||
| 035 | |a (OCoLC)150256503 | ||
| 035 | |a (DE-599)DNB 2007060782 | ||
| 040 | |a DE-604 |b ger |e aacr | ||
| 041 | 0 | |a eng | |
| 044 | |a xxu |c US | ||
| 049 | |a DE-703 |a DE-83 |a DE-11 |a DE-188 |a DE-824 | ||
| 050 | 0 | |a QA167 | |
| 082 | 0 | |a 516/.13 | |
| 084 | |a SK 170 |0 (DE-625)143221: |2 rvk | ||
| 084 | |a 05-06 |2 msc | ||
| 084 | |a 52-06 |2 msc | ||
| 245 | 1 | 0 | |a Geometric combinatorics |c Ezra Miller ... ed. |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2007 | |
| 264 | 1 | |a [Princeton, NJ] |b Inst. for Advanced Study | |
| 300 | |a XII, 691 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a IAS/Park City mathematics series |v 13 | |
| 650 | 4 | |a Analyse combinatoire | |
| 650 | 4 | |a Géométrie combinatoire | |
| 650 | 4 | |a Combinatorial geometry | |
| 650 | 4 | |a Combinatorial analysis | |
| 650 | 0 | 7 | |a Kombinatorische Geometrie |0 (DE-588)4140733-7 |2 gnd |9 rswk-swf |
| 655 | 7 | |0 (DE-588)1071861417 |a Konferenzschrift |2 gnd-content | |
| 689 | 0 | 0 | |a Kombinatorische Geometrie |0 (DE-588)4140733-7 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Miller, Ezra |e Sonstige |4 oth | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-1-4704-3912-5 |
| 830 | 0 | |a IAS/Park City mathematics series |v 13 |w (DE-604)BV010402400 |9 13 | |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u 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 |3 Inhaltsverzeichnis |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u 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 |3 Klappentext |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-016276835 | |
Datensatz im Suchindex
| _version_ | 1820889681915543552 |
|---|---|
| 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 |
| any_adam_object_boolean | 1 |
| building | Verbundindex |
| bvnumber | BV023073713 |
| callnumber-first | Q - Science |
| callnumber-label | QA167 |
| callnumber-raw | QA167 |
| callnumber-search | QA167 |
| callnumber-sort | QA 3167 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 170 |
| ctrlnum | (OCoLC)150256503 (DE-599)DNB 2007060782 |
| dewey-full | 516/.13 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516/.13 |
| dewey-search | 516/.13 |
| dewey-sort | 3516 213 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>00000nam a2200000zcb4500</leader><controlfield tag="001">BV023073713</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20240527</controlfield><controlfield tag="007">t|</controlfield><controlfield tag="008">080110s2007 xxuad|| |||| 00||| eng d</controlfield><datafield tag="010" ind1=" " ind2=" "><subfield code="a">2007060782</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">0821837362</subfield><subfield code="c">alk. paper</subfield><subfield code="9">0-8218-3736-2</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9780821837368</subfield><subfield code="9">978-0-8218-3736-8</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)150256503</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)DNB 2007060782</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">aacr</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="044" ind1=" " ind2=" "><subfield code="a">xxu</subfield><subfield code="c">US</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-703</subfield><subfield code="a">DE-83</subfield><subfield code="a">DE-11</subfield><subfield code="a">DE-188</subfield><subfield code="a">DE-824</subfield></datafield><datafield tag="050" ind1=" " ind2="0"><subfield code="a">QA167</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">516/.13</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 170</subfield><subfield code="0">(DE-625)143221:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">05-06</subfield><subfield code="2">msc</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">52-06</subfield><subfield code="2">msc</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Geometric combinatorics</subfield><subfield code="c">Ezra Miller ... ed.</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Providence, RI</subfield><subfield code="b">American Math. Soc.</subfield><subfield code="c">2007</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">[Princeton, NJ]</subfield><subfield code="b">Inst. for Advanced Study</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">XII, 691 S.</subfield><subfield code="b">Ill., graph. Darst.</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="1" ind2=" "><subfield code="a">IAS/Park City mathematics series</subfield><subfield code="v">13</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Analyse combinatoire</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Géométrie combinatoire</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Combinatorial geometry</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Combinatorial analysis</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Kombinatorische Geometrie</subfield><subfield code="0">(DE-588)4140733-7</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="655" ind1=" " ind2="7"><subfield code="0">(DE-588)1071861417</subfield><subfield code="a">Konferenzschrift</subfield><subfield code="2">gnd-content</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Kombinatorische Geometrie</subfield><subfield code="0">(DE-588)4140733-7</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Miller, Ezra</subfield><subfield code="e">Sonstige</subfield><subfield code="4">oth</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Erscheint auch als</subfield><subfield code="n">Online-Ausgabe</subfield><subfield code="z">978-1-4704-3912-5</subfield></datafield><datafield tag="830" ind1=" " ind2="0"><subfield code="a">IAS/Park City mathematics series</subfield><subfield code="v">13</subfield><subfield code="w">(DE-604)BV010402400</subfield><subfield code="9">13</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">Digitalisierung UB Bayreuth</subfield><subfield code="q">application/pdf</subfield><subfield code="u">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</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">Digitalisierung UB Bayreuth</subfield><subfield code="q">application/pdf</subfield><subfield code="u">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</subfield><subfield code="3">Klappentext</subfield></datafield><datafield tag="943" ind1="1" ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-016276835</subfield></datafield></record></collection> |
| genre | (DE-588)1071861417 Konferenzschrift gnd-content |
| genre_facet | Konferenzschrift |
| id | DE-604.BV023073713 |
| illustrated | Illustrated |
| index_date | 2024-07-02T19:34:07Z |
| indexdate | 2025-01-10T19:02:01Z |
| institution | BVB |
| isbn | 0821837362 9780821837368 |
| language | English |
| lccn | 2007060782 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016276835 |
| oclc_num | 150256503 |
| open_access_boolean | |
| owner | DE-703 DE-83 DE-11 DE-188 DE-824 |
| owner_facet | DE-703 DE-83 DE-11 DE-188 DE-824 |
| physical | XII, 691 S. Ill., graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | American Math. Soc. Inst. for Advanced Study |
| record_format | marc |
| series | IAS/Park City mathematics series |
| series2 | IAS/Park City mathematics series |
| 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 |