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 | Orlik P. Algebraic combinatorics: lectures at a summer school, Nordfjordeid, Norway, June, 2003 / Orlik P., Welker V. - Berlin; London: Springer, 2007. - 177 p. - ISBN 3540683755; ISBN 9783540683759
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PART I. LECTURES ON ARRANGEMENTS
Introduction .................................................... 5
1. Algebraic Combinatorics ..................................... 13
1.1. Chamber Counting ....................................... 13
1.2. Ranking Patterns ....................................... 17
1.3. Random Walks ........................................... 19
1.4. The Orlik-Solomon Algebra .............................. 23
1.5. The NBC Complex ........................................ 27
1.6. The Aomoto Complex ..................................... 33
1.7. Combinatorial Types .................................... 40
1.8. Formal Connections ..................................... 41
1.9. Multiplicities ......................................... 43
1.10.Ideal Invariance ....................................... 47
1.11.Examples ............................................... 52
1.12.Exercises .............................................. 58
2. Applications ................................................ 61
2.1. Topology ............................................... 61
2.2. Local System Cohomology ................................ 64
2.3. Resonance .............................................. 67
2.4. Moduli Spaces .......................................... 68
2.5. Gauss-Manin Connections ................................ 70
2.6. Exercises .............................................. 76
References ..................................................... 77
PART II.DISCRETE MORSE THEORY AND FREE RESOLUTIONS
1. Introduction ................................................ 83
1.1. Overview ............................................... 83
1.2. Enumerative and Algebraic Invariants of Simplicial
Complexes .............................................. 85
1.3. Cohen-Macaulay Simplicial Complexes .................... 92
1.4. Some Open Problems in the Field ........................ 97
2. Basic Definitions and Examples ............................. 107
2.1. Multigraded Free Resolutions .......................... 107
2.2. Basics of CW-Complexes ................................ 113
2.3. Basics of Cellular Homology ........................... 117
2.4. Cellular Chain Complexes and Cellular Resolutions ..... 120
2.5. Co-Artinian Monomial Modules. 123
3. Cellular Resolution ........................................ 125
3.1. When Does a CW-Complex Support a Cellular
Resolution ? .......................................... 125
3.2. Reading off The Betti Numbers ......................... 128
3.3. Examples of Cellular Resolutions ...................... 131
4. Discrete Morse Theory ...................................... 145
4.1. Forman's Discrete Morse Theory ........................ 146
4.2. Discrete Morse Theory for Graded CW-Complexes ......... 148
4.3. Minimizing Cellular Resolutions using Discrete
Morse Theory .......................................... 158
4.4. The Morse Differential ................................ 159
References .................................................... 169
Index ......................................................... 173
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