Proceedings of Gökova Geometry-Topology Conference 2005

01 Seven short stories on blowups and resolutions......Page 1
1. Introduction and examples......Page 2
2. Choosing the centers of blowup......Page 8
3. Transforms of ideals......Page 13
4. Properties of blowups......Page 17
5. Improving singularities by blowups......Page 22
6. The induction argument of resolution......Page 26
7. The resolution theorem and its proof......Page 33
1. Stable results......Page 49
2. Ambient results......Page 50
3. Real algebraic characteristic numbers......Page 51
4. Transcendental manifolds......Page 55
References......Page 58
1. Introduction......Page 59
2. Sheaves on fans......Page 64
3. The main construction on sheaves......Page 67
References......Page 74
04 Some remarks on G2-structures......Page 75
2. Algebra......Page 76
3. G2-structures......Page 83
4. Frame Bundle Calculations......Page 88
5. The Torsion-free Case......Page 100
6. Deformation and Evolution of G2-structures......Page 102
References......Page 108
1. Introduction......Page 110
2.1. Riemannian holonomy groups......Page 111
2.2. The holonomy group G2......Page 112
2.3. The holonomy group Spin(7)......Page 113
2.5. Relations between G2, Spin(7) and SU(m)......Page 114
3. Constructing G2-manifolds from orbifolds T7/......Page 115
3.2. Step 2: Resolving the singularities......Page 116
3.3. Step 3: Finding G2-structures with small torsion......Page 117
3.4. Step 4: Deforming to a torsion-free G2-structure......Page 118
3.5. Other constructions of compact G2-manifolds......Page 120
4. Compact Spin(7)-manifolds from Calabi--Yau 4-orbifolds......Page 121
4.1. Step 1: An example......Page 123
4.2. Step 3: Resolving R8/G......Page 124
4.3. Conclusions......Page 125
5.1. Calibrations and calibrated submanifolds......Page 126
5.2. Calibrated submanifolds and special holonomy......Page 127
5.3. Associative and coassociative submanifolds......Page 128
5.4. Examples of associative 3-submanifolds......Page 129
5.5. Examples of coassociative 4-submanifolds......Page 131
5.6. Cayley 4-folds......Page 132
6.1. Parameter counting and the local equations......Page 133
6.2. Deformation theory of coassociative 4-folds......Page 135
6.3. Deformation theory of associative 3-folds and Cayley 4-folds......Page 137
References......Page 138
06 Ricci-flat deformations of asymptotically cylindrical Calabi-Yau manifolds......Page 140
1. Asymptotically cylindrical manifolds......Page 141
2. Infinitesimal Ricci-flat deformations......Page 144
3. The moduli problem and a transverse slice......Page 146
4. Infinitesimal Ricci-flat deformations of asymptotically cylindrical Kähler manifolds......Page 150
4.1. Bounded harmonic forms and logarithmic sheaves......Page 151
5. The asymptotically cylindrical Ricci-flat deformations......Page 153
6.1. Rational elliptic surfaces......Page 154
6.2. Blow-ups of Fano threefolds......Page 155
References......Page 156
1. Introduction......Page 157
2. Deformations of special Lagrangian submanifolds......Page 158
References......Page 164
1. Introduction......Page 165
Acknowledgments......Page 167
2.1. The conormal knot......Page 168
2.2. Drawing the conormal knot front......Page 169
2.3. Nonhomotopic plane curves......Page 170
2.4. Loops of plane curves......Page 173
References......Page 174
0. Introduction......Page 175
1. Open book decompositions and contact structures......Page 176
2. An open book decomposition compatible with a contact (1)-surgery......Page 177
3. An open book decomposition compatible with a rational contact surgery......Page 179
5. An application......Page 182
References......Page 186
1. Introduction......Page 187
2.1. Link diagrams and Gauss diagrams......Page 189
2.2. From Gauss diagrams to virtual links......Page 190
2.4. Moves......Page 191
2.5. Three incarnations of virtual link theory......Page 192
2.6. Twisted virtual links......Page 194
2.7. Stripping of the third dimension......Page 196
3.1. Digression on Kauffman bracket of classical links......Page 197
3.2. Kauffman state sum in terms of Gauss Diagrams......Page 198
3.3. Blunted Gauss diagrams......Page 199
3.5. Exponents in the Kauffman bracket......Page 200
4.1. Orientation of a chord diagram......Page 201
4.3. Moves of chord diagrams......Page 202
4.4. Checkerboard coloring of a classical link diagram......Page 204
4.5. On orientable surfaces......Page 205
4.6. Alternatable virtual links......Page 206
4.7. On non-orientable surfaces......Page 207
4.8. Obstruction to orientability of a chord diagram......Page 208
5.1. Khovanov homology of classical links......Page 209
5.3. Failure in the non-orientable case......Page 210
References......Page 211