Buildings, Finite Geometries and Groups: Proceedings of a Satellite Conference, International Congress of Mathematicians, Hyderabad, India, 2010 ... (Springer Proceedings in Mathematics (10))

1. On Characterizing Designs By Their Codes (B. Bagchi).- 2. The Geometry of Extremal Elements in a Lie Algebra (A.M. Cohen).- 3. Properties of a 27-dimensional Space of Symmetric Bilinear Forms Acted on by E6 (R. Gow).- 4. On the Geometry of Global Function Fields, the Riemann-Roch Theorem, and Finiteness Properties of S-arithmetic Groups (R. Gramlich).- 5. Some Remarks on Two-Transitive Permutation Groups as Multiplication Groups of Quasigroups (G. Hiss, F. Lubeck).- 6. Curve Complexes Versus Tits Buildings: Structures and Applications (Lizhen Ji).- 7. On Isotypies Between Galois Conjugate Blocks (R. Kessar).- 8. Representations of Unitriangular Groups (T. Le, K. Magaard).- 9. Hermitian Vernonesean Caps (J. Schillewaert, H. Van Maldeghem).- 10. On a Class of c.F4-geometries (A. Pasini).- 11. Buildings and Kac-Moody Groups (B. Remy).- 12. Some Equations Over Finite Fields Related to Simple Groups of Suzuki and Ree Types (N.S. Narasimha Sastry).- 13. Oppositeness in Buildings and Simple Modules for Finite Groups of Lie Type (P. Sin).- 14. Modular Representations, Old and New (B. Srinivasan).- 15. The Use of Blocking Sets in Galois Geometries and in Related Research Areas (V. Pepe, L. Storme).- 16. Quadratic Actions (F.G. Timmesfeld).- Problem Set Cover......Page 1 Series: Springer Proceedings in Mathematics, Volume 10......Page 3 Buildings, Finite Geometries and Groups: Proceedings of a Satellite Conference, International Congress of Mathematicians, Hyderabad, India, 2010......Page 4 Copyright......Page 5 Preface......Page 6 Talks given in the Conference......Page 8 Participants......Page 10 Contents......Page 12 On Characterizing Designs by Their Codes......Page 14 1 Introduction......Page 15 2 Generalized Polygons......Page 16 3 Dual Codes of 2-Designs......Page 18 4 Codes of Finite Projective Spaces......Page 20 5 Projective Planes of Prime Order......Page 24 References......Page 27 1 Extremal Elements in Lie Algebras......Page 28 2 Geometry from Extremal Elements......Page 33 3 Root Filtration Spaces......Page 35 4 Parapolar Spaces......Page 41 5 Degenerate Root Filtration Spaces......Page 43 6 Conclusion......Page 45 References......Page 47 1 Introduction......Page 50 3 The Exceptional 27-Dimensional Jordan Algebra and the Group E6......Page 51 4 Subspaces with Special Rank Properties......Page 57 References......Page 59 1 Introduction......Page 62 2 Projective Varieties......Page 63 3 Curves Over Finite Fields Considered as Varieties......Page 64 4 Geometry of Numbers......Page 68 5 Curves Over Finite Fields Considered as Schemes......Page 70 6 Reduction Theory for Rationally Trivial Group Schemes......Page 76 7 Reduction Theory for Reductive Groups Over the Adèles......Page 81 8 Filtrations of Euclidean Buildings......Page 85 9.1 Finiteness Properties of S-Arithmetic Groups......Page 89 9.2 Isoperimetric Properties of S-Arithmetic Groups......Page 91 References......Page 92 1 Introduction......Page 94 2 Permutation Groups......Page 95 3 A Sufficient Condition......Page 97 4 Two-Transitive Groups......Page 98 4.1 The Almost Simple Two-Transitive Groups......Page 99 4.2 The Two-Transitive Groups with Elementary Abelian Socle......Page 101 5 Searching for Quasigroups By Computer......Page 103 References......Page 104 Curve Complexes Versus Tits Buildings: Structures and Applications......Page 106 1.2 The Origin of Tits Buildings......Page 107 1.3 The Origin of Curve Complexes......Page 108 2 Definition of Buildings......Page 109 2.1 A Geometric Definition of Tits Buildings via Symmetric Spaces......Page 110 2.2 Axioms for Spherical Buildings via Apartments......Page 111 2.3 Euclidean and Hyperbolic Buildings......Page 113 2.4 Rational Tits Buildings of Linear Algebraic Groups......Page 114 3 Definition of Curve Complexes......Page 115 4 Geometric and Topological Properties of Buildings......Page 117 5 Geometric and Topological Properties of Curve Complexes......Page 120 6.2 Mostow Strong Rigidity and Generalizations......Page 123 6.3 Compactifications of Locally Symmetric Spaces......Page 125 6.4 Cohomological Dimension and Duality Properties of Arithmetic Groups......Page 129 6.5 Simplicial Volumes of Locally Symmetric Spaces......Page 131 6.6 Asymptotic Cones of Symmetric Spaces and Locally Symmetric Spaces......Page 133 6.7 Applications of Euclidean Buildings......Page 134 6.8 Compactifications of Euclidean Buildings......Page 135 7 Applications of Curve Complexes......Page 138 7.1 Automorphism Groups of Curve Complexes......Page 139 7.3 Ending Lamination Conjecture of Thurston......Page 140 7.4 Quasi-Isometric Rigidity of Mapping Class Groups......Page 144 7.5 Finite Asymptotic Dimension of Mapping Class Groups and Novikov Conjectures......Page 145 7.7 Heegaard Splittings and Hempel Distance of 3-Manifolds......Page 149 7.8 Partial Compactifications of Teichmüller Spaces and Their Boundaries......Page 150 7.9 Cohomological Dimension and Duality Properties of Mapping Class Groups......Page 153 7.10 Tangent Cones at Infinity of Teichmüller Spaces, Moduli Spaces, and Mapping Class Groups......Page 154 7.11 Simplicial Volumes of Moduli Spaces......Page 155 7.12 Action of Modg,n on C(Sg,n) and Applications......Page 156 References......Page 158 1 Introduction......Page 166 2 Notation and Definitions......Page 167 2.2 Local Structure of Blocks......Page 168 2.3 Perfect Isometries and Isotypies......Page 169 3 Proofs......Page 170 References......Page 175 1 Introduction......Page 176 2 Character Degrees of UXr(q)......Page 177 3 The Number of Characters of a Given Degree......Page 180 4 Construction of Characters......Page 182 5 Outlook......Page 185 References......Page 186 1 Introduction......Page 188 2.1 Axiomatization of Projective Spaces......Page 189 2.2 Quadric Veronesean Caps......Page 190 2.3 Hermitian Veronesean caps......Page 191 3.1 The Projective Space Associated with the Cap......Page 192 3.2 The Basic Step......Page 194 3.3 The General Case......Page 197 4 An Application of Hermitian Veronesean Caps......Page 198 5 Another Application......Page 202 References......Page 204 1 Introduction......Page 206 1.1 Organization of the Paper......Page 207 1.2 Notation and Conventions......Page 208 2 Looking for the Right Hypotheses......Page 210 3.1 Preliminaries......Page 213 3.2 G(Fi22) and G(3Fi22) (t = 1)......Page 214 3.4 G(226F4(2)), G(E6(2)), G(2E6(2)) and G(32E6(2)) (t = 2)......Page 215 4.1 The Case of t=1......Page 216 4.2 The Case of t = 4......Page 219 4.3 The Case of t = 2......Page 220 5 A Geometric Construction of G(E6(2)) and More Af.F4-Geometries......Page 223 5.1 Preliminaries on Buildings of Type E6 and F4......Page 224 5.2 A Construction for G(E6(2))......Page 227 5.3 More Af.F4-Geometries......Page 230 6.1 Representations and Extensions......Page 231 6.2 GF(2)-Representations......Page 233 6.3 Projective Embeddings and Affine Extensions......Page 234 7.1 Affine Polar Spaces......Page 235 7.2 Standard Quotients of Affine Polar Spaces......Page 236 7.3 Minimal Standard Quotients as Affine Extensions or Tangent Geometries......Page 237 8.1 Shrinkings......Page 238 8.2 Geometries at Infinity......Page 239 9 A Lemma on Flag-Transitive F4-Geometries......Page 240 10 The Graph (F) of an F4-Building F......Page 241 References......Page 242 1 Introduction......Page 244 1.2 Conventions......Page 248 2.1 (Simplicial) Buildings......Page 249 2.2 Analogies with Lie Groups and Exotic Examples......Page 250 2.3 Kac-Moody Groups and Kac-Moody Buildings......Page 251 3 Simplicity and Rigidity......Page 252 3.2 Simplicity......Page 253 3.3 Rigidity......Page 255 4.1 Amenability......Page 257 4.2 Quasi-Homomorphisms......Page 259 References......Page 260 1 Introduction......Page 264 2 Preliminaries......Page 270 3 Structure of R When X=F4( q)......Page 271 4 Proofs......Page 277 5 Remarks......Page 283 References......Page 284 1 Introduction......Page 286 2 Oppositeness in Buildings......Page 288 3 Some Lemmas on Double Cosets......Page 290 4.1 Fundamental Endomorphisms of F......Page 291 4.2 Proof of Theorem 4.1......Page 292 5 Highest Weights......Page 294 6 Examples......Page 296 References......Page 298 1 Introduction......Page 300 2 Finite Groups......Page 301 3 Finite Reductive Groups......Page 304 4 Symmetric Groups, General Linear Groups, Finite Reductive Groups......Page 306 5 Weyl Groups, Cyclotomic Hecke Algebras, q-Schur Algebras......Page 308 6 Lie Algebras......Page 309 7 Modular Representations, New......Page 310 8 Introducing Lie Theory......Page 311 10 KLR-Algebras: The Diagrammatic Approach......Page 313 11 Graded Representation Theory......Page 314 13 End of Story?......Page 315 References......Page 316 1 Definitions and Introductory Results......Page 318 2 Maximal Partial Spreads......Page 321 3 Blocking Sets and Coding Theory......Page 324 4.1 The Griesmer Bound......Page 326 4.2 Minihypers and the Belov–Logachev–Sandimirov Construction......Page 328 5 Extension Results......Page 331 6 Blocking Sets and Cryptography......Page 332 7.1 Covers in Galois Geometries......Page 333 7.2 Minihypers and i-Tight Sets......Page 334 7.3 t-Fold k-Blocking Sets in PG(n,q)......Page 335 7.4 Blocking Sets and Semifields......Page 336 7.5 Open Problems......Page 337 References......Page 338 1 History......Page 342 2 Connections with Abstract Root Subgroups......Page 343 3 Quadratic Action Without Commuting Root Groups......Page 346 4 Quadratic Modules......Page 348 References......Page 349 Two Problems on Finite Sets......Page 350 Embedding the Dual of a Non-Embeddable Polar Space......Page 351 Known Examples of Small Minimal Blocking Sets......Page 352 ``Which (Families of) Finite Groups Admit Neat/Natural Gelfand Model?"......Page 353 This volume collects articles inspired by the Proceedings of the ICM 2010 Satellite Conference on "Buildings, Finite Geometries and Groups" organized at the Indian Statistical Institute, Bangalore, from August 29 - 31, 2010. These contributors include some of the most active researchers in areas related to finite simple groups, Chevalley groups and their generalizations: theory of buildings, finite incidence geometries, modular representations, Lie theory, and more. Contributions reflect the current major trends in research in the geometric and combinatorial aspects of the study of these groups. The unique perspective that the authors bring to their articles on current developments and major problems in their area is expected to be very useful to research mathematicians, graduate students and potential new entrants to these fields This is the Proceedings of the ICM 2010 Satellite Conference on “Buildings, Finite Geometries and Groups” organized at the Indian Statistical Institute, Bangalore, during August 29 – 31, 2010. This is a collection of articles by some of the currently very active research workers in several areas related to finite simple groups, Chevalley groups and their generalizations: theory of buildings, finite incidence geometries, modular representations, Lie theory, etc. These articles reflect the current major trends in research in the geometric and combinatorial aspects of the study of these groups. The unique perspective the authors bring in their articles on the current developments and the major problems in their area is expected to be very useful to research mathematicians, graduate students and potential new entrants to these areas.