This book presents a collection of invited articles by distinguished Mathematicians on the occasion of the Platinum Jubilee Celebrations of the Indian Statistical Institute, during the year 2007. These articles provide a current perspective of different areas of research, emphasizing the major challenging issues. Given the very significant record of the Institute in research in the areas of Statistics, Probability and Mathematics, distinguished authors have very admirably responded to the invitation. Some of the articles are written keeping students and potential new entrants to an area of mathematics in mind. This volume is thus very unique and gives a perspective of several important aspects of mathematics. Use of Resultants and Approximate Roots for Doing the Jacobian Problem (S S Abhyankar) Monodromy of Principal Bundles (I Biswas & A J Parameswaran) Oligomorphic Permutation Groups (P J Cameron) Descriptive Set Theory and the Geometry of Banach Spaces (G Godefroy) Multiplicity-Free Homogeneous Operators in the Cowen Douglas Class (A Kor??nyi & G Misra) The Standard Conjectures on Algebraic Cycles (M S Narasimhan) On the Classification of Binary Shifts on the Hyperfinite II1 Factor (G L Price) Symmetric and Quasi-Symmetric Designs and Strongly Regular Graphs (S S Sane) Perturbation Determinant, Krein's Shift Function and Index Theorem (K B Sinha) Zero Cycles and Complete Intersection Points on Affine Varieties (V Srinivas) Root Numbers and Rational Points on Elliptic Curves (R Sujatha) von Neumann Algebras and Ergodic Theory (V S Sunder) Gutzmer's Formula and the Segal Bargmann Transform (S Thangavelu) Finite Translation Generalized Quadrangles (J A Thas) Super Geometry as the Basis for Super Symmetry (V S Varadarajan)
Author(s): N. S. Narasimha Sastry
Year: 2009
Language: English
Pages: 280
Contents......Page 10
Foreword......Page 6
Preface......Page 8
1.1. Introduction......Page 12
1.2. Basic Technique......Page 14
1.3. Resultants and Discriminants......Page 15
1.4. Real Numbers and Approximate Roots......Page 20
References......Page 26
2.1. Introduction......Page 28
2.2. Tannakian Category......Page 30
2.3. A Tannakian Category for a Pointed Curve......Page 33
2.4. Monodromy of a Strongly Semistable Principal Bundles......Page 37
2.5. More on Monodromy......Page 42
2.6. Bundles on Higher Dimensional Varieties......Page 44
References......Page 45
3.1. Introduction......Page 48
3.1.1. Permutation groups......Page 49
3.1.2. Oligomorphic permutation groups......Page 51
3.1.3. Topology......Page 52
3.1.4. Cycle index......Page 53
3.2. Connections......Page 54
3.2.1. Model theory......Page 55
3.2.2. Combinatorial enumeration......Page 56
3.3.1. Direct and wreath products......Page 58
3.3.2. Other examples......Page 61
3.4. Growth Rates......Page 64
3.5. Graded Algebras......Page 66
3.6. Group Structure......Page 69
References......Page 71
4.1. Introduction......Page 74
4.2. A Short Survey on Analytic Sets......Page 76
4.3. Bossard’s Coding of Separable Banach Spaces......Page 80
4.4. Coanalytic Ranks......Page 84
4.5. A New Direction: The Converse Statements......Page 88
References......Page 91
5. Multiplicity-Free Homogeneous Operators in the Cowen- Douglas Class A. Kor´anyi and G. Misra......Page 94
5.1. Background Material......Page 95
5.2. Computation of the Multipliers for the Unit Disc......Page 100
5.3. Conditions Imposed by the Reproducing Kernel......Page 103
5.4. The Multiplicity-Free Case......Page 104
5.5. Examples......Page 111
References......Page 112
6.1. The Case of Complex Projective Varieties......Page 114
6.2. Standard Conjectures in Abstract Algebraic Geometry......Page 117
References......Page 120
7.1. Introduction......Page 122
7.2. Preliminaries......Page 124
7.3. Bitstreams and Polynomials......Page 128
7.4. Counting Polynomials with Symmetry......Page 132
7.5. Conjugacy Classes of Binary Shifts......Page 141
References......Page 146
8.1. Introduction and Preliminaries......Page 148
8.2. Symmetric Designs......Page 149
8.3. Strongly Regular Graphs......Page 156
8.4. Quasi-Symmetric Designs......Page 161
References......Page 166
9.1. Introduction......Page 168
9.2. Perturbation Determinant......Page 169
9.3. Witten Index and Its Invariance......Page 173
9.4. Krein’s Shift Function......Page 177
9.5. Application to Quantum Mechanics and Generalized Levinson’s Theorem......Page 179
References......Page 182
10. Zero Cycles and Complete Intersection Points on A.ne Varieties V. Srinivas......Page 184
References......Page 195
11.1. Elliptic Curves and the Birch and Swinnerton-Dyer Conjecture......Page 196
11.2. Congruent Number Problem......Page 197
11.3. Root Numbers and the Parity Conjecture......Page 200
11.4. Recent Results......Page 202
11.5. Examples and Applications......Page 204
References......Page 207
12. von Neumann Algebras and Ergodic Theory V. S. Sunder......Page 210
References......Page 218
13.1. Introduction......Page 220
13.2. Segal-Bargmann Transform on the Heisenberg Group......Page 224
13.3. Gutzmer Formulas and Their Applications......Page 228
References......Page 231
14.1.1. Finite generalized quadrangles......Page 234
14.1.2. Grids and dual grids......Page 235
14.1.3. The classical generalized quadrangles......Page 236
14.1.4. Ovals, hyperovals and ovoids......Page 237
14.1.5. The generalized quadrangles T2(O) and T3(O) of Tits......Page 238
14.1.6. The generalized quadrangles T. 2 (O)......Page 239
14.2.1. Translation generalized quadrangles......Page 240
14.2.2. The kernel of a translation generalized quadrangle......Page 241
14.2.3. T(n,m, q)s and translation generalized quadrangles......Page 242
14.3.1. Properties of O(n,m, q)......Page 244
14.3.2. Properties of pseudo-ovals......Page 245
14.4. Eggs O(n, 2n, q): Fundamental Results and Characterizations......Page 246
14.4.2. Fundamental results on eggs......Page 247
14.5.1. Old results......Page 249
14.5.2. New results......Page 251
References......Page 255
15.1. Introduction......Page 258
15.2. Super Geometry......Page 260
15.3. Super Symmetric Extensions of Relativistic Theories and Super Poincar´e Groups......Page 263
15.4. Classification of Super Particles......Page 265
15.5. UIR’s of Regular Super Semi Direct Products......Page 267
References......Page 271
Author Index......Page 274
Subject Index......Page 278
Contents of Part I......Page 280
