| Applications of Representation Theory to Harmonic Analysis of Lie Groups (and Vice Versa) | p. 1 |
| Basic Facts of Harmonic Analysis on Semisimple Groups and Symmetric Spaces | p. 2 |
| Structure of Semisimple Lie Algebras | p. 2 |
| Decompositions of Semisimple Lie Groups | p. 4 |
| Parabolic Subgroups | p. 5 |
| Spaces of Homogeneous Functions on G | p. 6 |
| The Plancherel Formula | p. 8 |
| The Equations of Mathematical Physics on Symmetric Spaces | p. 10 |
| Spherical Analysis on Symmetric Spaces | p. 10 |
| Harmonic Analysis on Semisimple Groups and Symmetric Spaces | p. 12 |
| Regularity of the Laplace-Beltrami Operator | p. 16 |
| Approaches to the Heat Equation | p. 18 |
| Estimates for the Heat and Laplace Equations | p. 18 |
| Approaches to the Wave and Schrodinger Equations | p. 20 |
| Further Results | p. 21 |
| The Vanishing of Matrix Coefficients | p. 22 |
| Some Examples in Representation Theory | p. 22 |
| Matrix Coefficients of Representations of Semisimple Groups | p. 24 |
| The Kunze-Stein Phenomenon | p. 27 |
| Property T | p. 28 |
| The Generalised Ramanujan-Selberg Property | p. 29 |
| More General Semisimple Groups | p. 31 |
| Graph Theory and its Riemannian Connection | p. 31 |
| Cayley Graphs | p. 32 |
| An Example Involving Cayley Graphs | p. 33 |
| The Field of p-adic Numbers | p. 34 |
| Lattices in Vector Spaces over Local Fields | p. 35 |
| Adeles | p. 36 |
| Further Results | p. 37 |
| Carnot-Caratheodory Geometry and Group Representations | p. 38 |
| A Decomposition for Real Rank One Groups | p. 38 |
| The Conformal Group of the Sphere in R[superscript n] | p. 38 |
| The Groups SU(1, n + 1) and Sp(1, n + 1) | p. 41 |
| References | p. 46 |
| Ramifications of the Geometric Langlands Program | p. 51 |
| Introduction | p. 51 |
| The Unramified Global Langlands Correspondence | p. 56 |
| Classical Local Langlands Correspondence | p. 61 |
| Langlands Parameters | p. 61 |
| The Local Langlands Correspondence for GL[subscript n] | p. 62 |
| Generalization to Other Reductive Groups | p. 63 |
| Geometric Local Langlands Correspondence over C | p. 64 |
| Geometric Langlands Parameters | p. 64 |
| Representations of the Loop Group | p. 65 |
| From Functions to Sheaves | p. 66 |
| A Toy Model | p. 68 |
| Back to Loop Groups | p. 70 |
| Center and Opers | p. 71 |
| Center of an Abelian Category | p. 71 |
| Opers | p. 73 |
| Canonical Representatives | p. 75 |
| Description of the Center | p. 76 |
| Opers vs. Local Systems | p. 77 |
| Harish-Chandra Categories | p. 81 |
| Spaces of K-Invariant Vectors | p. 81 |
| Equivariant Modules | p. 82 |
| Categorical Hecke Algebras | p. 83 |
| Local Langlands Correspondence: Unramified Case | p. 85 |
| Unramified Representations of G(F) | p. 85 |
| Unramified Categories [characters not reproducible]-Modules | p. 87 |
| Categories of G[[t]]-Equivariant Modules | p. 88 |
| The Action of the Spherical Hecke Algebra | p. 90 |
| Categories of Representations and D-Modules | p. 92 |
| Equivalences Between Categories of Modules | p. 96 |
| Generalization to other Dominant Integral Weights | p. 98 |
| Local Langlands Correspondence: Tamely Ramified Case | p. 99 |
| Tamely Ramified Representations | p. 99 |
| Categories Admitting ([characters not reproducible], I) Harish-Chandra Modules | p. 103 |
| Conjectural Description of the Categories of ([characters not reproducible], I) Harish-Chandra Modules | p. 105 |
| Connection between the Classical and the Geometric Settings | p. 109 |
| Evidence for the Conjecture | p. 115 |
| Ramified Global Langlands Correspondence | p. 117 |
| The Classical Setting | p. 117 |
| The Unramified Case, Revisited | p. 120 |
| Classical Langlands Correspondence with Ramification | p. 122 |
| Geometric Langlands Correspondence in the Tamely Ramified Case | p. 122 |
| Connections with Regular Singularities | p. 126 |
| Irregular Connections | p. 130 |
| References | p. 132 |
| Equivariant Derived Category and Representation of Real Semisimple Lie Groups | p. 137 |
| Introduction | p. 137 |
| Harish-Chandra Correspondence | p. 138 |
| Beilinson-Bernstein Correspondence | p. 140 |
| Riemann-Hilbert Correspondence | p. 141 |
| Matsuki Correspondence | p. 142 |
| Construction of Representations of G[subscript R] | p. 143 |
| Integral Transforms | p. 146 |
| Commutativity of Fig. 1 | p. 147 |
| Example | p. 148 |
| Organization of the Note | p. 151 |
| Derived Categories of Quasi-abelian Categories | p. 152 |
| Quasi-abelian Categories | p. 152 |
| Derived Categories | p. 154 |
| t-Structure | p. 156 |
| Quasi-equivariant D-Modules | p. 158 |
| Definition | p. 158 |
| Derived Categories | p. 162 |
| Sumihiro's Result | p. 163 |
| Pull-back Functors | p. 167 |
| Push-forward Functors | p. 168 |
| External and Internal Tensor Products | p. 170 |
| Semi-outer Hom | p. 171 |
| Relations of Push-forward and Pull-back Functors | p. 172 |
| Flag Manifold Case | p. 175 |
| Equivariant Derived Category | p. 176 |
| Introduction | p. 176 |
| Sheaf Case | p. 176 |
| Induction Functor | p. 179 |
| Constructible Sheaves | p. 179 |
| D-module Case | p. 180 |
| Equivariant Riemann-Hilbert Correspondence | p. 181 |
| Holomorphic Solution Spaces | p. 182 |
| Introduction | p. 182 |
| Countable Sheaves | p. 183 |
| C[superscript infinity]-Solutions | p. 185 |
| Definition of RHom[superscript top] | p. 186 |
| DFN Version | p. 189 |
| Functorial Properties of RHom[superscript top] | p. 190 |
| Relation with the de Rham Functor | p. 192 |
| Whitney Functor | p. 194 |
| Whitney Functor | p. 194 |
| The Functor RHom[characters not reproducible] | p. 195 |
| Elliptic Case | p. 196 |
| Twisted Sheaves | p. 197 |
| Twisting Data | p. 197 |
| Twisted Sheaf | p. 198 |
| Morphism of Twisting Data | p. 199 |
| Tensor Product | p. 200 |
| Inverse and Direct Images | p. 200 |
| Twisted Modules | p. 201 |
| Equivariant Twisting Data | p. 201 |
| Character Local System | p. 202 |
| Twisted Equivariance | p. 202 |
| Twisting Data Associated with Principal Bundles | p. 203 |
| Twisting (D-module Case) | p. 204 |
| Ring of Twisted Differential Operators | p. 205 |
| Equivariance of Twisted Sheaves and Twisted D-modules | p. 207 |
| Riemann-Hilbert Correspondence | p. 207 |
| Integral Transforms | p. 208 |
| Convolutions | p. 208 |
| Integral Transform Formula | p. 209 |
| Application to the Representation Theory | p. 210 |
| Notations | p. 210 |
| Beilinson-Bernstein Correspondence | p. 212 |
| Quasi-equivariant D-modules on the Symmetric Space | p. 214 |
| Matsuki Correspondence | p. 216 |
| Construction of Representations | p. 217 |
| Integral Transformation Formula | p. 219 |
| Vanishing Theorems | p. 221 |
| Preliminary | p. 221 |
| Calculation (I) | p. 222 |
| Calculation (II) | p. 224 |
| Vanishing Theorem | p. 226 |
| References | p. 229 |
| List of Notations | p. 231 |
| Index | p. 233 |
| Amenability and Margulis Super-Rigidity | p. 235 |
| Introduction | p. 235 |
| Amenability for Locally Compact Groups | p. 236 |
| Definition, Examples, and First Characterizations | p. 236 |
| Stability Properties | p. 239 |
| Lattices in Locally Compact Groups | p. 240 |
| Reiter's Property (P[subscript 1]) | p. 241 |
| Reiter's Property (P[subscript 2]) | p. 242 |
| Amenability in Riemannian Geometry | p. 244 |
| Measurable Ergodic Theory | p. 244 |
| Definitions and Examples | p. 244 |
| Moore's Ergodicity Theorem | p. 247 |
| The Howe-Moore Vanishing Theorem | p. 249 |
| Margulis' Super-rigidity Theorem | p. 252 |
| Statement | p. 252 |
| Mostow Rigidity | p. 252 |
| Ideas to Prove Super-rigidity, k = R | p. 253 |
| Proof of Furstenberg's Proposition 4.1 - Use of Amenability | p. 255 |
| Margulis' Arithmeticity Theorem | p. 256 |
| References | p. 257 |
| Unitary Representations and Complex Analysis | p. 259 |
| Introduction | p. 259 |
| Compact Groups and the Borel-Weil Theorem | p. 264 |
| Examples for SL(2, R) | p. 272 |
| Harish-Chandra Modules and Globalization | p. 274 |
| Real Parabolic Induction and the Globalization Functors | p. 284 |
| Examples of Complex Homogeneous Spaces | p. 294 |
| Dolbeault Cohomology and Maximal Globalizations | p. 302 |
| Compact Supports and Minimal Globalizations | p. 318 |
| Invariant Bilinear Forms and Maps between Representations | p. 327 |
| Open Questions | p. 341 |
| References | p. 343 |
| Quantum Computing and Entanglement for Mathematicians | p. 345 |
| The Basics | p. 346 |
| Basic Quantum Mechanics | p. 346 |
| Bits | p. 348 |
| Qubits | p. 349 |
| References | p. 350 |
| Quantum Algorithms | p. 351 |
| Quantum Parallelism | p. 351 |
| The Tensor Product Structure of n-qubit Space | p. 352 |
| Grover's Algorithm | p. 353 |
| The Quantum Fourier Transform | p. 354 |
| References | p. 355 |
| Factorization and Error Correction | p. 355 |
| The Complexity of the Quantum Fourier Transform | p. 356 |
| Reduction of Factorization to Period Search | p. 359 |
| Error Correction | p. 360 |
| References | p. 362 |
| Entanglement | p. 362 |
| Measures of Entanglement | p. 363 |
| Three Qubits | p. 365 |
| Measures of Entanglement for Two and Three Qubits | p. 367 |
| References | p. 368 |
| Four and More Qubits | p. 369 |
| Four Qubits | p. 369 |
| Some Hilbert Series of Measures of Entanglement | p. 374 |
| A Measure of Entanglement for n Qubits | p. 374 |
| References | p. 376 |
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