Geometric and spectral analysis CRM workshops on: Geometry of Eigenvalues and Eigenfunctions, June 4 - 8, 2012, Manifolds of Metrics and Probabilistic Methods in Geometry and Analysis, July 2 - 6, 2012, Spectral Invariants on Non-compact and Singular Spac

In 2012, The Centre De Recherches Mathématiques Was At The Center Of Many Interesting Developments In Geometric And Spectral Analysis, With A Thematic Program On Geometric Analysis And Spectral Theory Followed By A Thematic Year On Moduli Spaces, Extremality And Global Invariants. This Volume Contains Original Contributions As Well As Useful Survey Articles Of Recent Developments By Participants From Three Of The Workshops Organized During These Programs: Geometry Of Eigenvalues And Eigenfunctions, Held From June 4-8, 2012; Manifolds Of Metrics And Probabilistic Methods In Geometry And Analysis, Held From July 2-6, 2012; And Spectral Invariants On Non-compact And Singular Spaces, Held From July 23-27, 2012. The Topics Covered In This Volume Include Fourier Integral Operators, Eigenfunctions, Probability And Analysis On Singular Spaces, Complex Geometry, Kähler-einstein Metrics, Analytic Torsion, And Strichartz Estimates. This Book Is Co-published With The Centre De Recherches Mathématiques. Cover 1 Title page 4 Contents 6 Preface 8 A New Proof of a Bismut-Zhang Formula for Some Class of Representations 12 1. Introduction 12 2. The idea of the proof 13 3. The Milnor metric and the Farber-Turaev torsion 15 4. The Ray-Singer norm of the Refined Analytic Torsion 18 5. The Determinant Line Bundle over the Space of Representations 20 6. The Bismut-Zhang theorem for some non-unitary representations 21 References 23 Tunneling, the Quillen Metric and Analytic Torsion for High Powers of a Holomorphic Line Bundle 26 1. Introduction 26 2. Proofs of the main results 30 3. Further remarks on tunneling and outlook 37 References 41 Simple Spectrum and Rayleigh Quotients 44 1. Introduction 44 2. The Conformal class 46 3. The Kähler class 51 Acknowledgments 54 References 55 Smooth and Singular Kähler–Einstein Metrics 56 1. Introduction 56 2. The \KE equation 58 3. \K edge geometry 60 4. Existence and non-existence 78 5. Energy functionals 91 6. The Ricci continuity method 103 7. A priori estimates for Monge–Ampère equations 110 8. The asymptotically logarithmic world 123 9. The logarithmic Calabi problem 135 Acknowledgements 138 References 138 Complex b-Manifolds 150 1. Introduction 150 2. Complex b-structures 151 3. Holomorphic vector bundles 154 4. The boundary of a complex b-manifold 156 5. Local invariants 161 6. Indicial complexes 164 7. Underlying CR complexes 170 8. Spectrum 174 9. Indicial cohomology 176 Appendix A. Totally characteristic differential operators 181 References 182 Iterative Structures on Singular Manifolds 184 Introduction 184 1. Manifolds with singularities of higher order 185 2. The cone algebra 199 3. The edge algebra 207 4. Shapiro-Lopatinskij edge-ellipticity 217 5. Iterated Mellin quantisation 221 6. Toeplitz edge problems 224 References 228 The Fundamental Gap and One-Dimensional Collapse 234 1. Motivation and results 234 2. Collapsing triangular domains 238 3. Collapsing polygonal domains 246 4. The general case 250 Acknowledgements 255 References 255 The Logarithmic Singularities of the Green Functions of the Conformal Powers of the Laplacian 258 Introduction 258 1. The Heat Kernel of the Laplace Operator 260 2. The Bergman Kernel of a Strictly Pseudoconvex Domain 263 3. Ambient Metric and Conformal Powers of the Laplacian 266 4. Local Conformal Invariants 268 5. Scattering Theory and Conformal Fractional Powers of the Laplacian 270 6. Green Functions of Elliptic Operators 274 7. Green Functions and Conformal Geometry 276 8. Proof of Theorem 7.3 279 References 281 A Symbolic Calculus for Fourier Integral Operators 286 Introduction 286 1. Notation and definitions 287 2. Canonical transformations in T*M 289 3. Fourier integral operators 293 4. Symbolic calculus for FIOs 296 References 301 Strichartz Estimates on Exterior Polygonal Domains 302 1. Introduction 302 2. Global Strichartz Estimates for the Model Problems 304 3. Local Smoothing Estimates for X and on Euclidean Cones 304 4. Proof of Theorem 1 311 5. A brief comment on nonlinear applications 312 References 315 Probabilistic Sobolev Embeddings, Applications to Eigenfunctions Estimates 318 1. Introduction 318 2. Probabilistic estimates 321 3. Eigenbases 325 4. Other manifolds 326 5. Probability calculus on spheres 328 References 328 Random Smooth Functions on Compact Riemannian Manifolds 330 1. The setup 330 2. The white noise limit of the variational complexity 334 3. Statistical reconstruction of the geometry of M 337 4. A probabilistic interpretation of the Gauss-Bonnet theorem 338 5. Outline of proofs 339 References 345 On the Exponential Decay of Laplacian Eigenfunctions in Planar Domains with Branches 348 1. Introduction 348 2. Extension for Robin boundary condition 350 3. Illustrations 353 References 358 Bounds on the Principal Frequency of the p-Laplacian 360 1. Overview of the p-Laplacian 360 2. Introduction and main results 361 3. Lower bounds involving the inradius in higher dimensions 364 4. Lower bounds involving eigenvalues of the p-Laplace operator on manifolds 367 5. Proofs 369 Acknowledgements 374 References 374 Back Cover 378