Probabilistic Methods in Geometry, Topology and Spectral Theory.

Cover -- Title page -- Contents -- Preface -- A geometric treatment of log-correlated Gaussian free fields -- 1. Introduction -- 2. Abstract Wiener Space and Gaussian Free Fields -- 3. Regularization of GFF on ℝⁿ -- 4. Random Measure and KPZ in ℝ³ -- References -- Tangent nodal sets for random spherical harmonics -- 1. Introduction -- 2. Geometric Preliminaries -- 3. Calculating the Expectation -- Appendix A. Computation of Covariance Matrix -- References -- Formal Zeta function expansions and the frequency of Ramanujan graphs -- 1. Introduction -- 2. Main Results -- 3. Graph Theoretic Preliminaries -- 3.1. Graphs and Morphisms -- 4. Variants of the Zeta Function -- 5. The Expected Value of \cL_{ } -- 6. A Simpler Variant of the ᵢ and \cPᵢ -- 7. Random Graph Covering Maps and Other Models -- 8. Numerical Experiments -- References -- Rank and Bollobás-Riordan polynomials: Coefficient measures and zeros -- 1. Introduction -- 2. Tutte polynomial -- 3. Distribution of the coefficients: a priori results -- 4. Numerical experiments on the coefficient measure -- 5. Zeros of Tutte polynomials -- 6. Ribbon graphs: summary -- 7. Ribbon graphs and "left-hand turn" surfaces -- 8. Bollobás-Riordan polynomials -- 9. Random graphs with orientations -- 10. Convergence of the coefficient measures of Bollobás-Riordan polynomials -- 11. Numerical investigations: coefficients and zeros of Bollobás-Riordan polynomials -- 12. Conclusion -- Appendix: Computer code -- Acknowledgements -- References -- The Brownian motion on (ℝ) and quasi-local theorems -- 1. Introduction -- 2. Diffusion on \Aff(\R) and similar groups -- 3. Approximation of diffusion by random walks and associated return probability estimates -- 4. Quasi-Local Theorems -- Acknowledgments -- References -- Quantum limits of Eisenstein series in ℍ³ -- 1. Introduction.

2. Spectral Theory in \pslok\\uphs -- 3. Proofs -- References -- Observability and quantum limits for the Schrödinger equation on ^{ } -- 1. Introduction -- 2. Statement of the main results -- 3. Semiclassical measures and their invariance properties -- Acknowledgments -- References -- Random nodal lengths and Wiener chaos -- 1. Introduction -- 2. Random nodal lengths -- 3. Chaotic expansions -- 4. On the proof of Theorem 2.2 -- 5. Further related work -- Acknowledgments -- References -- Entropy bounds and quantum unique ergodicity for Hecke eigenfunctions on division algebras -- 1. Introduction -- 2. Notation -- 3. Bounds on the mass of tubes -- 4. Diophantine Lemmata -- 5. Bounds on the mass of tubes, II -- 6. The AQUE problem and the application of the entropy bound. -- Appendix A. Proof of Lemma A.1: how to construct a higher rank amplifier -- Acknowledgments -- References -- Back Cover.

This volume contains the proceedings of the CRM Workshops on Probabilistic Methods in Spectral Geometry and PDE, held from August 22-26, 2016 and Probabilistic Methods in Topology, held from November 14-18, 2016 at the Centre de Recherches Mathématiques, Université de Montréal, Montréal, Quebec, Canada. Probabilistic methods have played an increasingly important role in many areas of mathematics, from the study of random groups and random simplicial complexes in topology, to the theory of random Schrödinger operators in mathematical physics. The workshop on Probabilistic Methods in Spectral Geometry and PDE brought together some of the leading researchers in quantum chaos, semi-classical theory, ergodic theory and dynamical systems, partial differential equations, probability, random matrix theory, mathematical physics, conformal field theory, and random graph theory. Its emphasis was on the use of ideas and methods from probability in different areas, such as quantum chaos (study of spectra and eigenstates of chaotic systems at high energy); geometry of random metrics and related problems in quantum gravity; solutions of partial differential equations with random initial conditions. The workshop Probabilistic Methods in Topology brought together researchers working on random simplicial complexes and geometry of spaces of triangulations (with connections to manifold learning); topological statistics, and geometric probability; theory of random groups and their properties; random knots; and other problems. This volume covers recent developments in several active research areas at the interface of Probability, Semiclassical Analysis, Mathematical Physics, Theory of Automorphic Forms and Graph Theory.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.