About this Title
Alex H. Barnett, Dartmouth College, Hanover, NH, Carolyn S. Gordon, Dartmouth College, Dartmouth, NH, Peter A. Perry, University of Kentucky, Lexington, KY and Alejandro Uribe, University of Michigan, Ann Arbor, MI, Editors
Publication: Proceedings of Symposia in Pure Mathematics
Publication Year: 2012; Volume 84
ISBNs: 978-0-8218-5319-1 (print); 978-0-8218-9196-4 (online)
This volume contains the proceedings of the International Conference on Spectral Geometry, held July 19–23, 2010, at Dartmouth College, Dartmouth, New Hampshire.
Eigenvalue problems involving the Laplace operator on manifolds have proven to be a consistently fertile area of geometric analysis with deep connections to number theory, physics, and applied mathematics. Key questions include the measures to which eigenfunctions of the Laplacian on a Riemannian manifold condense in the limit of large eigenvalue, and the extent to which the eigenvalues and eigenfunctions of a manifold encode its geometry.
In this volume, research and expository articles, including those of the plenary speakers Peter Sarnak and Victor Guillemin, address the flurry of recent progress in such areas as quantum unique ergodicity, isospectrality, semiclassical measures, the geometry of nodal lines of eigenfunctions, methods of numerical computation, and spectra of quantum graphs. This volume also contains mini-courses on spectral theory for hyperbolic surfaces, semiclassical analysis, and orbifold spectral geometry that prepared the participants, especially graduate students and young researchers, for conference lectures.
Graduate students and research mathematicians interested in Riemannian geometry and analysis on manifolds.
Table of Contents
Part I. Expository Lectures
- David Borthwick – Introduction to spectral theory on hyperbolic surfaces
- Carolyn Gordon – Orbifolds and their spectra
- Alejandro Uribe and Zuoqin Wang – A brief introduction to semiclassical analysis
Part II. Invited Papers
- Nalini Anantharaman and Fabricio Macià – The dynamics of the Schrödinger flow from the point of view of semiclassical measures
- Gregory Berkolaiko and Peter Kuchment – Dependence of the spectrum of a quantum graph on vertex conditions and edge lengths
- Jeffrey Bouas, Stephen Fulling, Fernando Mera, Krishna Thapa, Cynthia Trendafilova and Jef Wagner – Investigating the spectral geometry of a soft wall
- Emily B. Dryden, Victor Guillemin and Rosa Sena-Dias – Equivariant inverse spectral problems
- Carolyn Gordon, William Kirwin, Dorothee Schueth and David Webb – Classical equivalence and quantum equivalence of magnetic fields on Flat Tori
- Victor Guillemin, Alejandro Uribe and Zuoqin Wang – A semiclassical heat trace expansion for the perturbed harmonic oscillator
- Andrew Hassell and Alex Barnett – Estimates on Neumann eigenfunctions at the boundary, and the “method of particular solutions” for computing them
- Peter Sarnak – Recent progress on the quantum unique ergodicity conjecture
- Hamid Hezari and Zuoqin Wang – Lower bounds for volumes of nodal sets: An improvement of a result of Sogge-Zelditch
- Chris Judge – The nodal set of a finite sum of Maass cusp forms is a graph
- Thomas Kappeler, Beat Schaad and Peter Topalov – Asymptotics of spectral quantities of Schrödinger operators
- Igor Wigman – On the nodal lines of random and deterministic Laplace eigenfunctions
- Steven Zelditch – Pluri-potential theory on Grauert tubes of real analytic Riemannian manifolds, I