Spectral Theory and its applications.

LMS-EPSRC short Course 26 June - 1 July 2011
University College London

Organizers: L. Parnovski, Y. Petridis

Course Schedule

Group Photo of the participants

Prof. Damanik uploaded his lecture notes in the arXiv. Find them here

All lectures take place in Room 505 in 25 Gordon Street, where the UCL Mathematics Department is located. Click here for information on how to reach the location.

The course will consist of four lecture courses, four or five lecture each. The lecture courses will be on the following subjects:

  • Spectral geometry by Iosif Polterovich, Université de Montréal
  • Spectral geometry is a relatively young and rapidly developing branch of mathematics, which is a blend of partial differential equations, differential geometry and functional analysis. Many problems in spectral geometry are motivated by questions arising in various fields of physics, such as acoustics, quantum mechanics, and hydrodynamics. The aim of the course is to present basic notions and fundamental results of spectral geometry, as well as to discuss some recent developments and open problems. Main topics include: variational principles for eigenvalues, eigenvalue inequalities, spectral asymptotics, and nodal geometry of eigenfunctions.

    Recommended preliminary reading:

    1. E. B. Davies, "Spectral Theory and Differential Operators", chapters 4, 6, 7.

    2. I. Chavel, "Eigenvalues in Riemannian Geometry", chapters 1, 2.

  • Random operators by David Damanik, Rice University
  • The course will cover the results and methods used in the spectral theory of random differential operators, both discrete and continuous. The lecturer will discuss the various random models and their relative advantages. The physical reasoning behind Anderson localization will be explained; a proof of Anderson localization will be provided in the simplest setting. Other situations when one can/cannot explain Anderson localization will be discussed.

    The aim is to understand the contents of the following papers:

    Localization for One Dimensional, Continuum, Bernoulli-Anderson Models David Damanik (UCI), Robert Sims (UAB), Gunter Stolz (UAB) (published in Duke, 2002)

    A Continuum Version of the Kunz-Souillard Approach to Localization in One Dimension David Damanik, Gunter Stolz (published in Crelle, 2011)

    However, the students are not required to study them before the course. Lecture notes and other material will be distributed during the course.

  • Spectral theory of locally symmetric spaces and applications to number theory by Akshay Venkatesh, Stanford University
  • This course will explain the importance of the study of the Laplace operator in the geometric framework of locally symmetric spaces. This is really nonabelian harmonic analysis on quotients of a semisimple or reductive group G modulo a discrete group. The lecturer will explain, starting from the simplest case of G = SL(2,R) the spectral decomposition, both in the compact and cofinite case, and its relation to irreducible representations of G. Starting from the Poisson summation formula as an example of a trace formula, the Selberg trace formula will be discussed. The lecturer will present applications to Weyl’s Law for counting eigenvalues together with number theoretic applications e.g. the distribution of class numbers in quadratic fields.

    Lecture 1: Eisenstein series

    Lecture 2: Eisenstein series

    Lecture 3: The trace formula

    Lecture 4: Applications

  • Spectral Theory of N-body Schrödinger operators by Jan Philip Solovej, University of Copenhagen
  • Lecture 1: General theory of many-body quantum Mechanics: Many-body operators with 2-body potentials, Particle statistics, Canonical and grand-canonical formalism. Fock spaces.

    Lecture 2: Examples: Non-interacting systems and the Lieb-Thirring inequality, Charged systems.

    The material used in lecture 2 can be found here.

    Lecture 3: Stability of Matter with an almost complete proof in a simple case.

    The material used in lectures 1-3 can be found here.

    Lecture 4: Semiclassics and the Weyl law for eigenvalue sums.

    The material used in lecture 4 can be found here.

    Practical matters

    Registration is at Ramsay Hall.

    Most of the participants will be accommodated in the Ramsay hall. More information can be found here.

    Some of the participants will be accommodated in the Ian Baker house. More information can be found here.

    Travel information: Participants should arrange to arrive Sunday, June 26 2011 in the afternoon. The departure is on Friday, July 1 2011.

    Breakfast and dinner is at Ramsay Hall. Coffee, lunch and refreshments will be in Room 502 with the exception of June 30, when they will take place in Room 707.

    The Banquet will take place at Navarro's

    The excursion will be at the British Museum in two groups: 3:30 and 4:00 PM on Wednesday.

    Link to the official information about the course, including the registration form.