Summer Graduate School

This two week school will focus on spectral theory of periodic, almost-periodic, and random operators.  The study of periodic problems is one of the most classical subjects of spectral theory. Many of the basic properties of periodic operators are now well understood due to the powerful method known as the Floquet-Bloch decomposition which performs a partial diagonalisation of the operator. Another type of problems mathematicians and physicists have been interested in is known under the broad name of ergodic problems; the most prominent among them being almost-periodic and random problems. These problems are much more complicated because no straightforward partial diagonalisation is known for them. The study of the spectral properties for ergodic problems developed to a large extent independently of the periodic theory, although occasionally periodic results were used in almost-periodic or even random settings. The interplay between almost-periodic and random problems has been more significant, but still, most of the methods have been specific to each of these two types. In the last few years however there emerged a number of methods which originated in one type of these problems but were successfully used to tackle problems from neighbouring areas (periodic methods used to deal with almost-periodic or random problems, etc). This suggests that these three lines of research have more in common than previously believed. The main aim of this school is to teach the students who work in one of these areas methods used in parallel problems, explain the similarities between all these areas and show them the ‘big picture’.

School Structure

There will be four introductory 3-4 hour mini-courses in the main subjects of the school: periodic, almost-periodic, random, and ergodic operators. We plan to start these courses at a pretty basic level. This will be followed by a series of more advanced courses. Each advanced course will concentrate on a particular method that can be used in at least one (but usually two or three) of these areas and will give particular examples of its applicability. Each of these advanced mini-courses will be 3-4 hours long. Some of the courses will also contain tutorials.

Basic courses:
Jake Fillman (Texas State University) --- Introduction to ergodic problems
Alexander Sobolev (University College London)  --- Introduction to periodic problems
Constanza Rojas-Molina (Université Cergy-Pointoise and Günter Stolz (University of Alabama, Birmingham) --- Introduction to random problems
Rui Han (Louisiana State University) --- Introduction to almost-periodic problems.

Advanced courses:
Jeffrey Galkowski (University College London) --- The method of Gauge transform
Charles Smart (Yale University) --- Localization for Bernoulli--Anderson model on the lattice
David Damanik (Rice University) --- Gap Labelling
Wencai Liu (Texas A&M) --- Sharp arithmetic localization
Ilya Kachkovskiy (Michigan State) --- Almost periodic localization through Green's function estimates
Lingrui Ge (Peking University) --- Quantitative global theory of one-frequency quasiperiodic operators

Open problem session: chaired by Yulia Karpeshina (University of Alabama, Birmingham).

Prerequisites

We expect students attending the school to have a basic knowledge of Spectral Theory, for which it is enough to read any of the standard Spectral Theory books, e.g., Mathematical Methods in Quantum Mechanics (alternate, errata) by Teschl, A First Course in Spectral Theory (alternate) by Lukic, or Methods of Modern Mathematical Physics I: Functional Analysis by Reed and Simon.  No other prerequisites are expected, but some further reading may be encouraged.

For eligibility and how to apply, see the Summer Graduate Schools homepage