Name | Affiliation |
---|---|
Juan Adame | California Institute of Technology |
Jean Bellissard | Georgia Institute of Technology |
David Damanik | Rice University |
Jake Fillman | Rice University |
Rui Han | UC Irvine |
Svetlana Jitomirskaya | UC Irvine |
Alexander Gordon | University of NC at Charlotte |
Anton Gorodetski | UC Irvine |
Silvius Klein | Universidade de Lisboa, Portugal |
Rostislav Kozhan | UC Los Angeles |
Michel L. Lapidus | UC Riverside |
Marius Lemm | California Institute of Technology |
Milivoje Lukic | Rice University |
Chris Marx | California Institute of Technology |
Rajinder Mavi | University of Virginia |
May Mei | UC Irvine |
Gabriel Maybrun | UC Irvine |
Paul Munger | Rice University |
Son Nguyen | UC Irvine |
John Rock | CalPoly Pomona |
Christophe Sabot | Université Lyon 1 |
Christian Sadel | University of British Columbia, Canada |
Yuki Takahashi | UC Irvine |
Sidney Tsang | UC Irvine |
Will Yessen | UC Irvine |
Zhenghe Zhang | Northwestern University |
Maxim Zinchenko | University of New Mexico |
Wannier Transform and Bloch Theory for Aperiodic repetitive FLC tilings. Jean V. Bellissard (Georgia Institute of Technology, School of Mathematics and School of Physics) |
This mini-course will be divided into four sections:
1) Background: Delone sets, repetitivity, aperiodicity, finite local complexity, Hull, Transversal, the Lagarias group. Groupoids: the groupoid of the transversal, representation, covariance, covariant fields of Hilbert spaces, unitary equivalence. 2) The Anderson Putnam complex, examples (Fibonnaci, Thue-Morse, Rudin-Shapiro, the octagonal lattice, the Penrose lattice), inflations (for the substitutive case or the general case), inverse limits and reconstruction of the Hull. Substitution tilings. Bratteli diagrams, horizontal edges, reconstruction of the groupoid of the transversal. 3) The Wannier transform for one AP-complex: quasi-momentum space, the periodic case (as a reminder), the case of FLC tilings, the Plancherel formula (the Annier transform as a unitary transformation between covariant fileds of Hilbert spaces). 4) The momentum space decomposition of the Schrodinger operator. Description of the boundary conditions, cohomological equation. Examples in one dimension, the higher dimensional case. The finite volume approximation: band spectrum, absolute continuity. Discussion and conjectures: the infinite volume limit. Renormalization and inverse limit, the substitution case, expected estimates. A conjecture: absolutely continuous spectrum in D > 2 in the perturbative regime, a route towards a proof. |
Trace Map Dynamics and spectral properties of the Fibonacci Hamiltonian. David Damanik (Rice University) and Anton Gorodetski (UC Irvine) |
1. Discrete one dimensional Schrodinger operators. Quantum dynamics, spectral measures, solutions and transfer matrices. Fibonacci Hamiltonian, recursions, Fibonacci Trace Map. Suto's Theorem.
2. Hyperbolic Dynamical Systems. Hyperbolicity of the Trace Map. Dynamically defined Cantor sets, their properties. Spectrum of the Fibonacci Hamiltonian as a dynamically defined Cantor set. 3. Fractal dimension of the spectrum. Gap labeling and gap opening. Transport exponents. Density of states measure. Square and cubic Fibanacci Hamitonian. Structure of spectrum and density of states measure. 4. Sums of Cantor sets and convolutions of singular measures. Absolute continuity of the density of states measure of square Fibonacci Hamiltonian in small coupling regime. Known results and open questions. |
Spectral Theory of self-similar lattices and dynamics of rational maps. Christophe Sabot (Université Lyon 1) |
These lectures will focus on the spectral theory of "finitely ramified" self-similar lattices and their continuous counterparts, self-similar fractals (e.g. Sierpinski gasket, Snowflake, etc). These lattices, which can be viewed as toy-models for quasi-crystals, exhibit interesting relations with the dynamics of rational maps with several variables. The first lectures will be devoted to elementary properties of these lattices and to a short overview of multidimensional rational dynamics and pluripotential theory. Then, we will describe the renormalization map associated with a self-similar lattice and the crucial point of its appropriate compactification. We relate some characteristics of the dynamics of its iterates with some characteristics of the spectrum of our operator. More specifically, we give an explicit formula for the density of states in terms of the Green current of the map, and we relate the indeterminacy points of the map with the so-called Neuman-Dirichlet eigenvalues which lead to eigenfunctions with compact support on the unbounded lattice. Depending on the asymptotic degree of the map the spectral properties of the operators are drastically different. |
Spectral theory and dynamics of quasi-periodic Jacobi cocycles. Svetlana Jitomirskaya (UC Irvine) and Chris Marx (California Institute of Technology) |
Recently, several problems from the spectral theory of 1
dimensional quasi-periodic Jacobi operators could be successfully solved
studying the dynamics of associated cocycles. Even though some of these
ideas are well known in dynamical systems, they are still less common in
the spectral theory community. In our lectures, we aim to bridge the gap,
presenting a survey of the dynamics of quasi-periodic cocycles from a
spectral theorists point of view.
Topics that will be discussed include: quasi-periodic Jacobi and higher-dimensional cocycles, Lyapunov exponent (LE) and Oseledets' theorem, Conjugacies - Aubry duality revisited, dominated splittings, (almost) reducibility and the absolutely continuous spectrum, complexified cocycles, continuity of the LE, Avila's global theory and spectral consequences. Our recent work on Extended Harper's Model, a generalization of almost Mathieu, will serve as an illustration for some of the presented methods. |