Periodic, almostperiodic, and random operators: instructional school
Monday 5th January 2015 to Friday 16th January 2015
09:00 to 09:55  Registration  
09:55 to 10:00  Welcome from John Toland (INI Director)  
10:00 to 11:00  General spectral properties of ergodic operators I  INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30  Introduction to periodic operators I  INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 14:30  Introduction to periodic operators II  INI 1  
14:30 to 15:00  Afternoon Tea  
15:00 to 16:00  Random operators: multiscale analysis I  INI 1  
16:00 to 17:00  Random operators: multiscale analysis II  INI 1  
17:00 to 18:00  Welcome Wine Reception 
09:00 to 10:00 
B Simon (CALTECH (California Institute of Technology)) Orthogonal polynomials I 
INI 1  
10:00 to 11:00 
B Simon (CALTECH (California Institute of Technology)) Orthogonal polynomials II 
INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30  General spectral properties of ergodic operators II  INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 14:30  General spectral properties of ergodic operators III  INI 1  
14:30 to 15:00  Afternoon Tea  
15:00 to 16:00  Introduction to periodic operators III  INI 1  
16:00 to 17:00  Introduction to periodic operators IV  INI 1 
09:00 to 10:00  General spectral properties of ergodic operators IV  INI 1  
10:00 to 11:00 
B Simon (CALTECH (California Institute of Technology)) Orthogonal polynomials III 
INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
B Simon (CALTECH (California Institute of Technology)) Orthogonal polynomials IV 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 14:30  Random operators: multiscale analysis III  INI 1  
14:30 to 15:00  Afternoon Tea  
15:00 to 16:00  Random operators: multiscale analysis IV  INI 1  
16:00 to 17:00  Introduction to periodic operators V  INI 1  
19:30 to 22:00  Conference Dinner at Emmanuel College 
09:00 to 10:00 
F Klopp (Université Pierre et Marie Curie) Random operators: many body problems I
The lectures will be devoted to the study of the thermodynamic limit of interacting fermions in a random background potential. We will restrict ourselves to the 0 temperature case. After discussing the case of non interacting fermions in a quite general setting, to understand the effect of interactions, we will specialize to various simple models where single particles are one dimensional.

INI 1  
10:00 to 11:00 
F Klopp (Université Pierre et Marie Curie) Random operators: many body problems II
The lectures will be devoted to the study of the thermodynamic limit of interacting fermions in a random background potential. We will restrict ourselves to the 0 temperature case. After discussing the case of non interacting fermions in a quite general setting, to understand the effect of interactions, we will specialize to various simple models where single particles are one dimensional.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
F Klopp (Université Pierre et Marie Curie) Random operators: many body problems III
The lectures will be devoted to the study of the thermodynamic limit of interacting fermions in a random background potential. We will restrict ourselves to the 0 temperature case. After discussing the case of non interacting fermions in a quite general setting, to understand the effect of interactions, we will specialize to various simple models where single particles are one dimensional.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 14:30  Random operators: multiscale analysis V  INI 1  
14:30 to 15:00  Afternoon Tea  
15:00 to 16:00  Random operators: multiscale analysis VI  INI 1  
16:00 to 17:00  Introduction to periodic operators VI  INI 1 
09:00 to 10:00 
F Klopp (Université Pierre et Marie Curie) Random operators: many body problems IV
The lectures will be devoted to the study of the thermodynamic limit of interacting fermions in a random background potential. We will restrict ourselves to the 0 temperature case. After discussing the case of non interacting fermions in a quite general setting, to understand the effect of interactions, we will specialize to various simple models where single particles are one dimensional.

INI 1  
10:00 to 11:00 
F Klopp (Université Pierre et Marie Curie) Random operators: many body problems V
The lectures will be devoted to the study of the thermodynamic limit of interacting fermions in a random background potential. We will restrict ourselves to the 0 temperature case. After discussing the case of non interacting fermions in a quite general setting, to understand the effect of interactions, we will specialize to various simple models where single particles are one dimensional.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
F Klopp (Université Pierre et Marie Curie) Random operators: many body problems VI
The lectures will be devoted to the study of the thermodynamic limit of interacting fermions in a random background potential. We will restrict ourselves to the 0 temperature case. After discussing the case of non interacting fermions in a quite general setting, to understand the effect of interactions, we will specialize to various simple models where single particles are one dimensional.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 14:30  Periodic operators: the method of gauge transform I  INI 1  
14:30 to 15:00  Afternoon Tea  
15:00 to 16:00  Random operators: multiscale analysis VII  INI 1  
16:00 to 17:00  Random operators: multiscale analysis VIII  INI 1 
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 14:30  Periodic operators: the method of gauge transform II  INI 1  
14:30 to 15:00  Afternoon Tea  
15:00 to 16:00  Periodic operators: the method of gauge transform III  INI 1  
16:00 to 17:00  Periodic operators: the method of gauge transform IV  INI 1  
17:00 to 18:30 
B Simon (CALTECH (California Institute of Technology)) Tales of Our Forefathers 
INI 1  
18:30 to 19:30  Wine Reception 
09:00 to 10:00 
B Simon (CALTECH (California Institute of Technology)) Orthogonal polynomials V 
INI 1  
10:00 to 11:00 
B Simon (CALTECH (California Institute of Technology)) Orthogonal polynomials VI 
INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Onedimensional quasiperiodic Schrödinger operators I
Quasiperiodic Schrödinger operators arise in solid state physics, describing the influence of an external magnetic field on the electrons of a crystal. In the late 1970s, numerical studies for the most prominent model, the almost Mathieu operator (AMO), produced the first example of a fractal in physics known as "Hofstadter's butterfly," marking the starting point for the ongoing strong interest in such operators in both mathematics and physics.
Whereas research in the first three decades was focussed mainly on unraveling the unusual properties of the AMO, in recent years a combination of techniques from dynamical systems with those from spectral theory has allowed for a more "global," modelindependent point of view. Intriguing phenomena first encountered for the AMO, notably the appearance of a critical phase corresponding to purely singular continuous spectrum, could be tested for their prevalence in general models.
This workshop will introduce the participants to some of the techniques necessary to understand the spectral properties of quasiperiodic Schrödinger operators. The presentation is of expository nature and will particularly emphasize the close ties to dynamical systems (``matrix cocycles''), which was successfully used to address several open problems (e.g. the ``Ten Martini problem'') and enabled abovementioned global perspective.
Topics included are: Basics: collectivity and regularity of spectral properties, matrix cocycles, arithmetic conditions Lyapunov exponent: positivity and continuity Supercritical behavior  Anderson localization Subcritical behavior  Duality, reducibility, and absolutely continuous spectrum Avila's global theory and critical behavior 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court 
11:30 to 12:30  KillipSimon problem and Jacobi flow on GSMP matrices  INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 14:30 
B Simon (CALTECH (California Institute of Technology)) Orthogonal polynomials VII 
INI 1  
14:30 to 15:00  Afternoon Tea  
15:00 to 16:00 
B Simon (CALTECH (California Institute of Technology)) Orthogonal polynomials VIII 
INI 1  
16:00 to 17:00 
Onedimensional quasiperiodic Schrödinger operators II
Quasiperiodic Schrödinger operators arise in solid state physics, describing the influence of an external magnetic field on the electrons of a crystal. In the late 1970s, numerical studies for the most prominent model, the almost Mathieu operator (AMO), produced the first example of a fractal in physics known as "Hofstadter's butterfly," marking the starting point for the ongoing strong interest in such operators in both mathematics and physics.
Whereas research in the first three decades was focussed mainly on unraveling the unusual properties of the AMO, in recent years a combination of techniques from dynamical systems with those from spectral theory has allowed for a more "global," modelindependent point of view. Intriguing phenomena first encountered for the AMO, notably the appearance of a critical phase corresponding to purely singular continuous spectrum, could be tested for their prevalence in general models.
This workshop will introduce the participants to some of the techniques necessary to understand the spectral properties of quasiperiodic Schrödinger operators. The presentation is of expository nature and will particularly emphasize the close ties to dynamical systems (``matrix cocycles''), which was successfully used to address several open problems (e.g. the ``Ten Martini problem'') and enabled abovementioned global perspective.
Topics included are: Basics: collectivity and regularity of spectral properties, matrix cocycles, arithmetic conditions Lyapunov exponent: positivity and continuity Supercritical behavior  Anderson localization Subcritical behavior  Duality, reducibility, and absolutely continuous spectrum Avila's global theory and critical behavior 
INI 1 
09:00 to 10:00  Periodic operators: the method of gauge transform V  INI 1  
10:00 to 11:00  Periodic operators: the method of gauge transform VI  INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Onedimensional quasiperiodic Schrödinger operators III
Quasiperiodic Schrödinger operators arise in solid state physics, describing the influence of an external magnetic field on the electrons of a crystal. In the late 1970s, numerical studies for the most prominent model, the almost Mathieu operator (AMO), produced the first example of a fractal in physics known as "Hofstadter's butterfly," marking the starting point for the ongoing strong interest in such operators in both mathematics and physics.
Whereas research in the first three decades was focussed mainly on unraveling the unusual properties of the AMO, in recent years a combination of techniques from dynamical systems with those from spectral theory has allowed for a more "global," modelindependent point of view. Intriguing phenomena first encountered for the AMO, notably the appearance of a critical phase corresponding to purely singular continuous spectrum, could be tested for their prevalence in general models.
This workshop will introduce the participants to some of the techniques necessary to understand the spectral properties of quasiperiodic Schrödinger operators. The presentation is of expository nature and will particularly emphasize the close ties to dynamical systems (``matrix cocycles''), which was successfully used to address several open problems (e.g. the ``Ten Martini problem'') and enabled abovementioned global perspective.
Topics included are: Basics: collectivity and regularity of spectral properties, matrix cocycles, arithmetic conditions Lyapunov exponent: positivity and continuity Supercritical behavior  Anderson localization Subcritical behavior  Duality, reducibility, and absolutely continuous spectrum Avila's global theory and critical behavior 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 17:00  informal discussion 
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 14:30 
Onedimensional quasiperiodic Schrödinger operators IV
Quasiperiodic Schrödinger operators arise in solid state physics, describing the influence of an external magnetic field on the electrons of a crystal. In the late 1970s, numerical studies for the most prominent model, the almost Mathieu operator (AMO), produced the first example of a fractal in physics known as "Hofstadter's butterfly," marking the starting point for the ongoing strong interest in such operators in both mathematics and physics.
Whereas research in the first three decades was focussed mainly on unraveling the unusual properties of the AMO, in recent years a combination of techniques from dynamical systems with those from spectral theory has allowed for a more "global," modelindependent point of view. Intriguing phenomena first encountered for the AMO, notably the appearance of a critical phase corresponding to purely singular continuous spectrum, could be tested for their prevalence in general models.
This workshop will introduce the participants to some of the techniques necessary to understand the spectral properties of quasiperiodic Schrödinger operators. The presentation is of expository nature and will particularly emphasize the close ties to dynamical systems (``matrix cocycles''), which was successfully used to address several open problems (e.g. the ``Ten Martini problem'') and enabled abovementioned global perspective.
Topics included are: Basics: collectivity and regularity of spectral properties, matrix cocycles, arithmetic conditions Lyapunov exponent: positivity and continuity Supercritical behavior  Anderson localization Subcritical behavior  Duality, reducibility, and absolutely continuous spectrum Avila's global theory and critical behavior 
INI 1  
14:30 to 15:00  Afternoon Tea  
15:00 to 16:00 
Onedimensional quasiperiodic Schrödinger operators V
Quasiperiodic Schrödinger operators arise in solid state physics, describing the influence of an external magnetic field on the electrons of a crystal. In the late 1970s, numerical studies for the most prominent model, the almost Mathieu operator (AMO), produced the first example of a fractal in physics known as "Hofstadter's butterfly," marking the starting point for the ongoing strong interest in such operators in both mathematics and physics.
Whereas research in the first three decades was focussed mainly on unraveling the unusual properties of the AMO, in recent years a combination of techniques from dynamical systems with those from spectral theory has allowed for a more "global," modelindependent point of view. Intriguing phenomena first encountered for the AMO, notably the appearance of a critical phase corresponding to purely singular continuous spectrum, could be tested for their prevalence in general models.
This workshop will introduce the participants to some of the techniques necessary to understand the spectral properties of quasiperiodic Schrödinger operators. The presentation is of expository nature and will particularly emphasize the close ties to dynamical systems (``matrix cocycles''), which was successfully used to address several open problems (e.g. the ``Ten Martini problem'') and enabled abovementioned global perspective.
Topics included are: Basics: collectivity and regularity of spectral properties, matrix cocycles, arithmetic conditions Lyapunov exponent: positivity and continuity Supercritical behavior  Anderson localization Subcritical behavior  Duality, reducibility, and absolutely continuous spectrum Avila's global theory and critical behavior 
INI 1  
16:00 to 17:00 
Onedimensional quasiperiodic Schrödinger operators VI
Quasiperiodic Schrödinger operators arise in solid state physics, describing the influence of an external magnetic field on the electrons of a crystal. In the late 1970s, numerical studies for the most prominent model, the almost Mathieu operator (AMO), produced the first example of a fractal in physics known as "Hofstadter's butterfly," marking the starting point for the ongoing strong interest in such operators in both mathematics and physics.
Whereas research in the first three decades was focussed mainly on unraveling the unusual properties of the AMO, in recent years a combination of techniques from dynamical systems with those from spectral theory has allowed for a more "global," modelindependent point of view. Intriguing phenomena first encountered for the AMO, notably the appearance of a critical phase corresponding to purely singular continuous spectrum, could be tested for their prevalence in general models.
This workshop will introduce the participants to some of the techniques necessary to understand the spectral properties of quasiperiodic Schrödinger operators. The presentation is of expository nature and will particularly emphasize the close ties to dynamical systems (``matrix cocycles''), which was successfully used to address several open problems (e.g. the ``Ten Martini problem'') and enabled abovementioned global perspective.
Topics included are: Basics: collectivity and regularity of spectral properties, matrix cocycles, arithmetic conditions Lyapunov exponent: positivity and continuity Supercritical behavior  Anderson localization Subcritical behavior  Duality, reducibility, and absolutely continuous spectrum Avila's global theory and critical behavior 
INI 1 