# Timetable (PEPW01)

## Periodic, almost-periodic, 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 One-dimensional quasi-periodic Schrödinger operators I Quasi-periodic 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 on-going 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," model-independent 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 quasi-periodic 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 above-mentioned 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 Killip-Simon 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 One-dimensional quasi-periodic Schrödinger operators II Quasi-periodic 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 on-going 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," model-independent 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 quasi-periodic 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 above-mentioned 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 One-dimensional quasi-periodic Schrödinger operators III Quasi-periodic 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 on-going 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," model-independent 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 quasi-periodic 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 above-mentioned 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 One-dimensional quasi-periodic Schrödinger operators IV Quasi-periodic 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 on-going 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," model-independent 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 quasi-periodic 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 above-mentioned 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 One-dimensional quasi-periodic Schrödinger operators V Quasi-periodic 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 on-going 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," model-independent 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 quasi-periodic 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 above-mentioned 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 One-dimensional quasi-periodic Schrödinger operators VI Quasi-periodic 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 on-going 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," model-independent 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 quasi-periodic 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 above-mentioned 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