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Timetable (PEPW01)

Periodic, almost-periodic, and random operators: instructional school

Monday 5th January 2015 to Friday 16th January 2015

Monday 5th January 2015
09:00 to 09:55 Registration
09:55 to 10:00 Welcome from John Toland (INI Director)
10:00 to 11:00 D Damanik (Rice University)
General spectral properties of ergodic operators I
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 P Kuchment (Texas A&M University)
Introduction to periodic operators I
INI 1
12:30 to 13:30 Lunch at Wolfson Court
13:30 to 14:30 P Kuchment (Texas A&M University)
Introduction to periodic operators II
INI 1
14:30 to 15:00 Afternoon Tea
15:00 to 16:00 A Klein (University of California, Irvine)
Random operators: multiscale analysis I
INI 1
16:00 to 17:00 A Klein (University of California, Irvine)
Random operators: multiscale analysis II
INI 1
17:00 to 18:00 Welcome Wine Reception
Tuesday 6th January 2015
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 D Damanik (Rice University)
General spectral properties of ergodic operators II
INI 1
12:30 to 13:30 Lunch at Wolfson Court
13:30 to 14:30 D Damanik (Rice University)
General spectral properties of ergodic operators III
INI 1
14:30 to 15:00 Afternoon Tea
15:00 to 16:00 P Kuchment (Texas A&M University)
Introduction to periodic operators III
INI 1
16:00 to 17:00 P Kuchment (Texas A&M University)
Introduction to periodic operators IV
INI 1
Wednesday 7th January 2015
09:00 to 10:00 D Damanik (Rice University)
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 A Klein (University of California, Irvine)
Random operators: multiscale analysis III
INI 1
14:30 to 15:00 Afternoon Tea
15:00 to 16:00 A Klein (University of California, Irvine)
Random operators: multiscale analysis IV
INI 1
16:00 to 17:00 P Kuchment (Texas A&M University)
Introduction to periodic operators V
INI 1
19:30 to 22:00 Conference Dinner at Emmanuel College
Thursday 8th January 2015
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 A Klein (University of California, Irvine)
Random operators: multiscale analysis V
INI 1
14:30 to 15:00 Afternoon Tea
15:00 to 16:00 A Klein (University of California, Irvine)
Random operators: multiscale analysis VI
INI 1
16:00 to 17:00 P Kuchment (Texas A&M University)
Introduction to periodic operators VI
INI 1
Friday 9th January 2015
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 A Sobolev (University College London)
Periodic operators: the method of gauge transform I
INI 1
14:30 to 15:00 Afternoon Tea
15:00 to 16:00 A Klein (University of California, Irvine)
Random operators: multiscale analysis VII
INI 1
16:00 to 17:00 A Klein (University of California, Irvine)
Random operators: multiscale analysis VIII
INI 1
Monday 12th January 2015
12:30 to 13:30 Lunch at Wolfson Court
13:30 to 14:30 A Sobolev (University College London)
Periodic operators: the method of gauge transform II
INI 1
14:30 to 15:00 Afternoon Tea
15:00 to 16:00 A Sobolev (University College London)
Periodic operators: the method of gauge transform III
INI 1
16:00 to 17:00 A Sobolev (University College London)
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
Tuesday 13th January 2015
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 C Marx (Oberlin College)
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
Wednesday 14th January 2015
11:30 to 12:30 P Yuditskii (Johannes Kepler Universität)
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 C Marx (Oberlin College)
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
Thursday 15th January 2015
09:00 to 10:00 A Sobolev (University College London)
Periodic operators: the method of gauge transform V
INI 1
10:00 to 11:00 A Sobolev (University College London)
Periodic operators: the method of gauge transform VI
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 C Marx (Oberlin College)
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
Friday 16th January 2015
12:30 to 13:30 Lunch at Wolfson Court
13:30 to 14:30 C Marx (Oberlin College)
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 C Marx (Oberlin College)
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 C Marx (Oberlin College)
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
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons