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Seminars (PEP)

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Event When Speaker Title Presentation Material
PEPW05 9th February 2014
14:00 to 15:00
The Point Scatterer: A Survey
PEPW01 5th January 2015
10:00 to 11:00
General spectral properties of ergodic operators I
PEPW01 5th January 2015
11:30 to 12:30
Introduction to periodic operators I
PEPW01 5th January 2015
13:30 to 14:30
Introduction to periodic operators II
PEPW01 5th January 2015
15:00 to 16:00
Random operators: multiscale analysis I
PEPW01 5th January 2015
16:00 to 17:00
Random operators: multiscale analysis II
PEPW01 6th January 2015
09:00 to 10:00
B Simon Orthogonal polynomials I
PEPW01 6th January 2015
10:00 to 11:00
B Simon Orthogonal polynomials II
PEPW01 6th January 2015
11:30 to 12:30
General spectral properties of ergodic operators II
PEPW01 6th January 2015
13:30 to 14:30
General spectral properties of ergodic operators III
PEPW01 6th January 2015
15:00 to 16:00
Introduction to periodic operators III
PEPW01 6th January 2015
16:00 to 17:00
Introduction to periodic operators IV
PEPW01 7th January 2015
09:00 to 10:00
General spectral properties of ergodic operators IV
PEPW01 7th January 2015
10:00 to 11:00
B Simon Orthogonal polynomials III
PEPW01 7th January 2015
11:30 to 12:30
B Simon Orthogonal polynomials IV
PEPW01 7th January 2015
13:30 to 14:30
Random operators: multiscale analysis III
PEPW01 7th January 2015
15:00 to 16:00
Random operators: multiscale analysis IV
PEPW01 7th January 2015
16:00 to 17:00
Introduction to periodic operators V
PEPW01 8th January 2015
09:00 to 10:00
F Klopp 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.
PEPW01 8th January 2015
10:00 to 11:00
F Klopp 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.
PEPW01 8th January 2015
11:30 to 12:30
F Klopp 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.
PEPW01 8th January 2015
13:30 to 14:30
Random operators: multiscale analysis V
PEPW01 8th January 2015
15:00 to 16:00
Random operators: multiscale analysis VI
PEPW01 8th January 2015
16:00 to 17:00
Introduction to periodic operators VI
PEPW01 9th January 2015
09:00 to 10:00
F Klopp 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.
PEPW01 9th January 2015
10:00 to 11:00
F Klopp 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.
PEPW01 9th January 2015
11:30 to 12:30
F Klopp 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.
PEPW01 9th January 2015
13:30 to 14:30
Periodic operators: the method of gauge transform I
PEPW01 9th January 2015
15:00 to 16:00
Random operators: multiscale analysis VII
PEPW01 9th January 2015
16:00 to 17:00
Random operators: multiscale analysis VIII
PEPW01 12th January 2015
13:30 to 14:30
Periodic operators: the method of gauge transform II
PEPW01 12th January 2015
15:00 to 16:00
Periodic operators: the method of gauge transform III
PEPW01 12th January 2015
16:00 to 17:00
Periodic operators: the method of gauge transform IV
PEPW01 12th January 2015
17:00 to 18:30
B Simon Tales of Our Forefathers
PEPW01 13th January 2015
09:00 to 10:00
B Simon Orthogonal polynomials V
PEPW01 13th January 2015
10:00 to 11:00
B Simon Orthogonal polynomials VI
PEPW01 13th January 2015
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

PEPW01 14th January 2015
11:30 to 12:30
Killip-Simon problem and Jacobi flow on GSMP matrices
PEPW01 14th January 2015
13:30 to 14:30
B Simon Orthogonal polynomials VII
PEPW01 14th January 2015
15:00 to 16:00
B Simon Orthogonal polynomials VIII
PEPW01 14th January 2015
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

PEPW01 15th January 2015
09:00 to 10:00
Periodic operators: the method of gauge transform V
PEPW01 15th January 2015
10:00 to 11:00
Periodic operators: the method of gauge transform VI
PEPW01 15th January 2015
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

PEPW01 16th January 2015
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

PEPW01 16th January 2015
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

PEPW01 16th January 2015
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

PEP 21st January 2015
16:00 to 17:00
M Lukic Applications of Jacobi matrices to propagation in the XY spin chain
Propagation in spin chains is usually described by Lieb-Robinson bounds, which are statements about exponential tails beyond a cone |x|
PEP 3rd February 2015
14:00 to 15:00
Discrete spectrum of Schroedinger operators with oscillating decaying potentials
We consider the Schroedinger operator $H_{\eta W} = -\Delta + \eta W$, self-adjoint in $L^2(R^d)$, $d \geq 1$. Here $\eta$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. We study the asymptotic behaviour of the discrete spectrum of $H_{\eta W}$ near the origin, and due to the irregular decay of $\eta W$, we encounter some non semiclassical phenomena; in particular, $H_{\eta W}$ has less eigenvalues than suggested by the semiclassical intuition.
PEP 5th February 2015
11:00 to 12:00
Behaviour of zero modes for a one-dimensional Dirac operator arising in models of graphene
A basic model for conduction within a potential channel in graphene leads naturally to a one-dimensional Dirac operator. The profile of the channel enters the operator as a potential, while zero modes (or zero energy eigenstates) of the operator correspond to conduction modes in the channel.

We consider the behaviour of these zero modes relative to the potential strength and (to a lesser extent) the transversal wave number;both cases can be rephrased as spectral problems for linear operator pencils.

Several results on the eigenvalues of these pencils are presented, in particular relating to their asymptotic distribution.We show that this depends in a subtle way on the sign variation and the presence of gaps or dips in the potential; somewhat more surprisingly, it also depends on the arithmetic properties of certain quantities determined by the potential.
PEP 9th February 2015
14:00 to 15:00
Z Rudnick The point scatterer: a survey
PEP 11th February 2015
14:00 to 15:00
Schrodinger or Pauli operators with self-generated magnetic fields
There is a huge literature on Schrodinger or Pauli operators with fixed exterior magnetic fields. If however the magnetic field is not given a-priori but is the field generated by the charged particles themselves we get the coupled Shrodinger-Maxwell or Pauli-Maxwell systems. I will discuss the spectral theory and what is known, but in particular what is not known for these systems. It is known that there is stability of matter for both the Schrodinger and Pauli systems. It is however an open problem if there is existence of the thermodynamic limit. Depending on the coupling strength to the magnetic field it is also known that the semiclassical Weyl law sometimes holds and sometimes does not. A semiclassical eigenvalue estimate that holds generally in all regimes is not known.

The work on stablity of matter is joint work with Lieb and Loss from 1995. The work on semiclassics is more recent work with Erdos and Fournais.
PEP 17th February 2015
14:00 to 15:00
Entanglement entropy of free fermions
PEP 18th February 2015
14:00 to 15:00
Exponential dynamical localization in N-particle Anderson models on graphs with long-range interaction via Fractional Moment Analysis
In this talk, we extend the techniques of the multi-particle variant of the Fractional Moment Method,developed by Aizenman and Warzel, to disordered quantum systems in general finite or countable graphs with polynomial growth of balls, in presence of an exponentially decaying interaction of infnite range. In the strong disorder regime, we prove complete exponential multi-particle strong dynamical localization. Prior results, obtained with the help of the multi-scale analysis, proved only a sub-exponential decay of eigenfunction correlators for such systems.

We also comment on recent results on exponential spectrallocalization in presence of a slower (sub-exponentially) decaying interaction,in discrete and continuous models.
PEP 24th February 2015
14:00 to 15:00
Accumulation of complex eigenvalues for a class of indefinite Sturm-Liouville operators
PEP 25th February 2015
14:00 to 15:00
S Morozov On the minimax principle for eigenvalues of Dirac operator with Coulombic singularities
I will review serveral minimax characterisations for the eigenvalues in the gap of the massive Dirac operator with electrostatic Coulombic singularities and present a new result which covers all subcritical coupling constants. The talk is based on a joint work with David Müller (LMU Munich).
PEP 3rd March 2015
14:00 to 15:00
Negative eigenvalues of two-dimensional Schroedinger operators
I will discuss estimates for the number of negative eigenvalues of a two-dimensional Schroedinger operator in terms of "L log L" type Orlicz norms of the potential. The obtained results prove a conjecture by N.N. Khuri, A. Martin and T.T. Wu (2002).
PEP 4th March 2015
14:00 to 15:00
Resolvent estimates for high-contrast elliptic problems with periodic coefficients
PEP 10th March 2015
14:00 to 15:00
The string density problem and the Camassa-Holm equation
PEP 12th March 2015
14:00 to 15:00
Uncertainty principles and spectral analysis of Schroedinger operators
PEP 17th March 2015
12:30 to 13:30
M Vogel Eigenvalue statistics for a class of non-self-adjoint operators under random perturbations
PEP 18th March 2015
16:40 to 17:40
Fractional Brownian Motion with zero Hurst index and GUE random matrices
PEP 19th March 2015
12:30 to 13:30
C Sadel Some results on $d \times d$ cocycles
I want to present some results on higher dimensional cocycles which may have applications for quasi-periodic operators on strips. The talk is split into two parts:

1.) On complex analytic one-frequency cocycles (joint work with A. Avila and S. Jitomirskaya) We classify dominated cocycles and show joint continuity in frequency and analytic cocycle function A(x) of all Lyapunov exponents at irrational frequencies. As a consequence we also obtain that for a given frequency there is a dense open set of cocycles that are odminated or have trivial Lyapunov spectrum.

2.) A Herman-Avila-Bochi formula for Hermitian symplectic or pseudo-unitary cocycles. I show a generalization of the Herman-Avila Bochi formula for SL(2,R) cocycles.

PEPW02 23rd March 2015
10:00 to 11:00
N Saveliev Index theory on end-periodic manifolds
Co-authors: Tomasz Mrowka (MIT), Daniel Ruberman (Brandeis University)

End-periodic manifolds are non-compact Riemannian manifolds whose ends are modeled on an infinite cyclic cover of a closed manifold; an important special case are manifolds with cylindrical ends. We extend some of the classical index theorems to this setting, including the Atiyah-Patodi-Singer theorem computing the index of Dirac-type operators. Our theorem expresses this index in terms of a new periodic eta-invariant which equals the classical eta-invariant in the cylindrical end setting.

PEPW02 23rd March 2015
11:30 to 12:30
Uncertainty relations and Wegner estimates for random breather potentials
Co-authors: Ivica Nakic (Zagreb University), Matthias Täufer (TU Chemnitz), Martin Tautenhahn (TU Chemnitz)

We present a new scale-free, quantitative unique continuation estimate for Schroedinger operators in multidimensional space. Depending on the context such estimates are sometimes called uncertainty relations, observations inequalities or spectral inequalities. To illustrate its power we prove a Wegner estimate for Schroedinger operators with random breather potentials. Here we encounter a non-linear dependence on the random coupling constants, preventing the use of standard perturbation theory. The proofs rely on an analysis of the level sets of the random potential, and can be extended to a rather general framework.

PEPW02 23rd March 2015
13:30 to 14:30
A Wegner estimate and localisation for alloy-type models with sign-changing exponentially decaying single-site potentials
Co-authors: Martin Tautenhahn (TU Chemnitz, Germany), Ivan Veselic (Tu Chemnitz, Germany), Karsten Leohardt (MPIPKS, Dresden, Germany)

In this talk, we will consider discrete Schroedinger operators on the d-dimensional Euclidean lattice with random potential of alloy-type. The single site potential is exponentially decaying and allowed to be sign changing. The main aim is to prove a Wegner estimate, which is polynomial in the size of the box and linear in the size of the energy interval. Our result generalises earlier ones obtained by Veselic. Our Wegner estimate is of a type which can be used for the multiscale analysis proof of localisation in all energy regions, where the initial scale estimate holds. Concerning localisation, it should be mentioned that Krueger has obtained localisation results for a class of discrete alloy-type models which include ours.

This is joint work with Karsten Leonhardt (MPIPKS, Dresden), Martin Tautenhahn (TU Chemnitz), and Ivan Veselic (TU Chemnitz).

Bibliography:

H. Krueger: Localization for random operators with non-monotone potentials with exponentially decaying correlations, Ann. Henri Poincare 13 (3), 543-598, 2012.

I. Veselic: Wegner estimates for discrete alloy-type models, Ann. Henri Poincaree 11 (5), 991-1005, 2010.

PEPW02 23rd March 2015
15:00 to 16:00
T Kappeler Spectral asymptotics of Zahkarov Shabat operators and their application to the nonlinear Schrödinger equation on the circle
In this talk I will present asymptotics of various spectral quantities of Zahkarov Shabat operators and show how to apply them for proving that the nonlinear Fourier transform of the defocusing nonlinear Schrödinger equation on the circle (Birkhoff map)is the linear Fourier transform up to a nonlinear part which is 1-smoothing.Various implications will be discussed. This is joint work with Beat Schaad and Peter Topalov.
PEPW02 23rd March 2015
16:00 to 17:00
Entanglement in the disordered XY spin chain and open problems for random block operators
Random block operators appear as effective one-particle Hamiltonians in the study of the anisotropic XY spin chain. We will discuss that dynamical localization of the effective Hamiltonian implies a uniform area law for the entanglement of all eigenstates of the XY chain in random field. An open problem in this context is the regularity of the Lyapunov exponents for the associated random block operators. A difficulty arises due to the break down of irreducibility of transfer matrices at zero energy.
PEPW02 24th March 2015
10:00 to 11:00
S Jitomirskaya Diophantine properties and the spectral theory of explicit quasiperiodic models
The development of the spectral theory of quasiperiodic operators has been largely centered around and driven by several explicit models, all coming from physics. In this talk we will review the highlights of the current state-of-the-art of the spectral theory for the following three models: almost Mathieu operator, extended Harper's model and Maryland model, focusing on arithmetically driven spectral transitions, measure of the spectrum, and the Cantor nature of the spectrum.

Those models all demonstrate interesting dependence on the arithmetics of parameters (even in some cases when the final conclusion does not have such dependence) and have traditionally been approached through KAM-type schemes. Even when the KAM arguments have been replaced by the non-perturbative ones allowing to treat more couplings, frequencies that are neither far from nor close enough to rationals presented a challenge as for them there was nothing left to perturb about. A remarkable relatively recent development concerning the explicit models is that very precise results have become possible: not only many facts have been established for a.e. frequencies and phases, but in many cases it has become possible to go deeper in the arithmetics and either establish precise arithmetic transitions or even obtain results for all values of parameters. More detailed talks on some of the covered topics will be given in the April workshop by J. You, Q. Zhou, R. Han, and W. Liu.

PEPW02 24th March 2015
11:30 to 12:30
Anderson localization for one-dimensional ergodic Schrödinger operators with piecewise monotonic sampling functions
Co-author: Svetlana Jitomirskaya (University of California, Irvine)

We consider the one-dimensional ergodic operator families \begin{equation} \label{h_def} (H_{\alpha,\lambda}(x) \Psi)_m=\Psi_{m+1}+\Psi_{m-1}+\lambda v(x+\alpha m) \Psi_m,\quad m\in \mathbb Z, \end{equation} in $l^2(\mathbb Z)$. Such operators are well studied for analytic $v$, where they undergo a metal-insulator transition from absolutely continuous spectra (for small $\lambda$) to purely point spectra with exponentially decaying eigenfunctions (for large $\lambda$); the latter is usually called Anderson localization. Very little is known for general continuous of smooth $v$. However, there are several well developed models with discontinuous $v$, such as Maryland model and the Fibonacci Hamiltonian.

We study the family $H_{\alpha,\lambda}(x)$ with $v$ satisfying a bi-Lipshitz type condition (for example, $v(x)=\{x\}$). It turns out that for every $\lambda$, for almost every $\alpha$ and all $x$ the spectrum of the operator $H_{\alpha,\lambda}(x)$ is pure point. This is the first example of pure point spectrum at small coupling for bounded quasiperiodic-type operators, or more generally for ergodic operators with underlying systems of low disorder.

We also show that the Lyapunov exponent of this system is continuous in energy for all $\lambda$ and is uniformly positive for $\lambda$ sufficiently (but nonperturbatively) large. In the regime of uniformly positive Lyapunov exponent, our result gives uniform localization, thus providing the first natural example of an operator with this property. This is a joint result with Svetlana Jitomirskaya, University of California, Irvine.

PEPW02 24th March 2015
13:30 to 14:30
Perturbative methods for Schrödinger operator: from periodic to quasiperiodic potentials.
Co-authors: Young-Ran Lee (Sogang University), Roman Shterenberg (UAB)

We consider Schrödinger operator in dimension two and discuss perturbative methods and spectral results for periodic, limit-periodic and quasi-periodic potentials. We start with methods for periodic potentials, and then discuss their development for limit-periodic potentials, and, eventually, multiscale analysis in the momentum space for quasi-periodic potentials.

PEPW02 24th March 2015
15:00 to 16:00
C Joyner Spectral statistics of Bernoulli matrix ensembles - a random walk approach
Co-author: Uzy Smilansky (Weizmann Institute of Science)

We investigate the eigenvalue statistics of random Bernoulli matrices, where the matrix elements are chosen independently from a binary set with equal probability. This is achieved by initiating a discrete random walk process over the space of matrices and analysing the induced random motion of the eigenvalues - an approach which is similar to Dyson's Brownian motion model but with important modifications. In particular, we show our process is described by a Fokker-Planck equation, up to an error margin which vanishes in the limit of large matrix dimension. The stationary solution of which corresponds to the joint probability density function of certain well-known fixed trace Gaussian ensembles.

PEPW02 24th March 2015
16:00 to 17:00
On Level Spacings for Jacobi Operators
The talk will review some results concerning the spacings of eigenvalues for restrictions of self-adjoint Jacobi operators to large "boxes" and their connections with other spectral properties of these operators. A central focus will be the phenomenon of "clock behavior" often associated with absolutely continuous spectrum.
PEPW02 25th March 2015
10:00 to 11:00
D Yafaev Surface waves and scattering by unbounded obstacles
Consider the Laplace operator $H=-\Delta$ in the exterior $\Omega$ of a parabolic region in ${\bf R}^d$, and let $H_{0}=-\Delta$ be the operator in the space $L^2 ({\bf R}^d)$. The wave operators for the pair $H_{0}$, $H$ exist for an arbitrary self-adjoint boundary condition on $\partial\Omega$. For the case of the Dirichlet boundary condition, the wave operators are unitary which excludes the existence of surface waves on $\partial\Omega$. For the Neumann boundary condition, the existence of surface waves is an open problem, and we are going to discuss it.
PEPW02 25th March 2015
11:30 to 12:30
Some connections between Weyl-Titchmarsh theory, oscillation theory, and density of states
We intend to discuss connections between Weyl-Titchmarsh theory, oscillation theory, and density of states for certain classes of one-dimensional Schroedinger operators.
PEPW02 25th March 2015
13:30 to 14:30
Local Density of States and the Spectral Function for Quasi-Periodic Operators
PEPW02 25th March 2015
15:00 to 16:00
Periodic spectral problem for the massless Dirac operator
Co-author: Michael Levitin (University of Reading)

Periodic spectral problems are normally formulated in terms of the Schrodinger operator. The aim of the talk is to examine issues that arise if one formulates a periodic spectral problem in terms of the Dirac operator.

The motivation for the particular model considered in the talk does not come from solid state physics. Instead, we imagine a single massless neutrino living in a compact 3-dimensional universe without boundary. There is no electromagnetic field in our model because a neutrino does not carry an electric charge and cannot interact (directly) with an electromagnetic field. The role of the electromagnetic covector potential is therefore taken over by the metric. In other words, we are interested in understanding how the curvature of space affects the energy levels of the neutrino.

More specifically, we consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. Our aim is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation (a peculiar feature of dimension 3). We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly. We also establish a relation between our asymptotic formu la and the eta invariant.

[1] R.J.Downes, M.Levitin and D.Vassiliev, Spectral asymmetry of the massless Dirac operator on a 3-torus, Journal of Mathematical Physics, 2013, vol. 54, article 111503.

PEPW02 25th March 2015
16:00 to 17:00
Chambers formulas and semiclassical analysis for generalized Harper's butterflies
If the first mathematical results were obtained more than 30 years ago with the interpretation of the celebrated Hofstadter butterfly, more recent experiments in Bose-Einstein theory suggest new questions. I will start with a partial survey on old results of Helffer-Sjöstrand and Kerdelhue´ and then discuss more recent questions related to generalized butterflies (Dalibard and coauthors, Hou, Kerdelhue´--Royo-Letelier). These new questions are strongly related to Harper on triangular or hexagonal lattices (in connection with the now very popular graphene). Our historics is focused on the mathematical results.
PEPW02 26th March 2015
10:00 to 11:00
Kinetic transport in crystals and quasicrystals
The Lorentz gas is one of the simplest, most widely used models to study the transport properties of rarified gases in matter. It describes the dynamics of a cloud of non-interacting point particles in an infinite array of fixed spherical scatterers. More than one hundred years after its conception, it is still a major challenge to understand the nature of the kinetic transport equation that governs the macroscopic particle dynamics in the limit of low scatterer density (the Boltzmann-Grad limit). Lorentz suggested that this equation should be the linear Boltzmann equation. This was confirmed in three celebrated papers by Gallavotti, Spohn, and Boldrighini, Bunimovich and Sinai, under the assumption that the distribution of scatterers is sufficiently disordered. In the case of strongly correlated scatterer configurations (such as crystals or quasicrystals), we now understand why the linear Boltzmann equation fails and what to substitute it with. A particularly striking featur e of the periodic Lorentz gas is a heavy tail for the distribution of free path lengths, with a diverging second moment, and superdiffusive transport in the limit of large times.

Joint work with A. Strombergsson and B. Toth.

PEPW02 26th March 2015
11:30 to 12:30
Two-scale 'micro-resonant' homogenisation of periodic (and some ergodic) problems
Co-author: Ilia Kamotski (University College London)

There has been lot of recent interest in composite materials whose macroscopic physical properties can be radically different from those of conventional materials, often due to effects of the so-called "micro-resonances". Mathematically this leads to studying high-contrast homogenization of (periodic or not) problems with a `critically’ scaled high contrast, where the resulting two-scale asymptotic behaviour appears to display a number of interesting effects. Mathematical analysis of these problems requires development of "two-scale" versions of operator and spectral convergences, of compactness, etc. We will review some background, as well as some more recent generalizations and applications. One is two-scale analysis of general "partially-degenerating" periodic problems, where strong two-scale resolvent convergence appears to hold under a rather generic decomposition assumptions, implying in particular (two-scale) convergence of se migroups with applications to a wide class of micro-resonant dynamic problems. Another is two-scale homogenization with random micro-resonances, which appears to yield macroscopic dynamics effects akin to Anderson localization. Some of the work is joined with Ilia Kamotski.

PEPW02 27th March 2015
10:00 to 11:00
T Suslina Operator error estimates for homogenization of elliptic systems with periodic coefficients
We study a wide class of matrix elliptic second order differential operators $A_\varepsilon$ in a bounded domain with the Dirichlet or Neumann boundary conditions. The coefficients are assumed to be periodic and depend on $x/\varepsilon$. We are interested in the behavior of the resolvent of $A_\varepsilon$ for small $\varepsilon$. Approximations of this resolvent in the $L_2\to L_2$ and $L_2 \to H^1$ operator norms are obtained. In particular, a sharp order estimate $$ \| (A_\varepsilon - \zeta I)^{-1} - (A^0 - \zeta I)^{-1} \|_{L_2 \to L_2} \le C\varepsilon $$ is proved. Here $A^0$ is the effective operator with constant coefficients.
PEPW02 27th March 2015
11:30 to 12:30
Wannier functions for periodic Schrödinger operators and harmonic maps into the unitary group
Co-author: Adriano Pisante ("La Sapienza" University of Rome)

The localization of electrons in crystalline solids is often expressed in terms of the Wannier functions, which provide an orthonormal basis of L2(Rd) canonically associated to a given periodic Schrödinger operator.

A very popular tool in theoretical and computational solid-state physics are the maximally localized Wannier functions, which are defined as the minimizers (in a suitable space of Wannier functions) of a localization functional introduced by Marzari and Vanderbilt in 1997. While early confirmed by numerical evidence, the exponential localization of such minimizers has remained an open question until recently.

In the talk, the concept of Wannier basis will be reviewed in detail, with emphasis on its geometric counterpart (Bloch frame). Then a recent result proving the existence and the exponential localization of the minimizers, under suitable assumptions, will be presented (joint work with A. Pisante). The proof exploits methods and techniques from the regularity theory of harmonic maps into the unitary group and the so-called "decomposition into unitons" of such maps.

PEPW02 27th March 2015
13:30 to 14:30
Everywhere discontinuous anisotropy of thin periodic composite plates
Co-author: Kirill Cherednichenko (University of Bath)

We consider an elastic periodic composite plate in full bending regime, i.e. when the displacement of the plate is of finite order. Both the thickness of the plate $h$ and the period of the composite structure $\varepsilon$ are small parameters. We start from the non-linear elasticity setting. Passing to the limit as $h, \varepsilon \to 0$ we carry out simultaneous dimension reduction and homogenisation to obtain an effective limit elastic functional which describes the asymptotic properties of the composite plate. We show, in particular, that in the regime $h

PEPW02 27th March 2015
15:00 to 16:00
Maryland equation, renormalization formulas and mimimal meromorphic solutions to difference equations
Co-author: Fedor Sandomirskyi (Saint Petersburg State University)

Consider the difference Schrödinger equation $\psi_{k+1}+\psi_{k-1}+\lambda\ {cotan} (\pi\omega k+\theta)\psi_k=E\psi_k,\quad k\in{\mathbb Z}$,where $\lambda$, $\omega$, $\theta$ and $E$ are parameters. If $\omega$ is irrational, this equation is quasi-periodic. It was introduced by specialists in solid state physics from Maryland and is now called the Maryland equation. Computer calculations show that, for large $k$, its eigenfunctions have a multiscale, "mutltifractal" structure. We obtained renormalization formulas that express the solutions to the input Marryland equation for large $k$ in terms of solutions to the Marryland equation with new parameters for bounded $k$. The proof is based on the theory of meromorphic solutions of difference equations on the complex plane, and on ideas of the monodromization met hod -- the renormalization approach first suggested by V.S.Buslaev and A.A. Fedotov.

Our formulas are close to the renormalization formulas from the theory of the Gaussian exponential sums $S(N)=\sum_{n=0}^N\,e^{2\pi i (\omega n^2+\theta n)}$, where $\omega$ and $\theta$ are parametrs. For large $N$, these sums also have a multiscale behavior. The renormalization formulas lead to a natural explanation of the famous mutiscale structure that appears to reflect certain quasi-classical asymptotic effects (Fedotov-Klopp, 2012).

PEPW02 27th March 2015
16:00 to 17:00
On the gaps in the spectrum of the periodic Maxwell operator
Co-authors: S. Cooper (University of Bath), V. Smyshlyaev (UCL)

We demonstrate the existence of the gaps in the spectrum of the periodic Maxwell operator with medium contrast coefficients. We discuss the location of the gaps and their dependence on the geometry of the media.

PEP 30th March 2015
12:30 to 13:30
J Breuer Jacobi Matrices and Central Limit Theorems in Random Matrix Theory
The notion of an orthogonal polynomial ensemble generalizes many important point processes arising in random matrix theory, probability and combinatorics. The most famous example perhaps is that of the eigenvalue distributions of unitary invariant ensembles (such as GUE) of random matrix theory. Remarkably, the study of fluctuations of these point processes is intimately connected to the study of Jacobi matrices. This talk will review our recent joint work with Maurice Duits exploiting this connection to obtain central limit theorems for orthogonal polynomial ensembles.
PEPW02 2nd April 2015
12:30 to 13:30
S Klein Continuity of Lyapunov Exponents via Large Deviations
Large deviation type (LDT) estimates for transfer matrices are important tools in the study of discrete, one dimensional, quasi-periodic Schrodinger operators. They have been used to establish positivity of the Lyapunov exponent, continuity properties of the Lyapunov exponent and of the integrated density of states, estimates on the Green's function, Anderson localization. We prove - in a general, abstract setting - that the availability of appropriate LDT estimates implies continuity of the Lyapunov exponents, with a modulus of continuity depending explicitly on the strength of the LDT. The devil is of course in the details, hidden here behind the words "availability" and "appropriate". We show that the study of the Lyapunov exponents associated with a band lattice quasi-periodic Schrodinger operator fits this abstract setting, provided the potential is a real analytic function of (one or of) several variables and that the frequency vector is Diophantine. Co-authored with: P Duarte
PEPW03 7th April 2015
10:00 to 11:00
J You Dry Ten Martini Problem in Non-Critical Case
We prove that the Dry Ten Martini Problem, i.e., all possible spectral gaps are open, holds for almost Mathieu operator with $(\lambda, \beta)\ne (\pm 1,0)$.
PEPW03 7th April 2015
11:30 to 12:30
Semi-classical analysis of non-self-adjoint transfer operators
Co-author: Margherita Disertori (Bonn)

We shall discuss the asymptotics of the top eigenvalues of non-self-adjoint integral operators in the semi-classical regime. The motivation comes from complex-valued one-dimensional statistical mechanics models, particularly, those arising in the study of random operators.

PEPW03 7th April 2015
13:30 to 14:30
Phase transitions for the almost Mathieu operator
Co-authors: Artur Avila (IMPA & Paris 7), Jiangong You (Nanjing University)

For the almost Mathieu operator with any fixed frequency, we locate the point where phase transition from singular continuous spectrum to pure point spectrum takes place, which settles Aubry-Andr\'e conjecture for all irrational frequencies, and also solves Avila and Jitomirskaya's conjectures. Together with former paper of Avila, we give a complete description of phase transitions for the almost Mathieu operator.

PEPW03 7th April 2015
15:00 to 15:25
Lyapunov exponents of quasi-periodic cocycles
Co-author: Pedro Duarte (University of Lisbon)

The purpose of this talk is to review some recent results concerning Lyapunov exponents of higher dimensional, analytic cocycles over a multi-frequency torus translation. Such cocycles appear naturally in the study of band lattice quasi-periodic Schrodinger operators.

The main new feature of this work is allowing a cocycle depending on several variables to have singularities, which requires a careful analysis involving pluri-subharmonic and analytic functions of several variables.

PEPW03 7th April 2015
15:30 to 15:55
W Liu Arithmetic Spectral Transitions for the Maryland Model
In this talk, I will give a precise description of spectra of the Maryland model $ (h_{\lambda,\alpha,\theta}u) _n=u_{n+1}+u_{n-1}+ \lambda \tan \pi(\theta+n\alpha)u_n$ for all values of parameters. For Almost Mathieu Operator (H_{\lambda,\alpha,\theta}u) _n=u_{n+1}+u_{n-1}+ \lambda \cos 2\pi(\theta+n\alpha)u_n, the Lyapunov exponent can almost determine its spectral types(A.Avila, S.Jitomirskaya, J.You, Q.Zhou). When turn to Maryland model, I introduce an arithmetically defined index $\delta (\alpha, \theta)$ and show that for $\alpha\notin\mathbb{Q},$ $\sigma_{sc}(h_{\lambda,\alpha,\theta})=\overline{\{e:\gamma_{\lambda}(e)
PEPW03 7th April 2015
16:00 to 16:25
Measure of the spectrum of the extended Harper's model
Co-author: Svetlana Jitomirskaya (University of California, Irvine)

In this talk, we will discuss the measure of the spectrum of the extended Harper's model(EHM). The measure of the spectrum of the Harper's model, which in mathematics is better known as the almost Mathieu operator(AMO), is know to be |4-2a| where a is the coupling constant. The way of calculating the measure mainly relies on the analysis of the AMO with rational frequency and the continuity argument. Here we focus on how to calculate the measure of the spectrum of the EHM with rational frequency, therefore implying the result of irrtional frequency.

PEPW03 8th April 2015
10:00 to 11:00
Dynamics of two quasi-periodically perturbed systems
We will present results concerning the dynamics of two different quasi-periodically perturbed systems. One of the systems we consider is the quasi-periodic Schrödinger cocycle.
PEPW03 8th April 2015
11:30 to 12:30
Recurrent random walks in random and quasi-periodic environments on a strip
This is joint work with D. Dolgopyat

We prove that a recurrent random walk (RW) in random environment (RE) on a strip which does not obey the Sinai law exhibits the Central Limit asymptotic behaviour.

We also show that there exists a collection of proper sub-varieties in the space of transition probabilities such that

1. If RE is stationary and ergodic and the transition probabilities are concentrated on one of sub-varieties from our collection then the CLT holds; 2. If the environment is i.i.d then the above condition is also necessary for the CLT.

As an application of our techniques we prove the CLT for quasi-periodic environments with Diophantine frequencies. One-dimensional RWRE with bounded jumps are a particular case of the strip model.

PEPW03 8th April 2015
13:30 to 14:30
Spectral Properties of Schroedinger Operator with a Quasi-periodic Potential in Dimension Two
Co-author: Roman Shterenberg (UAB)

We consider $H=-\Delta+V(x)$ in dimension two, $V(x)$ being a quasi-periodic potential. We prove that the spectrum of $H$ contains a semiaxis (Bethe-Sommerfeld conjecture) and that there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves $e^{i\langle \vec k,\vec x\rangle }$ at the high energy region. Second, the isoenergetic curves in the space of momenta $\vec k$ corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). It is shown that the spectrum corresponding to these eigenfunctions is absolutely continuous. A method of multiscale analysis in the momentum space is developed to prove the results.

PEPW03 8th April 2015
15:00 to 16:00
Continuity of Lyapunov Exponents and Cantor spectrum for a class of $C^2$ Quasiperiodic Schr\"odinger Cocycles
Co-author: Zhenghe Zhang (Rice University)

We show that for a class of $C^2$ quasiperiodic potentials and for any fixed \emph{Diophantine} frequency, the Lyapunov exponents of the corresponding Schr\"odinger cocycles are uniformly positive and weakly H\"older continuous as function of energies. Moreover, we show that the spectrum is Cantor. Our approach is of purely dynamical systems, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to general quasiperiodic $\mathrm{SL}(2,\R)$ cocycles.

PEPW03 8th April 2015
16:00 to 16:25
S Morozov High energy asymptotics of the integrated density of states of almost periodic pseudo-differential operators
The existence of complete asymptotic expansion for the integrated density of states in the high energy regime was long conjectured for periodic Schrödinger operators. I will discuss the history of the subject and present an eventual solution in the multidimensional situation. It turns out that the result applies to a big class of almost periodic pseudo-differential operators with smooth symbols. The proof is based on an application of the gauge transform discussed in the minicourse of A. Sobolev during the introductory workshop. The talk is based on a joint work with L. Parnovski and R. Shterenberg.
PEPW03 9th April 2015
10:00 to 11:00
S Nakamura Microlocal properties of scattering matrices
We consider scattering theory for a pair of operators $H_0$ and $H=H_0+V$ on $L^2(M,m)$, where $M$ is a Riemannian manifold, $H_0$ is a multiplication operator on $M$ and $V$ is a pseudodifferential operator of order $-\mu$, $\mu>1$. We show that a time-dependent scattering theory can be constructed, and the scattering matrix is a pseudodifferential operator on each energy surface. Moreover, the principal symbol of the scattering matrix is given by a Born approximation type function. The main motivation of the study comes from applications to discrete Schr\"odigner operators, but it also applies to various differential operators with constant coefficients and short-range perturbations on Euclidean spaces.
PEPW03 9th April 2015
11:30 to 12:30
M Shamis Wegner estimates for deformed Gaussian ensembles
Co-authors: Michael Aizenman (Princeton University), Ron Peled (Tel Aviv University), Jeff Schenker (Michigan State University ), Sasha Sodin (Tel Aviv University)

The deformed Gaussian ensembles are obtained by adding a deterministic Hermitian matrix to a random matrix drawn from the Gaussian Orthogonal, Unitary (or Symplectic) ensembles. We shall discuss several Wegner-type estimates for these models. As an application, we establish localization at strong disorder for the Wegner orbital model, with sharp dependence of the localization threshold on the number of orbitals.

PEPW03 9th April 2015
13:30 to 14:30
M Goldstein On gaps and bands of quasi-periodic operators
This talk will review results on quasi-periodic Sturm-Liouville operators developed in recent joint works with D.Damanik and M.Lukic and also on joint work with D.Damanik, W.Schlag, M.Voda. The central feature to be discussed is the following relations between gaps and bands.
PEPW03 9th April 2015
15:00 to 15:25
Random nonmonotonic multichannel Schr\"{o}dinger operators
Co-author: John Imbrie (University of Virginia)

Anderson localization is by now well understood for the standard random Schrodinger operator. On the other hand the motivation for the problem, which lies in many body systems still lacks a developed theory. For our part we consider several aspects arising in more-than-one body systems which prevent an immediate application of the methods of one body systems. In systems such as random Ising models, energy levels of the system may depend analytically on (finite truncations of) random parameters. Of course in the standard Anderson model the dependence of the energy levels on the random parameters is linear which leads to the celebrated Wegner estimate which allows the usual multiscale analysis. In our talk, we consider a single body model with potentials depending analytically on the random parameters. In multichannel Schrodinger models, the potentials at each site of the lattice are matrices which may depend analytically on the random parameters, eg, these models can be realized as tight binding models in $Z^D$ with dilute randomness. In the multichannel model, we utilize the transversality of the system's energies with respect to the random environment, this allows some control of the probabilities of resonances. Finally, we discuss new methods of localization proofs, for the multichannel model we obtain stretched exponential localization of eigenfunction correlations.

PEPW03 9th April 2015
15:30 to 15:55
C Rojas-Molina Ergodic properties and localization for Delone-Anderson models
Co-authors: F. Germinet (U. de Cergy-Pontoise) and P. Müller (Ludwig-Maximilians-Universität München)

Delone-Anderson models arise in the study of wave localization in random media, where the underlying configuration of impurities in space is aperiodic, as for example, in disordered quasicrystals. The lack of translation invariance in the model yields a break of ergodicity, and the loss of properties linked to it. In this talk we will present results on the existence of the integrated density of states, the ergodic properties of these models and results on dynamical localization.

PEPW03 9th April 2015
16:00 to 16:25
C Sadel Anderson transition at 2D growth-rate for the Anderson model on antitrees with normalized edge weights
An antitree is a discrete graph that is split into countably many shells $S_n$ consisting of finitely many vertices so that all vertices in $S_n$ are connected with all vertices in the adjacent shells $S_{n+1}$ and $S_{n-1}$. We normalize the edges between $S_n$ and $S_{n+1}$ with weights to have a bounded adjacency operator and add an iid random potential. We are interested in the case where the number of vertices $\# S_n$ in the $n$-th shell grows like $n^a$. In a particular set of energies we obtain a transition of the spectral type from pure point to partly s.c. to a.c. spectrum at $a=1$ which corresponds to the growth-rate in 2 dimensions.
PEPW03 10th April 2015
10:00 to 11:00
Reflectionless property and related problems on 1D Schrödinger operators
Reflectionless property for 1D Schrödinger operators is defined by using their Weyl functions or Green functions. The property is especially important when potentials of Schrödinger operators are ergodic, and it is proved that the reflectionless property holds on their absolutely continuous spectra. On the other hand Remling showed the deterministic version. They are related to the shift operation of potentials. In this talk we discus the capability of its extension to KdV equation and propose several open problems.
PEPW03 10th April 2015
11:30 to 12:30
Level statistics for 1-dimensional Schr\"odinger operator and beta-ensemble
A part of this talk is based on joint work with Prof. Kotani. We consider the following two classes of 1-dimensional random Schr\"odinger operators : (1) operators with decaying random potential, and (2) operators whose coupling constants decay as the system size becomes large. Our problem is to identify the limit $¥xi_{¥infty}$ of the point process consisting of rescaled eigenvalues. The result is : (1) for slow decay, $¥xi_{¥infty}$ is the clock process ; for critical decay $¥xi_{¥infty}$ is the $Sine_{¥beta}$ process, (2) for slow decay, $¥xi_{¥infty}$ is the deterministic clock process ; for critical decay $¥xi_{¥infty}$ is the $Sch_{¥tau}$ process. As a byproduct of (1), we have a proof of coincidence of the scaling limits of circular and Gaussian beta ensembles.
PEPW03 10th April 2015
13:30 to 14:30
A Gorodetski The Fibonacci Hamiltonian
Co-authors: David Damanik (Rice University), William Yessen (Rice University)

In the talk we will consider the discrete Schrodinger operator with potential given by the Fibonacci substitution sequence (the Fibonacci Hamiltonian) and provide a detailed description of its spectrum and spectral characteristics (namely, the optimal Holder exponent of the integrated density of states, the dimension of the density of states measure, the dimension of the spectrum, and the upper transport exponent) for all values of the coupling constant. In particular, we will establish strict inequalities between the four spectral characteristics in question, and discuss the exact small and large coupling asymptotics of these spectral characteristics. A crucial ingredient is the relation between spectral properties of the Fibonacci Hamiltonian and dynamical properties of the Fibonacci trace map (such as dimensional characteristics of the non-wandering hyperbolic set and its measure of maximal entropy as well as other equilibrium measures, topological entropy, multipliers of periodic orbits). We will establish exact identities relating the spectral and dynamical quantities, and show the connection between the spectral quantities and the thermodynamic pressure function.

PEPW03 10th April 2015
15:00 to 15:25
The isospectral torus of quasi-periodic Schrodinger operators via periodic approximations
Co-authors: David Damanik (Rice University), Michael Goldstein (University of Toronto)

This talk describes joint work with D.\ Damanik and M.\ Goldstein. We study quasi-periodic Schr\"odinger operators $H = -\frac{d^2}{dx^2} +V$ in the regime of analytic sampling function and small coupling. More precisely, the potential is \[ V(x)=\sum_{m\in \mathbb{Z}^\nu} c(m) \exp(2\pi i m \omega x) \] with $|c(m)|\le \epsilon \exp(-\kappa |m|)$. Our main result is that any reflectionless potential $Q$ isospectral with $V$ is also quasi-periodic and in the same regime, with the same Diophantine frequency $\omega$, i.e. \[ Q(x)=\sum_{m\in \mathbb{Z}^\nu} d(m) \exp(2\pi i m \omega x) \] with $|d(m)|\le \sqrt{2\epsilon} \exp(-\frac{\kappa}2 |m|)$. The proof relies on approximation by periodic potentials $\tilde V$, which are obtained by replacing the frequency $\omega$ by rational approximants $\tilde \omega$. We adapt the multiscale analysis, developed by Damanik--Goldstein for $V$, so that it applies to the periodic approximants $\tilde V$. This allows us to establish estimates for gap lengths and Fourier coefficients of $\tilde V$ which are independent of period, unlike the standard estimates known in the theory of periodic Schr\"odinger operators. Starting from these estimates, we obtain the main result by comparing the isospectral tori and translation flows of $\tilde V$ and $V$.

PEPW03 10th April 2015
15:30 to 15:55
Spectral packing dimension for 1-dimensional quasiperiodic Schrodinger operators
Co-author: Svetlana Jitomirskaya (UC Irvine)

In this talk, we are going to discuss the packing dimension of the spectral measure of 1-dimensional quasiperiodic Schrodinger operators. We prove that if the base frequency is Liouville, the packing dimension of the spectral measure will be one. As a direct consequence, we show that for the critical and supercritical Almost Mathieu Operator, the spectral measure has different Hausdorff and packing dimension.

PEPW03 10th April 2015
16:00 to 16:25
Homogeneous Spectrum for Limit-Periodic Operators
Co-author: Milivoje Lukic (University of Toronto)

We will discuss the spectra of limit-periodic Schr\"odinger operators. Specifically, the spectrum of a limit-periodic operator which obeys the Pastur-Tkachenko condition is homogeneous in the sense of Carleson. When combined with work of Gesztesy-Yuditskii, our theorem implies that the spectrum of a continuum Schr\"odinger operator with Pastur--Tkachenko potential has infinite gap length whenever the potential fails to be uniformly almost periodic.

PEPW03 10th April 2015
16:30 to 16:55
On the Homogeneity of the Spectrum for Quasi-Periodic Schroedinger Operators
Co-authors: David Damanik (Rice University), Michael Goldstein (University of Toronto), Wilhelm Schlag (University of Chicago)

I will discuss a recent result showing that the spectrum of discrete one-dimensional quasi-periodic Schroedinger operators is homogeneous in the regime of positive Lyapunov exponent. The homogeneity is in the sense of Carleson, as used in the study of the inverse spectral problem for reflectionless potentials. The talk is based on joint work with David Damanik, Michael Goldstein, and Wilhelm Schlag.

PEP 14th April 2015
14:00 to 16:00
A short introduction into multi-particle Anderson localization (a mini-course)
PEP 15th April 2015
15:00 to 16:00
J You On the Generic Cantor Spectrum problem
We prove that The spectrum is a Cantor set for generic one frequency analytic quasi-periodic Schr\"odinger operators, which gives an answer to Problem 6 of Barry Simon's list of conjectures in one frequency case. Some related problems will be posed in the talk. The talk is based on a joint work with R. Krikorian and Q. Zhou.
PEP 22nd April 2015
14:00 to 16:00
A short introduction into multi-particle Anderson localization (a mini-course)
PEP 23rd April 2015
14:00 to 15:00
Spectra of Sample Auto-Covariance Matrices
This work is based on R. Kühn and P. Sollich 2012 EPL 99 20008 doi:10.1209/0295-5075/99/20008

In this paper we compute spectra of sample auto-covariance matrices of stationary time series. The central result amounts to a generalization of Szeg"os theorem for spectra of Toeplitz matrices to the case with randomness due to finite sample effects. While the related problem of sample covariance matrices is well understood since the work of Marcenkov and Pastur, very little has been known about the sample auto-covariance problem.

PEP 28th April 2015
14:00 to 15:00
Non-linear generalisation of the May-Wigner instability transition (joint work with Yan Fyodorov)
PEP 28th April 2015
15:00 to 16:00
G Rozenblum Eigenvalues of the Schroedinger operator on infinite combinatorial and quantum graphs
PEP 6th May 2015
13:30 to 14:30
Magnetic wells in dimension two and three
this talks deals with semiclassical asymptotics of the two- or three-dimensional magnetic Laplacians in presence of magnetic confinement. Using generic assumptions on the geometry of the confinement, we exhibit semiclassical scales and their corresponding effective quantum Hamiltonians, by means of microlocal normal forms \textit{\`a la Birkhoff} or Grushin's problems. As a consequence, when the magnetic field admits a unique and non degenerate minimum, we are able to reduce the spectral analysis of the low-lying eigenvalues to a one-dimensional $\hbar$-pseudo-differential operators.
PEP 7th May 2015
13:30 to 14:30
Asymptotic behaviour of a sine-kernel determinant in the theory of the log-gas and random matrices
PEP 11th May 2015
16:00 to 17:00
Rothschild Distinguished Visiting Fellow Lecture: Disordered systems and related spectra
We discuss certain ideas and results of the theoretical physics of disordered solids and related developments of spectral theory and mathematical physics, in particular, spectral analysis of the Schrodinger operator with random potential and large random matrices.
PEP 13th May 2015
14:00 to 15:00
V Jaksic Conductance and absolutely continuous spectrum of 1D samples
In this talk I shall describe the characterization of the absolutely continuous spectrum of the one-dimensional Schr ̈odinger operators h = −∆ + v acting on 2 (Z + ) in terms of the limiting behavior of the Landauer-B ̈ uttiker and Thouless conductances of the associated finite samples. The finite samples are defined by restricting h to a finite interval [1, L] ∩ Z + and the conductance refers to the charge current across the sample in the open quantum system obtained by attaching independent electronic reservoirs to the sample ends. Our main result is that the conductances associated to an energy interval I are non-vanishing in the limit L → ∞ (physical characterization of the metallic regime) iff sp ac (h) ∩ I = ∅ (mathematical characterization of the metallic regime). This result is of importance for the foundations of quantum mechanics since it provides the first complete dynamical character- ization of the absolutely continuous spectrum of Schr ̈odinger operators. I shall also discuss its relation with Avila’s counterexample to the Schr ̈odinger Conjecture. This talk is based on a joint work with L. Bruneau, Y. Last, and C-A. Pillet.
PEP 13th May 2015
15:15 to 16:00
How to Place an Obstacle so as to Optimize the Dirichlet Eigenvalues in $\R^2$.
PEP 20th May 2015
11:30 to 12:30
J Schenker Dissipative Transport in the Localization Regime
PEP 20th May 2015
14:00 to 15:00
V Jaksic Jacobi Matrices and Non-equilibrium Statistical Mechanics
PEP 27th May 2015
14:00 to 15:00
Density of states and Lyapunov exponent in the heavy tail potentials
PEP 3rd June 2015
14:00 to 15:00
A Nota Derivation of the Fick's law for the Lorentz model in a low density regime
In this talk we consider a simple microscopic model given by the Lorentz gas, a system of non interacting light particles in a distribution of scatterers, in contact with two mass reservoirs. We show that, in a low density regime, there exists a unique stationary solution for the microscopic dynamics which converges to the stationary solution of the heat equation, namely to the linear profile of the density. In the same regime the macroscopic current in the stationary state is given by the Fick's law, with the diffusion coefficient determined by the Green-Kubo formula. These results are obtained in collaboration with G. Basile, F. Pezzotti and M. Pulvirenti.
PEP 3rd June 2015
15:10 to 16:10
N Filonov Uniqueness of the Leray-Hopf solution for a dyadic model
We consider the system of nonlinear differential equations

\label{1} \begin{cases} \dot u_n(t) + \la^{2n} u_n(t) - \la^{\be n} u_{n-1}(t)^2 + \la^{\be(n+1)} u_n(t) u_{n+1}(t) = 0,\\ u_n(0) = a_n, n \in \mathbb{N}, \quad \la > 1, \be > 0.

In this talk we explain why this system is a model for the Navier-Stokes equations of hydrodynamics. The natural question is to find a such functional space, where one could prove the existence and the uniqueness of solution. In 2008, A.~Cheskidov proved that the system (0.1) has a unique "strong" solution if $\be \le 2$, whereas the "strong" solution does not exist if $\be > 3$.

(Note, that the 3D-Navier-Stokes equations correspond to the value $\be = 5/2$.)

We show that for sufficiently "good" initial data the system (0.1)has a unique Leray-Hopf solution for all $\be > 0$.

PEP 4th June 2015
14:00 to 15:00
F Klopp Stark-Wannier ladders and cubic exponential sums
PEP 10th June 2015
14:00 to 15:00
The nodal count mystery
The beautiful nodal patterns of oscillating membranes, usually called by the (incorrect) name Chladni patterns, have been known for several centuries (Galileo, Leonardo, Hooke) and studied in the last hundred years by many leading mathematicians. In spite of that, many properties of these patterns remain a mystery. We will present the history and some of the recent advances in the area of counting the nodal domains. No prior knowledge of the subject is assumed.

This is a joint work with Gregory Berkolaiko (Texas A&M) and Uzy Smilansky (Weizmann Institute)

PEP 10th June 2015
15:10 to 16:10
Anderson localization in a multi-dimensional deterministic disorder I
PEP 11th June 2015
14:00 to 15:00
Compactness principles and convergence of spectra in double-porosity models
PEP 11th June 2015
15:10 to 16:10
S Pastukhova Homogenization of operators with quasi-periodic coefficients
PEP 12th June 2015
14:00 to 15:00
Anderson localization in a multi-dimensional deterministic disorder II
PEP 16th June 2015
14:00 to 15:00
N Filonov Absolute continuity returns: potentials that are periodic in some directions only
PEPW04 22nd June 2015
10:00 to 11:00
On non-smooth functions of Wiener-Hopf operators
We discuss trace formulae for the operator \begin{equation*} f(PAP) - Pf(A)P, \end{equation*} where $A$ is a pseudo-differential operator on $L^2(\mathbb R^d)$ with a smooth or discontinuous symbol, and $P$ is a multiplication by the indicator of a piece-wise smooth domain in $\mathbb R^d$. The function $f$ is not supposed to be smooth. The obtained formulae generalise results obtained by H. Widom in the 80's. These results are used to study the entanglement entropy of free fermions at positive temperature both in the low and high temperature limits.
PEPW04 22nd June 2015
11:30 to 12:30
On the two-dimensional random walk in an isotropic random environment
This is joint work with Erich Baur (Lyon) and Ofer Zeitouni (Weizmann Institute).

We report on work in progress on the standard model of a random walk in random environment in the critical dimension 2. We investigate exit distributions from large sets which are supposed to be essentially the same as those for ordinary random walks. Random walks in random environment are well understood in dimension one, and for small disorder in dimensions above 2, but the two-dimensional case is largely open.

PEPW04 22nd June 2015
13:30 to 14:30
From the mesoscopic to microscopic scale in random matrix theory
Co-authors: Laszlo Erdos (IST), Horng Tzer Yau (Harvard), Jun Yin (Wisconsin Madison)

Eugene Wigner has envisioned that the distributions of the eigenvalues of large Gaussian random matrices are new paradigms for universal statistics of large correlated quantum systems. These random matrix eigenvalues statistics supposedly occur together with delocalized eigenstates. I will explain recent developments proving this paradigm for eigenvalues and eigenvectors of random matrices. This is achieved by bootstrap on scales, from mesoscopic to microscopic. Random walks in random environments, homogenization and the coupling method play a key role.

PEPW04 22nd June 2015
15:00 to 16:00
On fluctuations of eigenvalues of random band matrices
We consider the fluctuation of linear eigenvalue statistics of random band $n$ dimensional matrices whose bandwidth $b$ is assumed to grow with n in such a way that $b/n$ tends to zero. Without any additional assumptions on the growth of b we prove CLT for linear eigenvalue statistics for a rather wide class of test functions. Thus we remove the main technical restriction $n>>b>>n^{1/2}$ of all the papers, in which band matrices were studied before. Moreover, the developed method allows to prove automatically the CLT for linear eigenvalue statistics of the smooth test functions for almost all classical models of random matrix theory: deformed Wigner and sample covariance matrices, sparse matrices, diluted random matrices, matrices with heavy tales etc.
PEPW04 22nd June 2015
16:00 to 17:00
Primitive Pythagorean triples and "near" quasicrystals
Counting things is a great favorite of children, and mathematicians as well, whatever the things are. In this talk, I discuss an old counting problem on primitive Pythagorean triples in view of modern theory of quasicrystals.
PEPW04 23rd June 2015
10:00 to 11:00
An eigensystem approach to Anderson localization
Co-author: Alexander Elgart (Virginia Tech)

We introduce a new approach for proving localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization) for the Anderson model at high disorder. In contrast to the usual strategy, we do not study finite volume Green's functions. Instead, we perform a multiscale analysis based on finite volume eigensystems, establishing localization of finite volume eigenfunctions with high probability. (Joint work with A. Elgart.)

PEPW04 23rd June 2015
11:30 to 12:30
An eigensystem approach to Anderson localization, part II
Co-author: Abel Klein (UC Irvine)

We introduce a new approach for proving localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization) for the Anderson model at high disorder. In contrast to the usual strategy, we do not study finite volume Green's functions. Instead, we perform a multiscale analysis based on finite volume eigensystems, establishing localization of finite volume eigenfunctions with high probability. (Joint work with A. Klein.)

PEPW04 23rd June 2015
13:30 to 14:30
J Schenker Dissipative transport in the localized regime
Co-author: Jürg Fröhlich (ETH)

A quantum particle moving in a strongly disordered random environment is known to be subject to Anderson localization, which results in the complete suppression of transport. However, localization can be broken by a small perturbation, such as thermal noise from the environment, resulting in diffusive motion for the particle. I will discuss this phenomenon in two models in which the Schroedinger equation for a particle in the strongly localized regime is perturbed by (1) a time dependent fluctuating random potential and (2) a Lindblad operator incorporating the interaction with a heat bath in the Markov approximation. In each case, it can be proved that diffusive motion results with a strictly positive and finite diffusion constant. Furthermore, the diffusion constant tends continuously to zero at a calculable rate, as the strength of the perturbation is taken to zero. (Partially based on joint work with J. Fröhlich.)

PEPW04 23rd June 2015
15:00 to 16:00
Y Suhov New properties of entropy and their consequences
Co-authors: Salimeh Yasaei Sekeh (University of Sao Paulo at Sao Carlos, Brazil), Srefan Zohren (University of Oxford, UK)

Based on the concept of the weighted entropy (both classical and quantum), I will report a number of new results involving inequalities and convergence in a variety of context.

PEPW04 23rd June 2015
16:00 to 17:00
J Imbrie Level Spacing for Non-Monotone Anderson Models
PEPW04 24th June 2015
10:00 to 11:00
Pure point spectrum in the regime of zero Lyapunov exponents
Co-author: Anton Gorodetski (UC Irvine)

In the study of ergodic Schrodinger operators, a central role is played by the Lyapunov exponent of the associated Schrodinger cocycle. We discuss a construction showing that the regime of zero Lyapunov exponents can contain pure point spectrum.

PEPW04 24th June 2015
11:30 to 12:30
Invariance of IDS under Darboux transformation and its application
The integrated density of states (IDS) is a crucial quantity for studying spectral properties of ergodic Schroedinger opearators. Especially in one dimension it determines most of the spectral properties. On the other hand, in relation to completely integrable systems, Darboux transformation has been investigated from various points of views. In this talk the invariance of IDS under Darboux transformation will be shown, and as a byproduct the invariance of IDS under KdV flow will be remarked.
PEPW04 24th June 2015
13:30 to 14:30
Behavior of the spectrum of the periodic Schrodinger operators near the edges of the gaps
Co-author: Leonid Parnovski (UCL)

It is a common belief that generically all edges of the spectrum of periodic Schrodinger operators are non-degenerate, i.e. are attained by a single band function at finitely many points of quasi-momentum and represent a non-degenerate quadratic minimum or maximum. We present the construction which shows that all degenerate edges of the spectrum can be made non-degenerate under arbitrary small perturbation. The corresponding perturbation is found in the class of potentials with larger (but proportional) periods; thus the final operator is still periodic but the lattice of periods changes.

PEPW04 24th June 2015
15:00 to 16:00
$L_1$-Estimates for Eigenfunctions of the Dirichlet Laplacian
Co-authors: Michiel van den Berg (U Bristol), J\"urgen Voigt (TU Dresden)

For $d \in {\bf N}$ and $\Omega \ne \emptyset$ an open set in ${\bf R}^d$, we consider the eigenfunctions $\Phi$ of the Dirichlet Laplacian $-\Delta_\Omega$ of $\Omega$. We do {\it not} require $\Omega$ to be of finite volume. % If $\Phi$ is associated with an eigenvalue below the essential spectrum of $-\Delta_\Omega$, we provide estimates for the $L_1$-norm of $\Phi$ in terms of the $L_2$-norm of $\Phi$ and suitable spectral data of $-\Delta_\Omega$. The main idea in obtaining such estimates consists in finding a---sufficiently small---subset $\Omega' \subset \Omega$ where $\Phi$ is localized in the sense that $\Phi$ decays exponentially as one moves away from $\Omega'$. These $L_1$-estimates are then used in the comparison of the heat content of $\Omega$ at time $t>0$ and the heat trace at times $t' > 0$, where a two-sided estimate is established. \vskip.5em This is joint work with Michiel van den Berg (Bristol) and J\"urgen Voigt (Dresden), with improvements by Hendrik Vogt (Dresden).

PEPW04 25th June 2015
10:00 to 11:00
Spectral theory of the Schr?dinger operators on fractals

Spectral theory of the Schrodinger operators on fractals (Stanislav Molchanov UNC Charlotte)

 

Spectral properties of the Laplacian on the fractals as well as related topics (random walks on the fractal lattices, Brownian motion on the Sierpinski gasket etc.) are well understood. The next natural step is the analysis of the corresponding Schrodinger operators and not only with random ”ergodic” potentials (Anderson type Hamiltonians) but also with the classical potentials: fast decreasing, increasing or ”periodic” (in an appropriate sense) ones. The talk will present several results in this direction. They include a) Simon – Spencer type theorem (on the absence of a.c. spectrum) and localization theorem for the fractal nested lattices (Sierpinski lattice) b) Homogenization theorem for the random walks with the periodic intensities of the jumps c) Quasi-classical asymptotics and Bargman type estimates for the Schr?dinger operator with the decreasing gasket d) Bohr asymptotic formula in the case of the increasing to infinity potentials e) Random hierarchical operators, density of states and the non-Poissonian spectral statistics Some parts of the talk are based on joint research with my collaborators (Yu. Godin, A. Gordon, E. Ray, L. Zheng).

PEPW04 25th June 2015
11:30 to 12:30
P Hislop Eigenvalue statistics for random Schrodinger operators
Certain natural random variables associated with the local eigenvalue statistics for generalized Anderson-type random Schrodinger operators on the lattice and the continuum in the localization region are distributed according to compound Poisson distributions. In the lattice case the Levy measure of the associated distribution has finite support. Other properties of these random variables will be presented.
PEPW04 25th June 2015
13:30 to 14:30
Localisation and ageing in the parabolic Anderson model
The parabolic Anderson problem is the Cauchy problem for the heat equation on the d-dimensional integer lattice with random potential. It describes the behaviour of branching random walks in a random environment (represented by the potential) and is being actively studied by mathematical physicists. One of the most important situations is when the potential is time-independent and is a collection of independent identically distributed random variables. We discuss the intermittency effect occurring for such potentials and consisting in increasing localisation and randomisation of the solution. We also discuss the ageing behaviour of the model showing that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time.
PEPW04 25th June 2015
15:00 to 16:00
The invariant measure for random walks on a strip
We explain the necessary and sufficient condition for existence of the invariant measure for the Markov chain on the space of ergodic environments in all regimes.
PEPW04 25th June 2015
16:00 to 17:00
Green's function asymptotic behavior near a non-degenerate spectral edge of a periodic operator
Co-authors: Minh Kha (Texas A&M University), Andrew Raich (University of Arkansas)

Green's function behavior near and at a spectral edge of a periodic operator is one of what was called by M. Birman and T. Suslina "threshold properties." I.e., it depends on the local behavior of the dispersion relation near the edge. The recent results are presented for the case of a non-degenerate spectral edge (which is conjectured to be the generic situation).

This is a joint work with Minh Kha (Texas A&M) and Andrew Raich (Univ. of Arkansas)

PEPW04 26th June 2015
10:00 to 11:00
Recent results and open problems on many-body localization
In the most recent decade the topic of many-body localization, understood as the absence of thermalization in interacting quantum many-body systems, has seen strong attention and rapid development in the physics literature. We will survey the relatively small number of mathematically rigorous results on MBL which have been obtained, in particular for disordered oscillator systems and some models of quantum spin chains. Specifically, we will consider manifestations of MBL such as absence of many-body transport, exponential decay of ground and thermal state correlations, as well as area laws for the entanglement of states. We will also mention some of the many open problems in this field.
PEPW04 26th June 2015
11:30 to 12:30
F Klopp Interacting quantum particles in a 1 D random background
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons