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On a time-dependent Kohn-Sham equation and related optimal control problemsMI 02.08.011 (Boltzmannstr. 3, 85748 Garching)

A Kohn-Sham equation is a system of nonlinear coupled Schrödinger equations that describe multi-particle quantum systems in the framework of the time-dependent density functional theory.

In the first part of this talk, existence, uniqueness and regularity of solutions to a time-dependent Kohn-Sham equation are investigated. Further, in view of control applications, the presence of a control function and of an inhomogeneity are also considered.

The second part of this talk is devoted to application models in quantum physics and chemistry that require to control many-electron systems to achieve a desired target configuration. In particular, the theory and numerical solution of optimal control problems governed by a Kohn-Sham model are discussed considering different objectives and a bilinear control mechanism. Results of numerical experiments demonstrate the computational effectiveness of the proposed control framework.

The sphere packing problem in dimensions 8 and 24.MI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

In this talk we will show that the sphere packing problem in dimensions 8 and 24 can be solved by a linear programing method. In 2003 N. Elkies and H. Cohn proved that the existence of a real function satisfying certain constrains leads to an upper bound for the sphere packing constant. Using this method they obtained almost sharp estimates in dimensions 8 and 24. We will show that functions providing exact bounds can be constructed explicitly as certain integral transforms of modular forms. Therefore, we solve the sphere packing problem in dimensions 8 and 24.

Singularity analysis for heavy-tailed random variablesMI 00.07.011 (Boltzmannstr. 3, 85748 Garching)

We provide a new proof of five theorems by A. V. Nagaev on sums of i.i.d. $\mathbb{N}_0$-valued random variables with stretched exponential tails. The proof method is based on complex analysis and is a natural extension of contour integral proofs for local limit laws; it provides a unifying framework encompassing local central limit theorems, moderate and large deviations theorems for integer-valued heavy-tailed random variables. (Joint work with Nicholas M. Ercolani and Daniel Ueltschi.)

The Z-invariant Ising model via dimersMI 00.07.011 (Boltzmannstr. 3, 85748 Garching)

The Z-invariant Ising model is an Ising model in dimension 2 defined on an embedded isoradial graph. Coupling constants satisfy Yang-Baxter equations; they depend on a parameter k, interpreted as the external temperature. For the specific value k=0, one recovers the critical Ising model. We study this Ising model through the corresponding dimer model, associated via Fisher's correspondence. We prove an explicit expression for probabilities of the dimer model, only depending on the local geometry of the graph. We prove an explicit and also local expression for the free energy of the Ising model. We establish an order 2 phase transition in the parameter k for the Ising model, and show that this phase transition is the same as that of the Z-invariant spanning forest model. This is joint work with Cédric Boutillier and Kilian Raschel.

On a random dynamical system generated by Volterra quadratic operatorsMI 02.08.011 (Boltzmannstr. 3, 85748 Garching)

At the center of this talk is a random dynamical system (RDS) generated by a special class of Volterra quadratic operators on the simplex \(S^{m1}\) that can be considered forward and backward in time. In contrast to the deterministic set-up the trajectories of the forward RDS converge to the vertices of \(S^{m-1}\) a.s. implying the absence of coexistence of species if one interprets the elements of the simplex as distributions on \(m\) types of species. We will discuss the existence of attractors for the RDS and in particular prove that the minimal point attractor for the forward system equals the set of all vertices.

The realizability problem for point processesMI 00.07.011 (Boltzmannstr. 3, 85748 Garching)

The realizability problem is an infinite dimensional version of the classical truncated moment problem which naturally arises from applied fields dealing with the analysis of complex systems, and which is still open in many of its aspects. I will present a short overview of the few characterization results known in literature for the realizability problem and then focus on a particular instance on the $d-$dimensional lattice. Namely, given two functions $\rho_1(i)$ and $\rho_2(i, j)$ non-negative and symmetric on $\mathbb{Z}^d$, we ask whether they are the first two correlation functions of a translation invariant point process. In a recent joint work with Emanuele Caglioti and Tobias Kuna we provide an explicit construction of such a realizing process for any $d\geq 2$ when the radial distribution has a specific form. We also derive from this construction a lower bound for the maximal realizable density and compare it with the already known lower bounds.

Friendly frogs and stable matchingsMI 00.07.011 (Boltzmannstr. 3, 85748 Garching)

Consider the following two-player game. A set of points in $\mathbb{R}^d$ is fixed - we can imagine (for the two-dimensional case) that these are locations of lilypads on a pond. There are two frogs and two players take turns to move a frog to an unoccupied lilypad in such a way that the distance between the frogs is strictly decreased. A player that cannot move loses. We analyze this game and some variants of it, discovering links to a range of models of stable matchings. We focus particularly on the case of random infinite sets, where we use invariance, ergodicity, mass transport and deletion-tolerance to determine game outcomes.

Interpolating between random walk and rotor walkMI 00.07.011 (Boltzmannstr. 3, 85748 Garching)

A rotor walk on a graph is a deterministic process in which the exits from each vertex follow a prescribed periodic sequence. Such walks capture in many aspects the expected behavior of simple random walks, but with significantly reduced fluctuations compared to a typical random walk trajectory. I will introduce a family of stochastic processes on the integers, depending on a parameter p and interpolating between the deterministic rotor walk (p=0) and the simple random walk (p=1/2). These walks will be called p-rotor walks; they are not Markov chains, but they have a local Markov property: for each vertex x the sequence of successive exits from x is a Markov chain. For p-rotor walks with random initial configuration of rotors, I will prove that their scaling limit is a doubly-perturbed Brownian motion. This is based on joint work with Wilfried Huss (Graz University of Technology) and Lionel Levine (Cornell University).

Pathwise duality of monotone and additive Markov processes and applicationsMI 00.07.011 (Boltzmannstr. 3, 85748 Garching)

Duality between two Markov processes is a very useful tool for studying the distribution (at a fixed time) of one process via the distribution of another process. Pathwise duality refers to a much stronger connection of the two processes in the sense that it gives a coupling on the same probability space, and thus allows for almost sure statements over time intervals. There is little general theory on determining (pathwise) dual processes to Markov processes with respect to appropriate duality functions. In this talk, we present a systematic treatment of monotonicity-based pathwise dualities for Markov processes taking values in (finite) partially ordered sets. We show that every such Markov process whose generator can be represented in monotone maps has a pathwise dual process. In the special setting of attractive spin systems this has been discovered earlier by Gray. We also show that the dual simplifies greatly when the state space is a lattice and all monotone maps satisfy an additivity condition. This leads to a unified treatment of several well-known dualities, including Siegmund's dual for processes with a totally ordered state space, duality of additive spin systems, and a duality due to Krone for the two-stage contact process, and allows for the construction of new dualities as well.

Travelling wave solutions to the KPP equation with branching noise and recurrence of the support of solutionsMI 00.07.011 (Boltzmannstr. 3, 85748 Garching)

The one-dimensional KPP-equation driven by space-time white noise, \[ \partial_t u = \partial_{xx} u + \theta u - u^2 + u^{\frac{1}{2}} dW, \qquad t>0, x \in \mathbb{R}, \theta>0, \qquad \qquad u(0,x) = u_0(x) \geq 0 \] is a stochastic partial differential equation (SPDE) that exhibits a phase transition for initial non-negative finite-mass conditions. This SPDE arises for instance as the high density limit of particle systems which undergo branching random walks and allow for extra death due to overcrowding. If $\theta$ is below a critical value $\theta_c$, solutions die out to $0$ in finite time, almost surely. Above this critical value, the probability of (global) survival is strictly positive. Let $\theta>\theta_c$. For initial conditions that are ‘’uniformly distributed in space’’, a complete convergence result holds, that is, the corresponding solutions are all in the domain of attraction of a unique non-zero stationary distribution. What can be said for solutions with finite initial mass if we condition on their survival?

Extremal individuals in branching systems2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

Branching processes have been subject of intense and fascinating studies for a long time. In my talk I will present two problems in order to highlight their rich structure and various technical approaches in the intersection of probability and analysis. Firstly, I will present results concerning a branching random walk with the time-inhomogeneous branching law. We consider a system of particles, which at the end of each time unit produce offspring randomly and independently. The branching law, determining the number and locations of the offspring is the same for all particles in a given generation. Until recently, a common assumption was that the branching law does not change over time. In a pioneering work, Fang and Zeitouni (2010) considered a process with two macroscopic time intervals with different branching laws. In my talk I will present the results when the branching law varies at mesoscopic and microscopic scales. In arguably the most interesting case, when the branching law is sampled randomly for every step, I will present a quenched result with detailed asymptotics of the maximal particle. Interestingly, the disorder has a slowing-down effect manifesting itself on the log level. Secondly, I will turn to the classical branching Brownian motion. Let us assume that particles move according to a Brownian motion with drift \mu and split with intensity 1. It is well-know that for \mu\geq \sqrt{2} the system escapes to infinity, thus the overall minimum is well-defined. In order to understand it better, we modify the process such that the particles are absorbed at position 0. I will present the results concerning the law of the number of absorbed particles N. In particular I will concentrate on P(N=0) and the maximal exponential moment of N. This reveals new deep connections with the FKPP equation. Finally, I will also consider -\sqrt{2}<\mu<\sqrt{2} and N_t^x the number of particles absorbed until the time t when the system starts from x. In this case I will show the convergence to the traveling wave solution of the FKPP equation for an appropriate choice of x,t->\infty. The results were obtained jointly with B. Mallein and with J. Berestycki, E. Brunet and S. Harris respectively.

Asset Pricing with Spatial InteractionB 349 (Theresienstr. 39, 80333 München)

We propose a spatial capital asset pricing model (S-CAPM) and a spatial arbitrage pricing theory (S-APT) that extend the classical asset pricing models by incorporating spatial interaction. We then apply the S-APT to study the comovements of Eurozone stock indices (by extending the Fama French factor model to regional stock indices) and the futures contracts on S&P Case-Shiller Home Price Indices. In both cases spatial interaction is significant and plays an important role in explaining cross-sectional correlations. This is a joint work with Xianhua Peng and Haowen Zhong.

Asset Pricing with Spatial InteractionB 349 (Theresienstr. 39, 80333 München)

We propose a spatial capital asset pricing model (S-CAPM) and a spatial arbitrage pricing theory (S-APT) that extend the classical asset pricing models by incorporating spatial interaction. We then apply the S-APT to study the comovements of Eurozone stock indices (by extending the Fama French factor model to regional stock indices) and the futures contracts on S&P Case-Shiller Home Price Indices. In both cases spatial interaction is significant and plays an important role in explaining cross-sectional correlations.

This is a joint work with Xianhua Peng and Haowen Zhong.

Many-body Localization from a random matrix perspectiveMI 03.10.011 (Boltzmannstr. 3, 85748 Garching)

Many-body Localization (MBL) generalizes Anderson localization to many-body systems of interacting particles. MBL has been studied intensively since a decade and the persistence of localization in the presence of interactions is now well established in some particular cases; disordered 1-dimensional lattice spin chains constitute prototypical examples where a localized phase can show up. Instead, the existence of MBL in d>1 or in the continuum remains a debated issue. MBL is usually approached through perturbation theory around the integrable non-interacting limit. However, disorder or energy fluctuations induce unavoidable ergodic spots in the system, requiring to go beyond a purely perturbative analysis. In this talk, I will actually look at the problem starting from a random matrix perspective. Taking energy conservation into account, I will recover the fact that a localized phase do indeed exists in one dimensional lattice systems, but I will also conclude that MBL is destroyed as soon as d>1 or even in d=1 if interactions are not exponentially decaying. In d=1, the theory provides also a description of the material in the vicinity of ergodic spots. The talk is based on work in progress with Wojciech De Roeck.

Can one hear the shape of a random walk?2.01.11 (Parkring 11, 85748 Garching-Hochbrück)

We consider a Gibbs distribution over random walk paths on the square lattice, proportional to a random weight of the path’s boundary . We show that in the zero temperature limit, the paths condensate around an asymptotic shape. This limit shape is characterized as the minimizer of the functional, mapping open connected subsets of the plain to the sum of their principle eigenvalue and perimeter (with respect to the first passage percolation norm). A prime novel feature of this limit shape is that it is not in the class of Wulff shapes.

Quantum Many-Body Systems: Particles and SpectraMI HS 3 (Boltzmannstr. 3, 85748 Garching)

The energy spectrum of a quantum many-body system and its physical properties at low temperature can often be understood in terms of a particle description of the eigenvectors of the Hamiltonian of the system. These (quasi-)particles derive some of their most important properties from the local structure of the interactions and the physical space in which the system resides. I will review old and new results in mathematical physics that describe and exploit this particle structure in order to understand physical properties of the system, including recent results and open problems regarding topologically ordered phases of matter.

Assessing dependence between extreme risksMI HS 3 (Boltzmannstr. 3, 85748 Garching)

Dependence between rare events is of prime concern in risk management. For example, extreme comovements of prices or huge operational losses in different business lines represent a substantial risk for financial institutions. Severe losses or insurance claims can also result from the simultaneous occurrence of violent storms, fires, earthquakes or floods. Modelling such dependencies is one of the objectives of extreme-value theory, an area of research that stands at the crossroads between analysis, statistics, and probability. In this talk, I will show how dependencies between extreme risks can be quantified using copula-based models and illustrate some techniques of inference relevant to this issue.

Der Einsatz eines bestimmten Beweistyps: eine Frage der persönlichen Präferenz der Lehrperson oder ein Ausdruck adaptiver Unterrichtsgestaltung? A 027 (Theresienstr. 39, 80333 München)

http://www.mathematik.uni-muenchen.de/~mathkoll/vortraege/ss16/brunner.php

Stochastic Compactness of Levy Processes2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

We characterize stochastic compactness of a Levy process at \large times", i.e., as t ! 1, and \small times", i.e., as t ! 0, by properties of its associated Levy measure. We shall say that a Levy process Xt; t 0, is in the Feller class at innity (stochastically compact at innity) if there exist nonstochastic functions B(t) > 0, A(t) such that every sequence tk ! 1 contains a subsequence tk0 ! 1 with Xtk0

Non-Negative Dimensionality Reduction in Signal SeparationMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

In this talk, we discuss the application of (non-negative) dimensionality reduction methods in signal separation. In single-channel separation, the decomposition techniques as e.g. non-negative matrix factorization (NNMF) or independent component analysis (ICA) are typically applied to time-frequency data of the mixed signal obtained by a signal transform.

Starting from this classical separation procedure in the time-frequency domain, we considered an additional preprocessing step, in which the dimension of the data is reduced in order to facilitate the computation. Depending on the separation methods, different properties of the dimensionality reduction technique are required. We focused on the non-negativity of the low-dimensional data or - since the time-frequency data is non-negative - rather on the non-negativity preservation beyond the reduction step, which is mandatory for the application of NNMF.

Finally, we discuss the application of the developed non-negative dimensionality reduction techniques to signal separation. We present some numerical results when using our non-negative PCA (NNPCA) and compare its performance with other versions of PCA and different separation techniques, namely NNMF and ICA.

Quantum Transport in Quasicrystalline EnvironmentsB 134 (Theresienstr. 39, 80333 München)

In this talk we discuss how quantum states evolve in simple mathematical models of quasicrystals. The central model is given by the Fibonacci chain, for which we discuss in detail the recent progress that has been obtained regarding rigorous results about the spectrum, the eigenstates and the transport properties. We will describe the known results and some of the mechanisms leading to them. Higher-dimensional models are discussed as well, especially those for which the study may be informed by the recent progress in one dimension.

A Generalization of the Lemma of Littlewood and Offord 02.12.20 (Boltzmannstr. 3, 85748 Garching)

The Lemma of Littlewood and Offord is an important tool for estimating small ball probabilities for of the form \[ \mathbb{P}\left[\sum_{j=1}^{n}\epsilon_{j}x_{j}\in I \right], \] where $\epsilon_{j}$ are independent copies of a Bernoulli random variable with $\epsilon$ taking values in $\lbrace \pm 1 \rbrace$ with equal probability, $x$ is an arbitrary vector with $\vert \min_{j} x_{j}\vert >c$ and $I$ is an open interval of length at most $2c$. This problem has been solved by Erdös and could later be generalized to random variables $\epsilon$ taking values in $\lbrace \pm 1 \rbrace$ with unequal probability via the LYM inequality at the cost of replacing the assumption $\vert \min_{j} x_{j}\vert >c$ with the stronger assumption $\min_{j} x_{j}>c$. In this talk, I will present a generalization of this result for vectors $x$ with arbitrary sign patterns. The proof heavily relies on the combinatorial concept of Sperner families, which also were an important ingredient in the original proof. If time permits, we will also see a generalization to sets $I$ which are the union of $k$ open intervals which is stronger than applying a union bound and an even stronger estimate in the case where it additionally holds that $I=-I$.

Growth and PatternsRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

The formation of spatial patterns is often intrinsically linked to growth processes, as can be observed in phyllotactic plant patterns, or drying coffee spills. The growth mechanism often acts as a selection mechanism for the patterns observed in the bulk. We will survey old and new mathematical results that aim at a universal description of growth processes, characterizing growth speeds and selected patterns in model-independent fashions.

The importance of being just late Room 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

Nematic phase in a system of long hard rodsB 005 (Theresienstr. 39, 80333 München)

We consider a two-dimensional lattice model for liquid crystals consisting of long rods interacting via purely hard core interactions, with two allowed orientations defined by the underlying lattice. For this model, we rigorously prove the existence of a nematic phase, i.e., we show that at intermediate densities the system exhibits orientational order, either horizontal or vertical, but no positional order. This is joint work with A. Giuliani.

The maximum particle of branching random walk in spatially random branching environmentB 005 (Theresienstr. 39, 80333 München)

Branching Brownian motion and branching random walk have been the subject of intensive research during the last decades. We consider branching random walk and investigate the effect of introducing a spatially random branching environment; in this context, we are primarily interested in the positions of (the median of) the maximum particle and the so-called ‘breakpoint'. On an analytic level this corresponds to investigating the fronts of the solutions to a randomized Fisher-KPP equation and to the parabolic Anderson model, as well as their relative backlog.

Local explosion in growth-fragmentation processesB 005 (Theresienstr. 39, 80333 München)

Growth-fragmentation processes describe a family of particles which can grow larger or smaller with time, and occasionally split in a conservative manner. In the self-similar case, it is known that a simple Malthusian condition ensures that the process does not locally explode, in the sense that for all times, the masses of all the particles can be listed in non-increasing order. We shall present here the converse: when this Malthusian condition is not verified, then the growth-fragmentation process explodes almost surely. Our proof involves using the additive martingale to bias the probability measure and obtain a spine decomposition of the process, as well as properties of self-similar Markov processes. (based on a joint work with Robin Stephenson)

tbaMI 02.04.011 (Boltzmannstr. 3, 85748 Garching)

tba

Mean-field bound on the 1-arm exponent for Ising model in high dimensionsB 251 (Theresienstr. 39, 80333 München)

Arm exponents characterize critical behaviour of spatial models in Statistical Mechanics. In this talk I present results concerning the 1-arm exponent for the Ising model in high dimensions. I shall give a comparison with the 1-arm exponent for percolation, which has been obtained recently. Subsequently, I explain our approach for the mean-field bound in the Ising model.

Finite Element Approximations for a fractional Laplace EquationGebäude 33, Raum 0201 (Werner-Heisenberg-Weg 39, 85577 Neubiberg)

In this talk we focus on a direct finite element approximation for the Dirichlet homogeneous problem of the so called integral fractional Laplacian. Namely, we deal with basic analytical aspects required to convey an “almost” complete Finite Element analysis of the problem \[ \mbox{(1)} (-\Delta)^s u = f~\mbox{in}~\Omega, ~~u=0~\mbox{in}~\Omega^c \] where \(\Omega \rightarrow \mathbb{R}^n \) is a bounded domain and the fractional Laplacian of order \( s \) is defined by \[ (-\Delta)^s u(x)= C(n,s) P.V.\int_{\mathbb{R}^n} \frac{u(x)-u(y)}{|x-y|^{n+2s}} \] being \( C(n;s) \) a normalization constant.

Independently of the Sobolev regularity of the source \(f \), solutions of (1) are not expected to be in a better space than \( H^{s+min\{ s, 1/2-\epsilon \}} (\Omega) \) (see[3] and [5]). However, exploiting Hölder estimates developed in [4], we describe how to obtain further regularity results in a novel framework of weighted fractional Sobolev spaces, leading to a priori estimates in terms of the Hölder regularity of the data [2].

After developing a suitable polynomial interpolation theory in these weighted fractional spaces, optimal order of convergence in the energy norm for the standard linear finite element method is proved for graded meshes. We show some numerical experiments which are in full agreement with our theoretical predictions, and illustrate the optimality of the aforementioned estimates. We also devote some words to discuss basic aspects of the implementation code [1].

References

[1] G. Acosta, F.M.Bersetche, J.P. Borthagaray. A short FEM implementation for a 2D homogeneous Dirichlet problem of a fractional Laplacian. (work in progress)

[2] G. Acosta and J. P. Borthagaray. A fractional Laplace equation: regularity of solutions and Finite Element approximations. Preprint available at http://arxiv.org/abs/1507.08970, 2015.

[3] G. Grubb. Fractional Laplacians on domains, a development of Hörmander's theory of \(\mu\) –transmission pseudodifferential operators. Advances in Mathematics, 268:478-528, 2015.

[4] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary. Journal de Mathematiques Pures et Appliquees, 101(3):275-302, 2014.

[5] M. I. Vishik and G. I. Eskin, Convolution equations in a bounded region. Uspehi Mat. Nauk, 20:89-152, 1965. English translation in Russian Math. Surveys, 20:86-151, 1965.

Finite element approximations for a fractional Laplace equationGebäude 33, Raum 0201 (Werner-Heisenberg-Weg 39, 85577 Neubiberg)

https://www.unibw.de/bauv1/veranstaltung/vortrag/20160620_acosta/at_download/file

The Zak transform on proper $G$-spaces and its application in radiation designMI 02.10.011 (Boltzmannstr. 3, 85748 Garching)

The Zak transform on $\mathbb{R}^d$ is an important tool in condensed matter physics, signal processing, time-frequency analysis, and harmonic analysis in general. In this talk, I introduce a generalization of the Zak transform to a class of locally compact $G$-spaces, where $G$ is either a locally compact abelian or a second countable unimodular type I group. This framework unifies previously proposed generalizations of the Zak transform. The Zak transform has invariance properties analog to the classic case and is a Hilbert space isomorphism between the space of $L^2$-functions and a direct integral of Hilbert spaces that can explicitly be determined via a Weil formula for $G$-spaces and a Poisson summation formula for compact subgroups.

As an application, I will discuss the appearance of the Zak transform in radiation design. More precisely, I will show that certain classes of highly symmetric structures can be reconstructed from the diffraction patterns of symmetry-adapted plane waves up to a vectorial phase problem. This generalizes the results in [2,3] on the reconstruction of helical structures from twisted X-ray diffraction patterns.

[1] D. Jüstel: The Zak transform on strongly proper G-spaces and its applications, submitted, arXiv:1605.05168, 2016.

[2] G. Friesecke, R. D. James, D. Jüstel: Twisted X-rays: incoming waveforms yielding discrete diffraction patterns for helical structures, SIAM J. Appl. Math., accepted, arXiv:1506.04240, 2016.

[3] D. Jüstel, G. Friesecke, R. D. James: Bragg-Von Laue diffraction generalized to twisted X-rays, Acta Cryst. A72, 190-196, 2016.

A path From Discrete to Topological groupsA 027 (Theresienstr. 39, 80333 München)

Question: Given a map of discrete groups f:G---->H when is this map induced from a normal subgroup inclusion of topological groups X\subseteq Y by taking path components of G and H? For example, the trivial map on the integers Z--->0 arise in this way via the inclusion $Z\subseteq R$ of the integers in the additive group of real numbers. However, the trivial map on a non commutative group G---> 1 cannot arise from such a normal inclusion of topological groups by taking the induced map on path components. In considering this question one builds from the given map f of discrete groups an associated topological space Q:= H//G= H_hG the "homotopy quotient", and attempts to put a group structure on Q. This is not possible in general, for example if Q has non abelian fundamental group. We will discuss similar constructions which are standard in topology and show that a full answer to the above question is given by a simple extra structure on the given map G-->H and lies with a well known notion of crossed module and an observation due to Quillen. ( joint work with Y. Segev.)

Magic Point Empirical Interpolation: Parametric Integration and ApplicationsMI 02.10.011 (Boltzmannstr. 3, 85748 Garching)

We derive analyticity criteria for explicit error bounds and an exponential rate of convergence of the magic point empirical interpolation method introduced by Barrault et al. (2004). Furthermore, we investigate its application to parametric integration. We find that the method is well-suited to Fourier transforms and has a wide range of applications in such diverse fields as probability and statistics, signal and image processing, physics, chemistry and mathematical finance. To illustrate the method, we apply it to the evaluation to recurrent option pricing problems in finance. Our numerical experiments display convergence of exponential order, even in cases where the theoretical results do not apply.

Fault-tolerance in quantum information processing (Antrittsvorlesung)MI HS 3 (Boltzmannstr. 3, 85748 Garching)

Classical information theory provides quantitative answers to basic questions about communication and computation. In the presence of quantum effects, its basic tenets need to be reassessed as fundamentally novel information-processing primitives become possible. Their potential appears promising, but their realization hinges on our ability to construct mechanisms protecting information against unwanted noise.

In this talk, I will consider two problems associated with communication and computation. First, I will discuss the additivity problem for classical capacities. I will review some of the more recent results in this direction: for continuous-variable channels, quantum generalizations of certain geometric inequalities yield operationally relevant statements. Second, I will discuss the problem of performing gates on information encoded in an error-correcting code, and explain how this relates to automorphisms of the latter.

TBA (Antrittsvorlesung)MI HS 3 (Boltzmannstr. 3, 85748 Garching)

Abstract folgt.

Robust and nonparametric detection of change-points in time series using U-statistics and U-quantiles144 (Ludwigstr. 33, 80539 München)

Tests for detecting change-points in weakly dependent (more precisely: near epoch dependent) time series are studied. Our theory covers many standard models of time series analysis, such as ARMA and GARCH processes. The presentation gives certain emphasis to the basic problem of testing for an abrupt shift in location, but other questions like changes in variability are also considered. The popular CUSUM test is not robust to outliers and can be improved in case of non-normal data, particularly for heavy-tails. The talk investigates CUSUM-type tests based on the 2-sample Wilcoxon statistic or the 2-sample Hodges-Lehmann estimator, which is the median of all pairwise differences between the samples, by analyzing asymptotical properties and by comparing the performance in finite samples via simulation experiments. The 2-sample Hodges-Lehmann estimator is highly robust and has a high efficiency under normality. The asymptotics of the new change-point tests are established under general conditions without any moment assumptions. Both tests offer similarly good power against shifts in the center of the data, but the test based on the Hodges-Lehmann estimator performs superior if a shift occurs far from the center.

Minimising a relaxed Willmore functional for graphs subject to Dirichlet boundary conditionsRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

Den Abstract finden Sie unter: https://www.math.uni-augsburg.de/de/prof/ana/aktuelles/abstract_grunau.pdf

Entanglement in Disordered Quantum Many-Body SystemsA 027 (Theresienstr. 39, 80333 München)

The introduction of disorder into a quantum system gives rise to a localized regime. For one-particle systems, described by random Schroedinger-type operators, the meaning of this is well understood and several essentially equivalent characteristics have been proven for broad classes of examples. Much less settled is the proper description of localization phenomena in interacting many-body systems. In this talk we will focus on upper bounds on quantum entanglement as a possible manifestation of many-body localization. In particular, we will discuss area laws for the entanglement of eigenstates as well as for the time evolution under quantum quenches in some simple models of quantum lattice systems.

Geometric criteria for the uniqueness of the approximative traceRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

The notion of the approximative trace allows to associate boundary traces to individual Sobolev functions in W^{1,p}(\Omega) on general open domains \Omega in R^d. It was introduced by Arendt and ter Elst to study the Dirichlet-to-Neumann operator on rough domains and draws from prior works by Maz'ya, Daners, Warma and Biegert, amongst others. If \Omega is Lipschitz, then the approximative trace agrees with the classical trace operator. For sufficiently irregular domains, however, the approximative trace exhibits a curious non-uniqueness phenomenon: the zero function in W^{1,p}(\Omega) can have a multitude of different approximative traces. In this talk we present novel geometric criteria for the uniqueness of the approximative trace. In particular, the approximative trace is unique on open sets with continuous boundary and on arbitrary connected domains in R^2. Furthermore, we provide an example that shows that the uniqueness of the approximative trace depends on p. These results answer several open questions.