Straightening and moving the snake - localised clusters in passive and active Phase Field Crystal modelsMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

First, we consider the bifurcation structure of localised solutions for the standard (passive or thermodynamic) phase field crystal (PFC) model [aka conserved Swift-Hohenberg (cSH) equation with cubic nonlinearity] that may be obtained as a local approximation of a dynamical density functional theory (DDFT). It is thought to provide the simplest microscopic continuum description of the thermodynamic transition from a fluid state to a crystalline state and is frequently used to model colloidal crystallisation [1].

After introducing the cSH equation as an example of a conserved dynamics, its steady states and their bifurcations are analysed. In particular, We focus on the variety of spatially localized structures, that are found in addition to periodic structures. The location of these structures in the temperature versus mean concentration plane is determined using a combination of numerical continuation in one and two dimensions and direct numerical simulation in two and three dimensions. Localized states are found in the region of thermodynamic coexistence between the homogeneous and structured phases, and may lie outside of the binodal for these states. The results are related to the phenomenon of slanted snaking but take the form of standard aligned homoclinic snaking when the solutions are plotted employing the chemical potential as main control parameter [2]. We also show how the localised states are related to the Maxwell construction of the crystallisation isotherm.

Second, We consider the active phase field crystal (aPFC) model [3] that results as a combination of the passive PFC model and the Toner-Tu theory for self-propelled particles. It is considered a simple model that describes the transition between resting and traveling ('living') crystals. In the linear regime, analytical expressions for the transitions from the liquid state to both types of space-filling crystals are obtained and the resulting phase-diagram is discussed [4]. In the nonlinear regime, we describe a variety of localized resting and travelling clusters that may exist besides spatially extended crystals. We provide a semi-analytical criterion for the onset of motion in the nonlinear regime, that corresponds to drift pitchfork and drift transcritical bifurcations [5]. Numerical continuation is applied to follow resting and traveling localized states while varying the activity and the mean concentration as well as to determine their regions of existence. Based on this information we finally discuss teh scattering behaviour of the found travelling localised states.

[1] H. Emmerich, H. Löwen, R. Wittkowski, T. Gruhn, G. Toth, G. Tegze, and L. Granasy, Adv. Phys. 61, 665 (2012). [2] U. Thiele, A. J. Archer, M. J. Robbins, H. Gomez, and E. Knobloch, Phys. Rev. E. 87, 042915 (2013). [3] A. M. Menzel and H. Löwen, Phys. Rev. Lett. 110, 055702 (2013). [4] A. I. Chervanyov, H. Gomez and U. Thiele, EPL 115, 68001 (2016). [5] L. Ophaus, S. V. Gurevich and U. Thiele, PRE 98, 022608 (2018).

Finite-Difference Discretization of the Ambrosio-Tortorelli FunctionalsMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

Motivated by applications to image reconstruction, in this talk we present some results on the asymptotic behavior of a finite-differences discretization of the Ambrosio-Tortorelli functionals. These functionals provide an approximation via Gamma-convergence of the Mumford-Shah functional. Denoted by epsilon the approximation parameter of the Ambrosio-Tortorelli functionals and by delta the discretization step-size, we fully describe the relative impact of epsilon and delta in terms of Gamma-limits for the corresponding discrete functionals, in the three possible scaling regimes. We show, in particular, that when varepsilon and delta are of the same order, the underlying lattice structure affects the Gamma-limit, which turns out to be an anisotropic free-discontinuity functional. This is joint work with Caterina Ida Zeppieri (Münster) and Andrea Braides (Rome).

Variational Convergence of Discrete Euler ElasticaMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

In this talk, I am going to give a brief exposition of my recent work on discrete Euler elastica (joint work with Sebastian Scholtes and Max Wardetzky). Discrete Euler elastica are critical points of the Euler-Bernoulli bending energy subject to inextensibility constraints and clamped boundary conditions.

A very popular discrete bending energy for polygonal lines has been proposed by Hencky; it basically consists of a weighted sum of squared turning angles. We show that, under mesh refinement, a certain subset of almost minimizers of this energy (subject to inextensibility constraints and clamped boundary conditions) converges to the set of smooth minimizers: When using piecewise-linear interpolation as reconstruction operator, we obtain Hausdorff-convergence in the space of Lipschitz curves. However, the proof hinges on a reconstruction operator that embeds discrete curves into $BV^3$ (the space of curves whose third derivative is a measure of bounded variation). Modulo this operator, we even obtain Hausdorff-convergence in the Sobolev space $W^{2,p}$, $p <\infty$.

The main observation needed for the proof is that the intersections of a compactum with a decreasing sequence of sublevel sets of a lower semi-continuous function converges in Hausdorff distance to the set of minimizers. In order to exploit this observation, we have to take the following steps: (i) a~priori analysis of the minimizers in \emph{both} the smooth and discrete setting, (ii) estimation of the energies' consistency errors and of the constraint violations due to discretization/smoothing, and (iii) repairing these constraint violations by means of the Newton-Kantorovich theorem.

If time allows, I will briefly discuss modifications to this machinery that are needed for treating the convergence analysis of discrete minimal surfaces.

Physics-Informed Learning Machines for Physical SystemsMI HS 3 (Boltzmannstr. 3, 85748 Garching)

In this talk, we will present a new approach to develop a data-driven, learning-based framework for predicting outcomes of physical systems and for discovering hidden physics from noisy data.

A key concept is the seamless fusion and integration of data of variable fidelity into the predictive models. First, we will present a Bayesian approach using Gaussian Process Regression (GPR), and subsequently a deep learning approach based on neural networks (NNs) and generative adversarial networks (GANs). Unlike other approaches that rely on big data, here we “learn” from small data by exploiting the information provided by the physical conservation laws, which are used to obtain informative priors or regularize the neural networks. We will also make connections between GPR and NNs and discuss the new powerful concept of meta-learning.

Wild solutions to the Navier-Stokes equationMI HS 3 (Boltzmannstr. 3, 85748 Garching)

We consider the uniqueness question for weak solutions of the 3D Navier-Stokes equation. We prove that solutions are not unique within the class of weak solutions with finite kinetic energy. Moreover, we prove that any continuous weak solutions of the 3D Euler equations may be obtained as a strong vanishing viscosity limit.

A minimization approach (in De Giorgi's style) to the wave equation on time-dependent domainsMI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

We study weak solutions to the wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals in space-time.

Verschiedene Aspekte von Differentialgleichungen in arithmetischer Geometrie - Mathematisches Kolloquium der LMUA027 (Haus B , 80333 Theresienstr. 39)

Wir geben einen Überblick über die Reduktion modulo einer Primstelle von Differentialgleichungen, mit Hinblick auf Grothendiecks p-Krümmung Vermutung (noch offen) und Simpsons motivische Vermutungen (einige Punkte sind hier seit kurzem bekannt).

Alle Interessierten sind hiermit herzlich eingeladen. Eine halbe Stunde vor dem Vortrag gibt es Kaffee und Tee im Sozialraum (Raum 448) im 4. Stock.

High-dimensional optimization tasks with tensor network state methods MI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

Tensor network states and specifically matrix-product states have proven to be a powerful tool for simulating ground states of strongly correlated spin and fermionic models. In this contribution, we overview tensor network states techniques that can be used for the treatment of high-dimensional optimization tasks used in many-body quantum physics with long range interactions and ab initio quantum chemistry. We will also discuss the controlled manipulation of the entanglement, which is in fact the key ingredient of such methods, and which provides relevant information about correlations. We will present recent developments on fermionic orbital optimization, tree-tensor network states, multipartite entanglement, externally corrected coupled cluster density matrix renormalization group (TCCSD-DMRG).

Bidimensional Election Systems: Apportionment in theory and practiceMI 02.04.011 (Boltzmannstr. 3, 85748 Garching)

In Scandinavia, as well as in many other countries, the election systems to the national assemblies are in a way bidimensional: Seats are apportioned within constituencies but with respect to the national outcome of the parties. For that purpose, the seats are divided into proper constituency seats and adjustment seats. The allocation of the latter is mathematically interesting but politically controversial. Balinski and Demange have presented fairness properties which allocation methods of this kind (i.e. of the adjustment seats) should respect. They prove that this demand leads to one and only one method – given a specific underlying one-dimensional divisor rule like that of d’Hondt or Sainte-Laguë. This optimal solution can also be formulated as a simple linear optimal assignment problem. Pukelsheim has managed to convince law makers in the Canton of Zürich, to adopt this optimal allocation method (based on dual multipliers). The speaker, who has been advising the Parliament and Governments in Iceland (one of the Scandinavian countries) for over a quarter of a century on election systems, has however the experience that politicians, lawyers and political scientists will only accept recursive algorithms for seat apportionments. Iterative methods are not agreeable but the optimal solution calls for iterations. Consequently, practical allocation methods for bidimensional election systems are inevitably only approximations to the optimal method. In the talk several near optimal allocation methods will be presented, many of which are derived from heuristics for the classical transportation problem (Monge, Vogel, Russel). To test the practicality and quality of these methods a simulation model has been developed and will be presented in the talk. This model generates random election outcomes (with user-given averages, e.g. actual election results). The seats are then allocated using the different heuristic methods. The quality of each method is measured using different indicators, classical and new, thus enabling a ranking of the tested methods, in particular in comparison with the optimal method.

Renormalization of the hierarchical Anderson modelMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

We will discuss Anderson localization in the context of a Dyson-hierarchical approximation to the classical lattice model. The hierarchical structure of this model enables an exact renormalization group in the spirit of Wilson and Fisher. From this perspective, the localization question amounts to studying the trajectories of a discrete-time dynamical system in the space of probability densities. This talk is based on joint work with Simone Warzel.

Indirect Measurements in Quantum Mechanics B 133 (Theresienstr. 39, 80333 München)

We consider a quantum dot in a semi-conductor device $P$ and a train of probes sent to it. A projective measurement is applied to each probe, just after it interacts with $P$. This produces a discrete density-matrices valued stochastic process $rho_n$. We prove that the density matrices, $rho_n$, purify as n tends to infinity. Our results can be used to mathematically understand situations in the spirit of experiments of Haroche and Wineland (which led to the Nobel Prize in Physics 2012).

Scaling limit of a self-avoiding walk interacting with spatial random permutatinosB 252 (Theresienstr. 39, 80333 München)

We consider a self-avoiding walk conditioned to cross a large box and weighted with an energy proportional to the number of steps it takes to do so. We embed this walk into a background of self-avoiding nearest-neighbour loops or, equivalently, nearest-neighbour spatial random permutations.

Low regularity integrators for the Schrödinger equationRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

A large toolbox of numerical schemes for the nonlinear Schrödinger equation has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). In many situations these classical schemes allow a precise and efficient approximation. This, however, drastically changes whenever "non-smooth" phenomena enter the scene such as for problems at low-regularity and high oscillations. Classical schemes fail to capture the oscillatory parts within the solution which leads to severe instabilities and loss of convergence. In this talk I present a new class of Fourier integrators for the nonlinear Schrödinger equation at low-regularity. The key idea in the construction of the new schemes is to tackle and hardwire the underlying structure of resonances into the numerical discretization. These terms are the cornerstones of theoretical analysis of the long time behaviour of differential equations and their numerical discretizations and offer the new schemes strong geometric structure at low regularity.

Models for ramified transportation networks and their approximationsRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

A small number of models for transportation networks (modelling street, river, or vessel networks, for instance) has been studied intensely during the past decade, in particular the so-called branched transport and the so-called urban planning. They assign to each network the total cost for transporting material from a given initial to a prescribed final distribution and seek the cost-optimal network. Typically, the considered transportation cost per mass is smaller the more mass is transported together, which leads to highly patterned and ramified optimal networks. We present a joint framework for a broad class of transportation network models (including existing models) based on the geometric measure notion of flat 1-chains, which more easily allows to prove equivalence of various existing model formulations. In addition, based on this setting we present multiple phasefield approximations, some of which had been developed in the literature as phenomenological models without awareness of the connection to branched transport.

The Geometry of Model Uncertainty - Mathematisches Kolloquium, LMUA027 (A027 , 80333 Theresienstr. 39)

Starting with the "Black Monday" 1987 (if not earlier) and down to the present day, the over-confidence in mathematical models and the failure to account for model uncertainty have frequently been blamed for their infamous role in financial crisis. Remarkably, it remains an open challenge to quantify the effects of model uncertainty in a coherent way. From a mathematical persepctive, this is a delicate issue which touches on deep classical problems of stochastic analysis. Recent work has established new links to the field of optimal mass transport. This yields a powerful geometric approach to model uncertainty and a number of related problems in the theory of stochastic processes.

A mechanism for holography for non-interacting fields on anti-de Sitter spacetimesMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

The first goal of the talk will be to provide an introduction to linear classical and quantum fields on asymptotically anti-de Sitter spacetime. I will discuss in particular Klein-Gordon fields with Dirichlet, Neumann or Robin boundary conditions. Then, I will explain how holography results for classical fields can be used to prove the inclusion of quantum bulk observables in an algebra of boundary observables (based on joint works with Wojciech Dybalski and Oran Gannot).

Removing velocity superselection with infravacuum representationsMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

In nonrelativistic QED, the electron as an infraparticle exhibits velocity superselection, namely plane-wave configurations of the electron with different velocities give rise to inequivalent representations of the algebra of the asymptotic electromagnetic field in the infrared limit. Moreover, as another feature of the infrared problem, the Hamiltonian has no well-defined ground state in this realm. These properties make the construction of scattering states of electrons a difficult task. We approach these problems by viewing the electron on a new background state, the infravacuum state, which generates a new class of representations. In a model of one spinless electron interacting with the quantized electromagnetic field, we present two implementations of the infravacuum picture. The first one leads to the absence of velocity superselection, while in the second one such property persists and disappears only at the level of conjugate sectors, a situation which is unusual in the conventional DHR superselection theory.

Interacting dynamics for multi-time wave functionsMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

Transferring the idea of a wave function to relativistic quantum mechanics naturally leads to the concept of a multi-time wave function psi(t_1,x_1,...,t_N,x_N), proposed by Dirac already in 1932. The dynamics of psi is governed by a system of N Schrödinger/Dirac equations. I present mathematical results from my PhD thesis about interacting dynamics for such a multi-time wave function. I explain why the usual formulation of interaction via potentials is not viable for multi-time wave functions. The reason for this is an integrability condition necessary for the existence of a common solution to the system of equations. Then, I give the rigorous formulation of a model by Dirac, Fock, Podolsky, which achieves interaction by a second-quantized field and for which existence and uniqueness of solutions can be proven.

Phase Retrieval from Local Correlation Measurements with Fixed Shift LengthMI 02.10.011 (Boltzmannstr. 3, 85748 Garching)

We consider the natural extension of phase retrieval with local correlation measurements with shifts of length one to any fixed length size. As a result, we provide algorithms and recovery guaranties for extended model, as well as suitable measurements constructions.

Max-Linear Graphical Models via Tropical Geometry2.02.01 (Parkring 11, 85748 Garching-Hochbrück)

Motivated by extreme value theory, max-linear graphical models have been recently introduced and studied as an alternative to the classical Gaussian or discrete distributions used in graphical modeling. We present max-linear models naturally in the framework of tropical geometry. This perspective allows us to shed light on some known results and to prove others with algebraic techniques, including conditional independence statements and maximum likelihood parameter estimation. This is joint work with Claudia Klüppelberg, Steffen Lauritzen and Ngoc Tran.

A GSPT approach to perturbed SIR and SIRWS modelsMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

In this seminar we use tools from geometric singular perturbation theory to analyse two perturbed versions of the classic SIR model, and a modified version of the SIRWS model proposed by Dafilis et al. in 2012. The main tool we exploit is the entry-exit function, as presented by De Maesschalck and Schecter in 2015, which gives details regarding the behaviour of the orbit near the critical manifold.

Geometry of Gaussian free field sign clusters and random interlacementsB 252 (Theresienstr. 39, 80333 München)

We consider two fundamental percolation models with long-range correlations: Level set percolation for the Gaussian free field (GFF) and percolation of the vacant set of random interlacements. Both models have been the subject of intensive research during the last decades. In this talk we focus on structural properties of the level set percolation of the GFF. In particular, we establish the non-triviality of a phase in which two infinite sign clusters dominate, and their complement only has small connected components. While the respective results are new for $\Z^d$ as the underlying graph, we also cover more intricate geometries such as transient graphs with subdiffusive random walk behavior. As a consequence, we answer an open problem on the non-triviality of the phase transition of the vacant set of Random Interlacements on such geometries.

Second-order optimality conditions in optimal control problemsGebäude 33, Raum 1401 (Werner-Heisenberg-Weg 39, 85577 Neubiberg)

In the talk we give a review on second-order optimality conditions for optimal control problems subject to partial differential equations and variational inequalities. Such conditions imply, e.g., local optimality, local uniqueness of critical points, and stability of solutions with respect to perturbations. In addition, they are crucial in proofs of a-priori and a-posteriori error estimates for numerical approximations.

Some recent results on growth of Sobolev norms for time dependent Schödinger evolutionsMI 03.10.011 (Boltzmannstr. 3, 85748 Garching)

When a time independent Schrödinger Hamiltonian H0 is perturbed by a time dependent potential V (t) the modes of H0 are not preserved by the evolution generated by the quantum Hamiltonian H0 + V (t) and exchange of energies may occur. We shall present a general approach connecting spectral properties of H0, the size and the oscillations in time of the perturbation V (t).

Charaktere endlicher Gruppen - Mathematisches Kolloquium, LMUA027 (A027 , 80333 Theresienstr. 39)

Obwohl die Charaktertheorie durch Frobenius bereits vor mehr als 100 Jahren begruendet wurde, gibt es in diesem wichtigen Gebiet der Algebra weiterhin viele grundlegende offene Fragen. In meinem Vortrag werde ich einige recht einfach zu formulierende Vermutungen, die von mehr kombinatorischen bis zu sehr strukturellen Aussagen reichen, vorstellen und neue Resultate dazu erlaeutern.

Generic Chaining Applied to the RIP of Structured Random MatricesMI 02.10.011 (Boltzmannstr. 3, 85748 Garching)

The approaches by Haviv/Regev and Rudelson/Vershynin show a restricted isometry property of bounded orthonormal systems. Both of these approaches rely on generic chaining. This technique can be shown to be optimal up to constants. However, when it is applied to specific situations, often non-sharp estimates are used that lead to logarithmic factors in the overall result. Our goal is to analyze such estimates in the aforementioned proofs and develop approaches that can improve logarithmic factors in the number of required rows of the matrices.

OPTIMAL RISK SHARING IN PUBLIC PENSION SCHEMESBC1 2.02.01 (Parkring 11, 85748 Garching)

Traditionally, public pension schemes, organized in a social security framework, use a pay as you go technique (PAYG); from the benefit point of view, they are based on a Defined Benefit (DB) or a Defined Contribution (DC) approach. This dichotomy follows two extreme philosophies of risk spreading between the stakeholders: in DB, the organizer of the plan bears the risks; in DC (including the Notional accounts – NDC), the affiliates must bear the risks. Especially applied to social security, this traditional polar view can lead to unfair intergenerational equilibrium in both cases. The purpose of this presentation is to propose, in PAYG, alternative hybrid architectures based on a mix between DB and DC, in order to achieve simultaneously financial sustainability and social adequacy in a stochastic environment. Using different stochastic models for the risk factors, we propose different levels of optimality in terms of architecture of the pension scheme.

Extreme Value Analysis of Multivariate Time Series: Multiple Block sizes and Overlapping BlocksBC1 2.02.01 (Parkring 11, 85748 Garching)

The core of the classical block maxima method in (multivariate) extreme value statistics consists of fitting an extreme value distribution to a sample of maxima over blocks extracted from an underlying time series. Traditionally, the maxima are taken over disjoint blocks of observations of a fixed size. Alternatively, the blocks can be chosen to be of varying size and to slide through the observation period, yielding a larger number of overlapping blocks. Nonparametric estimation of extreme value copulas based on sliding blocks is found to be more efficient than estimation based on disjoint blocks.

On Kendall's regressionBC1 2.02.01 (Parkring 11, 85748 Garching)

Conditional Kendall's tau is a measure of dependence between two random variables, conditionally on some covariates. We assume a regression-type relationship between conditional Kendall's tau and some covariates, in a parametric setting with a large number of transformations of a small number of regressors. This model may be sparse, and the underlying parameter is estimated through a penalized criterion. We prove non-asymptotic bounds with explicit constants that hold with high probabilities. We derive the consistency of a two-step estimator, its asymptotic law and some oracle properties. We show how the problem of estimating conditional Kendall's tau can be rewritten as a classification task. We detail specific algorithms adapting usual machine learning techniques, including nearest neighbors, decision trees, random forests and neural networks, to the setting of the estimation of conditional Kendall's tau. Finite sample properties of these estimators and their sensitivities to each component of the data-generating process are assessed in a simulation study. Finally, we apply all these estimators to a dataset of European stock indices.

Hamiltonian monodromy and its generalisationsMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

Hamiltonian monodromy was introduced by Duistermaat as an obstruction to the existence of global action-angle coordinates in integrable Hamiltonian systems. Since then, this invariant was found to be non-trivial in various specific examples of such systems; for instance, in the spherical pendulum, the Lagrange top, and the hydrogen atom in crossed fields.

In the present talk, we shall discuss Hamiltonian monodromy and some of its generalisations. In particular, we shall discuss the so-called fractional monodromy, which generalises integer Hamiltonian monodromy to the case of singular fibers, and show how one can compute this invariant in integrable systems with a global circle action.

This is a report on a joint work with K. Efstathiou.

The Henderson problemB 252 (Theresienstr. 39, 80333 München)

The inverse Henderson problem of statistical mechanics concerns classical particles in continuous space which interact according to a pair potential depending on the distance of the particles. Roughly stated, it asks for the interaction potential given the equilibrium pair correlation function of the system. In 1974 Henderson proved that this potential is uniquely determined in a canonical ensemble and he claimed the same result for the thermodynamical limit of the physical system. Here we provide a rigorous proof of a slightly more general version of the latter statement using Georgii's version of the Gibbs variational principle.

Introduction to pde2path (and some recent additions) MI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

pde2path is a Matlab package for numerical continuation and bifurcation analysis of partial differential equations. Its main design goals are flexibility (to be able to treat large classes of problems), easy usage, and hackability (easy customization). It is based on the finite element method and arclength continuation. Currently, it can handle branches of steady solutions and time periodic orbits for various classes of systems of PDEs, and the associated bifurcations.

We first briefly review the basic setup using the Allen-Cahn equation as a simple example, and then explain some more advanced features such as bifurcations of higher multiplicity, including some connections to geometry by considering problems of pattern formation on curved surfaces.

Long Time Behaviour of Heat Kernels - Mathematisches Kolloquium, LMUA027 (A027 , 80333 Theresienstr. 39)

We study long time behaviour of heat kernels and show convergence of the semigroup to the ground state and convergence of suitably average logarithms of kernels to the ground state energy. The results hold for arbitrary selfadjoint positivity improving semigroup. This framework includes Laplace operator on manifolds, on graphs and on quantum graphs. (Joint work with Matthias Keller, Hendrik Vogt, Radoslaw Wojciechowski)