We consider a family of multivariate autoregressive stochastic sequences that restart when hit a neighbourhood of the origin, and study the distributional limit of their stationary distributions when the autoregressive coefficient tends to one, the noise scaling parameter tends to zero, and the neighbourhood size varies. We obtain a non-standard limit theorem where the limiting distribution is a mixture of an atomic distribution and an absolutely continuous distribution whose marginals, in turn, are mixtures of distributions of signed absolute values of normal random variables. Several examples are discussed.

Disordered systems such as glasses and spin glasses pose a challenge to theoretical physics. As a simplification mean-field models where the geometry of interactions is induced by a random graph have been introduced. These models turn out to be intimately related to deep questions in combinatorics that have been studied in their own right for well over half a century. Some of these questions have exciting applications in statistics and computer science. In this talk I am going to give an overview of this interdisciplinary research area and of the new contributions that physics insights have enabled.

Consider Bessel, Jacobi, and Dunkl processes on RN which depend on several positive coupling constants k . These processes describe interacting particle systems of Calogero-Moser-Sutherland type with N particles. For some cases, these processes are related with β -Hermite, -Laguerre, and -Jacobi ensembles. Moreover, for the frozen case k=∞ , the processes degenerate to deterministic or pure jump processes. We show how the generators for these processes lead to analogues of Wigner's semicircle and Marchenko-Pastur limit laws for N→∞ for the empirical distributions of the particles with arbitrary initial empirical distributions by using free convolutions. In the frozen cases, our approach leads to a new proof of well known limit laws for the empirical measures of the zeroes of Hermite, Laguerre, and Jacobi polynomials. The talk is based on joint work with J. Woerner and M. Auer.

A great achievement of physics in the second half of the twentieth century has been the prediction of conformal symmetry of the scaling limit of critical statistical physics systems. Around the turn of the millennium, the mathematical understanding of this fact has progressed tremendously in two dimensions with the introduction of the Schramm-Loewner Evolution and the proofs of conformal invariance of the Ising model and dimers. Nevertheless, the understanding is still restricted to very specific models. In this talk, we will gently introduce the notion of conformal invariance of lattice systems by taking the example of percolation models. We will also explain some recent and partial progress in the direction of proving conformal invariance for a large class of such models.

The assumption that high dimensional data is Gaussian is pervasive in many statistical procedures, due to the ease of analytic tractability this special distribution provides. We explore the relaxation of the Gaussian assumption in Shrinkage estimation, Stein's Unbiased Risk Estimate, and Single Index Models, using two tools that originate in Stein's method: Stein kernels, and the zero bias distribution. Taking this approach leads to measures of discrepancy from the Gaussian that arise naturally from the form of the procedures considered, and result in performance bounds that apply in contexts not restricted to the Gaussian. The resulting bounds typically include an additional term that reflects the cost of deviation from the Gaussian, and that vanishes for the Gaussian, thus recovering this paricular special case. (with Xiaohan Wei, Max Fathi, Gesine Reinert, and Adrien Samaurd)

In this talk, we discuss some limitations of the quantum graph approach in defining Dirichlet forms, Laplacians and/or Brownian motions in case when the graph has vanishing edge lengths. Our model set is the stretched Sierpinski gasket, SSG for short. It is the space obtained by replacing every branching point of the Sierpinski gasket by an interval. (It has also been called ``Hanoi attractor``.) As a result, it is the closure of a countable union of intervals and one might expect that a diffusion on SSG is essentially a kind of gluing together the Brownian motions on the intervals. In fact, there have been several works in this direction. However, there still remains, ``reminiscence`` of the Sierpinski gasket in the geometric as well as analytic structure of SSG and the same should therefore be expected for diffusions. This is a joint work with Patricia Alonso Ruiz (Texas A&M University) and Jun Kigami (Kyoto University).

List of speakers: Jussi Behrndt, Peter Benner, Marko Erceg, Vjekoslav Kovač, Matthias Langer, Martin Lazar, Matko Ljulj, Igor Mezič, Cristhian Montoya, Delio Mugnolo, Boris Muha, Jan Pietschmann, Max von Renesse, Albrecht Seelmann, Peter Stollmann, Josip Tambača, Zoran Tomljanovic, Ninoslav Truhar, Matthias Täufer, Marko Vrdoljak

Bei diesem Vortrag soll dem Hörer zunächst leicht verständlich die Verarbeitung von Kunststoffprozessen nahegebracht werden. Im zweiten Teil des Vortrags wird auf digitale Strategien auf der Grundlage von 3D CFD Simulationen zur Prozessbeschreibung, Prozessanalyse und Prozessoptimierung in der Kunststoffindustrie eingegangen. Hierzu wird an konkreten Industriebeispielen aufgezeigt, welche Möglichkeiten die digitale Prozessoptimierung allgemein und insbesondere im Bezug auf die Ankopplung und Nutzung von KI-Systemen bietet. Mithilfe dieser Beispiele soll dem Hörer das allgemeine Potential dieser Technologien im Hinblick auf die ressourcenschonende Prozessentwicklung und die Produktion von Morgen aufgezeigt und gemeinsam diskutiert werden.

In this talk we consider double phase problems with nonlinear boundary conditions possibly exhibiting a gradient dependence. Under quite general assumptions we prove existence results for such problems, where the perturbations satisfy a suitable behavior in the origin and at infinity. In the second part we focus on a class of Kirchhoff type problems set on a double phase framework, for which the existence of infinitely many solutions is ensured. This talk is based on joint works with S. El Manouni, A. Fiscella, A. Pinamonti and P. Winkert.

Wir betrachten Systeme x'(t) = Ax(t) + Bu(t), x(0) = x_0 mit einem Erzeuger A einer C_0 -Halbgruppe auf einem Banachraum X und B\in \mathcal{L}(U,X), wobei U ebenfalls ein Banachraum ist. Wir studieren zum einen das Konzept Stabilisierbarkeit, d.h. Existenz eines K\in \mathcal{L}(X,U), so dass A+BK eine exponentiell stabile Halbgruppe erzeugt; anders formuliert, die Kontrolle u(t) = Kx(t) stabilisiert das System. Zum anderen betrachten wir schwache Beobachtbarkeit, d.h. Y ist ein weiterer Banachraum, C\in\mathcal{L}(X,Y), und y(t) = Cx(t), dann soll \|x(t)\| kontrollierbar durch eine L_r-Norm von y und durch die Norm von x_0sein. Wir stellen die Beziehung zwischen den beiden Konzepten und sowie hinreichende Kriterien für sie dar.

In this talk we discuss recent results on the asymptotic behaviour of the spectral gap of one-dimensional Schrodinger operators in the limit of large intervals. The spectral gap, being defined as the difference between the first two eigenvalues, is a classical object in the spectral theory of operators but it also appears frequently in more applied settings. As a main objective, we will derive upper and lower bounds on the spectral gap for certain (and quite general) classes of potentials. Doing this, we will also come across some relatively surprising findings (partially, this is based on joint work with M. Täufer).

Random energy models (REM) suggested by B. Derrida turned out to be instrumental in the rigorous progress on statistical physics of disordered systems. Building on their success and motivated by interference phenomena, quantum physics, the Lee-Yang theory of phase transitions, the conjectured links with the Riemann zeta function and random matrix theory, we study several complex-valued versions of the REM. Specifically, we focus on the REM, the so-called generalized REM and on the so-called branching Brownian motion energy model. These are energy-based models of disordered systems, where the energy function is a complex-valued Gaussian process on a tree. In the two latter models, the energy function has strong correlations, while in the REM, the energy function is white noise. This allows us to explore the universality classes of the models by varying the strength of correlations. In all models, we identify the phase diagram, the fluctuations and the distribution of the zeros of the random partition function. To this end, we prove a range of non-standard limit theorems for the sums of strongly correlated random variables. Time permitting, we will discuss an ongoing work on models with microscopic interactions. The talk is based on joint works with Zakhar Kabluchko and Lisa Hartung.

Kurzüberblick:
1. Dr. Guido Zamponi, Evonik Operations, Process Technology & Engineering
Verfahrenstechnische Prozessinnovation im Kontext der Herausforderungen zur Nachhaltigkeit

2. Prof. Dr. Stefan Turek, Dekan der Fakultät für Mathematik an der TU Dortmund
Energy-efficient High Performance Computing Techniques and Technology
Abstract: Jede(r) Studierende in MINT-Fächern (und auch jeder/-e Schüler/-in) befasst sich im Rahmen des Studiums mit dem 'Lösen von linearen Gleichungssystemen', einer Kernaufgabe bei heutigen Computersimulationen. Wir entwickeln und analysieren zusammen die Basis-Algorithmen und erläutern die Hauptprobleme, insbesondere beim Einsatz zur Simulation von 'realistischen' Problemen und das insbesondere auf modernen Computerplattformen ... bis hin zu ganz aktueller Forschung, die versucht, moderne (Hochleistungs)Rechner bzgl. Numerik-, Hardware- und auch Energieeffizienz adäquat ausnutzen zu können.

3. Prof. Dr. Christian Holz-Rau, Fakultät für Raumplanung an der TU Dortmund
Verkehrswende - ein Städtevergleich

In England, the National Centre for Excellence in the Teaching of Mathematics (NCETM) has developed and is implementing a national 'Teaching for Mastery' (TfM) programme to improve mathematics teaching in both primary (ages 5 to 11) and secondary (ages 11 – 16) schools. The programme is informed by mathematics teaching padagogy used in the far east, developed and tailored for England’s primary and secondary schools.
The programme started in 2014 through exchange visits between teachers of maths from schools in England and Shanghai.
The NCETM has also developed a national network of school-led ‘Maths Hubs’ across England and coordinates their work. The Maths Hubs support the professional development of mathematics teachers in their regions and enable the national implementation of the TfM programme.
Both the NCETM and the Maths Hubs are funded centrally by the Department for Education.
The talk will describe the development of the TfM programme and what has been achieved so far.

Vorträge:
Mark Ainsworth (Brown University):
Mass Conserving, High Order, Mixed Finite Element Schemes for Incompressible Flow: Stability, Approximation & Preconditioning
Lutz Tobiska (Universität Magdeburg):
Pressure Robust Discretisations for the Generalised Stokes Problem

A function of two variables and is called a Hardy kernel, if it is homogeneous of degree minus one: . Hardy kernels are usually associated with integral operators on the positive semi-axis. Because of the homogeneity of the kernel, such operators are easily diagonalisable by the Mellin transform, and so the spectral theory of this class of operators is very simple. In the talk, I will discuss spectral properties of matrices , obtained by restrictions of Hardy kernels onto integers. Of course, after restricting to integers, the homogeneity property ``disappears`` and it is no longer clear how to diagonalise such matrices. Nevertheless, it turns out that some progress can be made in their spectral analysis. I will state some general theorems and consider in detail some examples.

We investigate the emergence of rigid polycrystalline structures from atomistic particle systems. The atomic interaction is governed by a suitably normalized pair interaction energy, where the `sticky disk’ interaction potential models the atoms as hard spheres that interact when they are tangential. The discrete energy is frame invariant and no underlying reference lattice on the atomistic configurations is assumed. By means of Gamma-convergence, we characterize the asymptotic behavior of configurations with finite surface energy scaling in the infinite particle limit. The effective continuum theory is described in terms of a piecewise constant field delineating the local orientation and micro-translation of the configuration. The limiting energy is local and concentrated on the grain boundaries, i.e., on the boundaries of the zones where the underlying microscopic configuration has constant parameters. The corresponding surface energy density depends on the relative orientation of the two grains, their microscopic translation misfit, and the normal to the interface. Joint work with Leonard Kreutz (Münster) and Bernd Schmidt (Augsburg).

In the talk we discuss an arbitrary metric graph, to which we glue a graph with edges of lengths proportional to a small parameter . On such graph, we consider a general self-adjoint second order differential operator with varying coefficients subject to general vertex conditions; all coefficients in differential expression and vertex conditions are supposed to be holomorphic in . We introduce a special operator on a special graph obtained by rescaling the aforementioned small edges and assume that it has no embedded eigenvalues at the threshold of its essential spectrum. Under such assumption, we show that that certain parts of the resolvent of the original operator are holomorphic in and we show how to find effectively all coefficients in their Taylor series. This allows us to represent the resolvent of by an uniformly converging Taylor-like series and its partial sums can be used for approximating the resolvent up to an arbitrary power of . In particular, the zero-order approximation reproduces recent convergence results by G. Berkolaiko, Yu. Latushkin, S. Sukhtaiev and by C. Cacciapuoti, but we additionally show that next-to-leading terms in -expansions of the coefficients in the differential expression and vertex conditions can contribute to the limiting operator producing the Robin part at the vertices, to which small edges are incident.

A challenging mathematical problem is to introduce meaningful curvature notions on discrete spaces like graphs (which do not have any smooth structure). An analytic Ricci type curvature, based on Bochner's identity involving a Laplacian (or more generally the generator of a Markov semigroup) and going back to Bakry and Emery in 1985, was initially studied in the context of graphs by Elworthy, Schmuckenschlaeger, Lin/Yau and others. The first aim of this talk is to motivate and to introduce this curvature notion. We will then present a graph theoretical version of Buser's inequality. Buser's inequality links non-negative Ricci curvature, a fundamental isoperimetric constant named after Jeff Cheeger, and the first positive eigenvalue of the Laplace operator. Buser provided back in 1982 two proofs of his inequality (both of them geometric in flavour - one was using Geometric Measure Theory and the Heintze-Karcher inequality while the other was more elementary). An alternative analytical proof in the graph setting and based on the heat semigroup was later given by Klartag/Kosma/Ralli/Tetali in 2016. They utilized a heat semigroup reformulation of Bakry-Emery curvature and their proof is analogous to a corresponding analytical proof by Ledoux in 2004 in the manifold case. This is one of the many instances when the seemingly innocent heat equation is of central importance in completely different and new contexts. If everything goes according to plan, I will finish the talk with an optimal ratio estimate between higher eigenvalues and the first Laplace eigenvalue by combining an improved Cheeger inequality for graphs due to Kwok/Lau/Lee/Oveis Gharan/Trevisan (2013) with the above mentioned Buser inequality. This talk is a review of joint work with Shiping Liu (USTC, Hefei, China).

Speakers: G. Berkolaiko, L. Boulton, H. Cornean, L. Grubisic, E. Korotyaev, P. Kuchment, M. Langer, M. Loss, M. Marletta, S. Morozov, I. Nakic, L. Parnovski, M. Plum, G. Rozenblum, A. Sagiv, L. Schimmer, E. Sere, T. Suslina, M. Täufer, V. Tewary, C. Tretter, K. Veselic

Oscillatory solutions to models of inviscid fluids may be described by probability distribution of certain phase variables in terms of the Young measure. Using the analytical results of Banach - Sachs and Komlos, we propose a method how to visualize/compute effectively these objects. Convergence of the approximate/numerical measure-valued solutions is strong in space and time with respect the standard Wasserstein distance.

The Euler and the Navier-Stokes equations describe the motion of fluids and a smooth solution can be ensured only for small times or large viscosites. It is open if a smooth solution existence for all times or if a break down can appear in finite time. In the formulation of the problem the initial data shall be Schwartz functions. We show in arXiv:1912.11705 that then the vorticity remains a Schwartz function as long as the smooth solution exists. We provide explicit bounds for all seminorms. This provides new break down criteria for the Euler and the Navier-Stokes. Our approach is a Montel space based splitting method since the set of Schwartz functions is a Montel space. Contrary to other (especially Sobolev and weak solution) methods our approach is not affected by the viscosity, i.e., the result holds for the hyperbolic Euler and the parabolic Navier-Stokes equations.

Introducing an interpolation method we estimate the spectral gap for Brownian motion on general domains with sticky-reflecting boundary diffusion associated to the first nontrivial eigenvalue for the Laplace operator with corresponding Wentzell-type boundary condition. In the manifold case our proofs involve novel applications of the celebrated Reilly formula. (Joint work with Victor Marx and Vitalii Konarovskyi.)

The impact of the Pauli principle, as expressed by the antisymmetry of many-particle wave functions, depends on the regularity of Sobolev function and hence on the space dimension, right? - Yes and no! On the one hand, all forms of so-called contact interactions are excluded in all space dimensions d >1 but not for d=1. On the other hand, the extent to which short-range forces of a certain normalized strength are weakened by the Pauli principle is independent of the space dimension. This is the result of a surprising interplay between spectral theory and Sobolev regularity. - Joint work with Michael Hofacker.

In einer – notwendig subjektiven – Rekonstruktion dessen, was die Mathematik- didaktik als wissenschaftliche Disziplin ausmacht, möchte die Vortragende ent- werfen, welches Gerüst von spannenden Leitfragen durch neunhundert und ei- nen Kolloquiumsvortrag hindurch tragen könnte, wie sich diese harmonisch aus Grundzügen der Mathematik selbst ergeben und welche begrenzenden Span- nungsfelder sich dabei auftun.

The optimal matching problem is one of the classical random optimization problems. The macroscopic behaviour is well understood since the work of Ajtai, Komlos, Tusnady, Talagrand and others in the 80s and early 90s. A few years ago, Caracciolo et al. proposed a new ansatz, based on a linearisation of the Monge-Ampere equation to the Poisson equation, to get refined estimates for this problem. I will show how one can justify this approach in a quantitative way leading to a large scale regularity theory for the optimal transport problem. This in turn can then be used to get new information on the matching problem, like convergence to a Gaussian field which we will shortly discuss at the end of the talk.

We present an overview of the dynamical aspects of old and new first-order methods used in optimization and variational analysis, and how inertial features and relaxation can help improve their performance. Special attention will be paid to inertial and overrelaxed primal-dual methods, as an illustration.

The Covid-19 pandemic still causes severe impacts to the society and the economy. This paper studies excess mortality during the pandemic years 2020 and 2021 in Germany empirically, with a special focus for the life insurer's perspective. Our conclusions are based on official counts of German governmental offices on the living and deaths of the entire population.

Weak solution concepts for mean curvature flow have been investigated since the seminal work of Brakke in the 70’s. We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are coupled to the phase volumes by a simple transport equation. First, we show that, in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417-461, (1993)], any limit point of solutions to the Allen-Cahn equation is a varifold solution in our sense. Second, we prove that any calibrated flow in the sense of Fischer et al. [arXiv:2003.05478] - and hence any classical solution to mean curvature flow - is unique in the class of our new varifold solutions. This is in sharp contrast to the case of Brakke flows, which a priori may disappear at any given time and are therefore fatally non-unique. Finally, we propose an extension of the solution concept to the multi-phase case which is at least guaranteed to satisfy a weak-strong uniqueness principle. This is joint work with Sebastian Hensel (U Bonn).

In this talk we present a quantitative geometric rigidity estimate in dimensions d = 2; 3 generalising a celebrated result by Friesecke, James and Müller to the setting of variable domains. Loosely speaking, we show that for each function y 2 H1(U;R3) and for each connected component of an open bounded set U(Rd, the L2-distance of ry from a single rotation can be controlled up to a constant by its L2-distance from the group SO(d), with the constant not depending on the precise shape of U, but only on an integral curvature functional related to @U. We further show that for linear strains the estimate can be refined, leading to a uniform control independent of the set U. The estimate can be used to establish compactness in the space of generalized special functions of bounded deformation (GSBD) for sequences of displacements related to deformations with uniformly bounded elastic energy. As an application, we rigorously derive linearized models for nonlinearly elastic materials with free surfaces by means of T-convergence. In particular, we study energies related to epitaxially strained crystalline films and to the formation of material voids inside elastically stressed solid

Proof is arguably what distinguishes mathematics most markedly from other do- mains of knowledge. Is it obvious what a proof is? Mathematicians have asked some incisive questions about the nature and role of proof. Research results on students’ conceptions of proof appear inconsistent: While beginning the transition to deductive reasoning is possible already in ele- mentary school, most high school students still have a vague notion of proof, at best. And many teachers are uncertain how to relate to non-algebraic arguments such as verbal, visual and generic ones.

We investigate the convergence of the proximal gradient method applied to control problems with non-smooth and non-convex control cost. Here, we focus on control cost functionals that promote sparsity, which includes functionals of $L^p$-type for $p \in [0, 1)$. We prove stationarity properties of weak limit points of the method. These properties are weaker than those provided by Pontryagin’s maximum principle and weaker than L-stationarity.

Der Vortrag zu Ehren Herrn Prof. Manfred Reimers anläßlich seines 88. Geburtstages und seiner Beiträge zur angewandten Mathematik wird zwei Fächer der multivariaten Approximationstheorie zum Thema haben.
Zum einen werden wir über sogenannte radiale Basisfunktionen berichten, und zwar zur Approximation auf Kugeloberflächen, die Interpolation auf gestreuten Datenpunkten auf kompakten Mengen erlauben und deren Approximationen durch Shifts von Kernen erzeugt werden.
Hierbei geht es vor allem um die Beschreibung von Bedingungen, die die Regularität der Interpolationsaufgabe garantieren, und auch um die Erzeugung von neuen Klassen radialer Basisfunktionen, die diese Eigenschaften bereitstellen. Es handelt sich beispielsweise um Bedingungen vollständiger oder mehrfacher Monotonie.
Des weiteren behandeln wir die Partition von kompakten Mengen in sehr allgemeinem Kontext, beispielsweise zum Zwecke der Quadratur auf Mannigfaltigkeiten. Die Arbeiten sind gemeinsam mit Janin Jäger aus Gießen bzw. Feng Dai und Yeli Niu aus Edmonton.

We will discuss the homogenization of inclusions of infinite conductivity, randomly stationary distributed inside a homogeneous conducting medium. A now classical result by Zhikov shows that, under a logarithmic moment bound on the minimal distance between the inclusions, an effective model with finite homogeneous conductivity exists. Relying on ideas from network approximation, we will provide a relaxed criterion ensuring homogenization. Several examples not covered by the previous theory will be presented. This is joint work with A. Girodroux-Lavigne.

siehe PDF-File

We introduce a Scott–Zhang type projection operator mapping to Lagrange elements for any polynomial order. Additionally to the usual properties, this operator is compatible with negative Sobolev norms. More specifically, it is stable in these weaker norms and leads to optimal rates of convergence. We discuss alternative operators and apply the operator to prove interpolation error estimates for parabolic problems. This is joint work with Tabea Tscherpel and Johannes Storn.