I will present an investigation starting from a very concrete problem in experimental and computational physics, namely molecular structure determination from X-FEL experiments, to the abstract problem of nonlinear stochastic tomography and the theoretical limits of optimization on measure spaces.

In this talk, we will consider inverse problems for parabolic systems with dynamic boundary conditions. Our main aim is to prove stability estimates for the determination of some unknown parameters appearing in the system, namely, source terms, potentials and initial data, from internal measurements. First, we prove a Lipschitz stability result for a class of source terms by establishing a suitable Carleman estimate. Then we study a nonlinear inverse problem involving the recovery of two radiative potentials simultaneously with initial temperatures. We prove a Lipschitz stability estimate for the potentials using Carleman estimates along with some positivity results of the solutions. Then we infer a logarithmic stability result for the initial temperatures by a logarithmic convexity method. Afterwards, we prove a general logarithmic stability result for initial data in an abstract parabolic system. To do so, we use a general logarithmic convexity result. As an application, we consider the Ornstein-Uhlenbeck equation. Then, we will study the identification of source terms from the final overdetermination in the heat equation with dynamic boundary conditions. For this purpose, we combine a weak solution approach and a gradient method with an adjoint problem. We perform some numerical simulations for the reconstruction of an unknown source term in the 1-D equation. Finally, some conclusions and possible extensions will be mentioned.

We investigate the complexity and develop algorithms for the rounding problem occurring during the final step of the combinatorial integral approximation approach used in Mixed-integer optimal control. In particular, we are interested in solving the rounding problem involving switching costs.

Im Zentrum des Vortrags steht das gemeinsame Lernen. Dazu braucht es reich haltige Aufgaben, bei denen bereits bei der Bearbeitung sowie auch später beim Austausch die Lernenden gemeinsam Mathematik «machen». Die zu bewälti- gende Mathematik «entsteht» dabei durch individuelle Entscheide und Interaktion in der Gruppe. Die neu entwickelten Aufgaben bieten Lerngelegenheiten für alle Lernniveaus. Der Vortrag geht von konkreten Beispielen aus und plädiert für eine verstärkt «lustvolle» Mathematik. Auch wenn der Ansatz, Mathematik «kooperativ zu rahmen» bisher nur in kleinem Rahmen erforscht wurde, sind erste Ergebnisse durchaus ermutigend.

Merriman, Bence and Osher's thresholding scheme is a time discretization of mean curvature flow. I restrict to the two-phase setting and mean convex initial conditions. In the sense of the minimizing movements interpretation of Esedoglu and Otto I show the time-integrated energy of the approximation to converge to the time-integrated energy of the limit. As a corollary, the conditional strong convergence results of Laux and Otto become unconditional in this case. The results are general enough to handle the extension of the scheme to anisotropic flows for which a non-negative kernel can be chosen.

In the beginning we will review the classical approaches to obtain exponential or algebraic rates of convergence towards minimizers of the energy in the case of gradient flows in (strictly) convex energy landscapes. We will discuss advantages and disadvantages of this method generalized to mildly non-convex energies and in particular try to understand in which cases this approach yields optimal rates. In the second part of the talk we will consider a more robust, but related method that relies on L^1-type distances instead of the ``natural`` distance associated to the gradient flow. By means of example we will consider the Cahn-Hilliard equation in one dimension and look at solutions with two transition layers. Here, in contrast to the case of one transition layer, the long-time limit is not fixed by mass conservation and in order to control the dynamics, a detailed understanding of the movement of the transition layers is needed. This talk is based on joint work with Sarah Biesenbach and Maria G. Westdickenberg.

In this talk we introduce a general class of singularly-perturbed elliptic functionals Fε and we study their asymptotic behaviour as the perturbation parameter ε > 0 vanishes. Under suitable assumptions, which in particular allow us to bound Fε by the Ambrosio-Tortorelli functionals, we show that the functionals Fε Γ-converge to a free-discontinuity functional of brittle type. This is a consequence of a volume-surface decoupling effect. If time permits, in the last part we will discuss the application of the general convergence result to the setting of stochastic homogenization. This result was obtained in collaboration with Annika Bach & Caterina Ida Zeppieri.

I will discuss the regularity of solutions to quasilinear systems satisfying a Legendre-Hadamard ellipticity condition. For such systems it is known that weak solutions may which fail to be C^1 in any neighbourhood, so we cannot expect a general regularity theory. However if we assume an a-priori regularity condition of the solutions we can rule out such counterexamples. Focusing on solutions to Euler-Lagrange systems, I will present an improved regularity results for solutions whose gradient satisfies a suitable BMO / VMO condition. Ideas behind the proof will be presented in the interior case, and global consequences will also be discussed.

Was haben das Feierabendbier in der Eckkneipe, der Signal Iduna Park und der Wetterbericht gemeinsam? Überall profitieren Menschen von digitalen Systemen. Das gut positionierte Reinigungsgerät in der Eckkneipe, das die Ansteckungsgefahr reduziert, die ideale Ausleuchtung des Spielfeldes für höchste Bildqualität bei der Fernsehübertragung oder die Berechnung des Wetters von morgen, um auf Wetter-ereignisse schnell reagieren zu können. 3D CFD Simulationen und allgemein digitale Zwillinge werden in vielen neuen Berei-chen eingesetzt und erfahren immer mehr an Bedeutung. Bei diesem Vortrag werden zunächst Technologien, wie digitale Zwillinge, 3D CFD Simulationen, verschiedene KI-Ansätze kurz und überblickend vorgestellt. Anschlie-ßend soll dem Hörer leicht verständlich und anhand von konkreten Beispielen aufge-zeigt werden, welche Möglichkeiten mit der Technologie einhergehen und wo sie be-reits effektiv eingesetzt werden und zukünftig eingesetzt werden können. Abschließend sollen Möglichkeiten und Grenzen dieser Technologien in unterschied-lichen Marktsegmenten vorgestellt und gemeinsam diskutiert werden. Der Vortrag richtet sich an Studierende (Bachelor, Master, Promotion) der TU Dort-mund, vorzugsweise in den Fachrichtungen Mathematik, Informatik oder allgemein den Naturwissenschaften. Vorkenntnisse sind ausdrücklich nicht erforderlich. Dem Vortrag werden auch ``nicht-deutschsprachige`` Interessierte folgen können, da die Vortragsfolien bei Bedarf (nach vorheriger Anmeldung) in Englisch sein werden.

The main topic of the talk are dynamical processes where the dynamics is occurring on a set of entities (nodes in a network) that are connected to each other via edges in some nontrivial way. This leads to a natural question of how such underlying nontrivial connectivity affects dynamical processes. We are in particular interested in the following question: if one is to disturb one or more nodes in order to disturb the dynamics of the whole network, which node(s) to choose? As our performance measure we use the concept of $H_\infty$ norm from control theory and analyse a class of networks where the dynamics of nodes are modeled by a 2nd order differential operator. We apply abstract results to the case of a wave equation on a metric graph. The talk is based on joint works with A. Bátkai, M. Kramar-Fijavž, I. Palunko, D. Tolić and Z. Tomljanović.

Limitedness describes my view of the environment of teaching and learning mathematics in South Africa. Mathematics education has been battling with: questionable content knowledge; high numbers of ‘out-of-field’ and ‘out-of-phase’ teachers; and limited curricular resources. The current presentation zooms in on limitedness around the use of curricular resources by these teachers. An important question for consideration over and above the appropriateness of existing curricular resources, is what it would take to develop these teachers’ capacities for designing productive lessons. The presentation will outline research which has been conducted with a group of teachers on their relationships with their textbooks; on critical factors influencing teachers’ use of curricular resources; and, on the fitness for purpose of available curricular resources.

Existential definability of subsets of fields, in particular of subrings, has been studied since the 1940'ies, and has seen several applications to problems of existential decidability (generalising Hilbert's 10th Problem). To establish existential definability of valuation rings, quadratic forms often play a substantial role. In this talk, we illustrate the connection between existential definability and quadratic form theory by giving some historical examples, and then discuss a recent result relying on modern developments in quadratic form theory, like a local-global principle for function fields over complete valued fields.

We introduce the notion of viscosity solutions for second order fully nonlinear elliptic/parabolic equations. A viscosity solution, u, is defined through the condition that whenever a test function, φ, touches u from below/above at a point then φ should be a classical super/sub-solutions of the equation at this point. Since u is apriori merely continuous, it is natural to study whether the equation finally impose any regularizing effects on u. We start by giving an overview of the theory that have been developed the last decades on the interior regularity properties of viscosity solutions, introducing the main tools used, namely the ABP and Harnack inequalities. Then we shall concentrate on oblique derivative problems (where the oblique derivative condition is understood only in the viscosity sense as well) and discuss about the up to the boundary (first order) Hölder regularity of the solutions giving quantitative bounds for the corresponding Hölder norms in terms of the data.

In this talk, we present recent mathematical results in Maxwell variational inequalities with technological applications in high-temperature superconductivity and EM-shielding. The established well-posedness opens a way to analyze the corresponding numerical approximations, optimal design and inverse problems, which are mainly complicated by the hyperbolic Maxwell structure and the non-smooth character. Results in these directions are presented.

We consider a sequence of linear-quadratic optimization problems defined in an abstract functional framework. Each problem is accompanied by the constraint of reaching a given target with some precision. We show that problems are well posed and we characterise their solutions. The main result provides the conditions under which these solutions converge to the minimizer of the limit problem. The theory can be applied to a wide range of problems: elliptic, parabolic, control ones etc. In the talk, the following examples will be presented: 1. optimal controllability of the heat equation 2. optimal solvability of the Poisson equation.

Given a set of departments and the pairwise transport weights between them, the Single-Row Facility Layout Problem (SRFLP) asks for an arrangement of the departments on one side of a path such that the weighted sum of the distances between the departments is minimized. The best integer linear programming approach for the SRFLP is based on the betweenness model (Amaral, 2009).
In this talk we present new classes of inequalities for the associated betweenness polytope and show their validity using combinatorial arguments. Using a cutting plane approach allows us to solve more and larger instances of the SRFLP to optimality than the lower bounding approach of Amaral. In the second part of the talk we consider the Multi-Bay Facility Layout Problem and Facility Layouts with Pier Type Patterns which are special Combined Cell Layout Problems (CCLP). Here the path structure is more complicated. We solve these problems exactly by enumerating over all assignments of the departments to the cells and solving several CCLP with fixed-cell assignment using the betweenness model in each of the subproblems. We show how to reduce the number of distinguishable cell assignments significantly by merging two cells of type single-row.

Copulas and doubly stochastic measures are mostly well known in the statistical community due to Sklar's famous theorem saying that copulas are the link between multivariate distribution functions and their marginals. Copulas, however, exhibit various interesting and surprising mathematical aspects going far beyond statistics. The talk will discuss some of those aspects, in particular some interrelations with functional analysis, geometry, number theory, and fractals.

Nonlinear infinite-dimensional systems are nonlinear dynamical systems whose state components lie in an infinite-dimensional space, typically a function space. Such systems, which are also called distributed parameter systems, are ubiquitous in real-life since they are able to model many physical processes, going from conservative mechanical systems to dissipative phenomena. A lot of questions may arise when dealing with such classes of systems. For instance, the well-posedness in terms of existence and uniqueness of solutions as well as the study of the equilibria, their stability and their control are paramount steps when studying these dynamical systems. For that class of systems, we study the field of tracking control, and especially funnel control, which constitutes an appropriate tool for the output of a system to track a class of reference signals. As a main contribution on this topic, we extend the available results that allow to consider linear infinite-dimensional systems as internal dynamics to the nonlinear setting. We prove that a class of nonlinear infinite-dimensional systems that satisfy some standard assumptions admits a differential relation between the input and the output that is conducive for funnel control. Our theoretical results are illustrated by means of examples and numerical simulations.

Positivity-preserving schemes are of high importance and gained more and more attention in recent years. However, they often result in nonlinear iteration schemes even when applied to a linear test problem y′ = Ay. Hence, a stability analysis becomes more complex. Moreover, the scheme often has to preserve additionally other properties of the analytic solution such as steady states and linear invariants. As a result, steady states of the test problem become non-hyperbolic fixed points of the method. In general, the stability analysis thereby turns into a case by case study.
In this talk, we present sufficient conditions for stability as well as local convergence, and hence, overcome this case by case study for a large class of steady state preserving schemes to which reasonable positivity-preserving schemes belong.
To illustrate the theoretical results, we consider so-called modified Patankar-type schemes that are positive and conservative numerical methods when applied to a positive and conservative production-destruction system. Applying the main theorem we are able to define stability functions which are the basis for the stability analysis. Finally, numerical experiments are presented to confirm the theoretical results.

Classroom discussions—dialogic interactions in which the ideas of students and teachers are collectively shared and developed—can increase student learning in mathematics classrooms. Yet, multilingual students in US schools often experience mathematics classrooms where most talk is teacher-led lecture. In this seminar, I will report on results from a five-year design research intervention in linguistically diverse 9th grade mathematics classrooms in the US-Mexico border region. Our goal was to promote student participation in dialogic interactions. I will 1) Outline the design framework, 2) Describe the design principles that were developed in the course of the research, and 3) Present key qualitative and quantitiave findings from the research. I will close with a discussion of how language policies and language ideologies shape the quality of student participation in classroom discussions, and consider future directions for work in this area of investigation.

In this talk, we will focus on the evolution equations associated with nonselfadjoint quadratic differential operators. On the one hand, it will be explained how the non-commutation phenomena between the selfadjoint and skew-selfadjoint parts of these operators allow the evolution equations studied to enjoy smoothing and localization properties in some specific directions of the phase space, that will be precisely described. On the other hand, it will be found that the properties of null-controllability of the Ornstein-Uhlenbeck equations (which are quadratic evolution equations) are related to a notion of integral thickness condition associated to the skew-adjoint part of the operators involved. These different properties will be deduced from a fine description of the polar decomposition of the evolution operators associated to the studied equations. These results come from a series of works with J. Bernier (LMJL) and J. Martin (IRMAR).

Am Beginn des Mathematikstudiums stellt der Umgang mit mathematischen Beweisen in der Hochschulmathematik für Studierende eine neuartige Anforderung dar. Diese liegt einerseits im eigenständigen Führen von Beweisen und andererseits im Verstehen von vorgegebenen Beweisen. Der Vortrag konzentriert sich auf Letzeres, d.h. das Verstehen von Beweisen, die Studierenden in Vorlesungen und in Lehrtexten präsentiert werden. Der Vortragende stellt zwei Ansätze aus seiner eigenen Lehrpraxis vor, mit der Studierende beim Beweisverstehen unterstützt werden sollen: Peer Instruction und Audioguides. Die zwei Ansätze sind für unterschiedliche Settings entworfen: Während sich Peer Instruction -- besonders in der Variante der ``Mid-Proof Peer Instruction`` -- für den Einsatz in synchronen Lehrveranstaltungen eignet, sind Audioguides eine Möglichkeit zur Unterstützung in der asynchronen Lehre. Im Vortrag werden Hintergrund und Einsatzweisen sowie Ergebnisse aus Untersuchungen zu beiden Methoden vorgestellt.

The Poisson hyperplane tessellation in Euclidean space is a classical and well-studied model of Stochastic Geometry. In this talk we will consider its counterpart(s) in hyperbolic space. As it turns out, Euclidean hyperplanes have more than one analogue in hyperbolic space, corresponding, roughly speaking, to thinking of hyperplanes as either totally geodesic hypersurfaces, equidistant sets from other hyperplanes, or spheres of infinite radius. In hyperbolic space these notions give rise to a one-parametric family of hyperplane-like hypersurfaces, known collectively as lambda-geodesic hypersurfaces, for lambda between 0 and 1. We will consider the isometry-invariant Poisson process of lambda-geodesic hyperplanes, and in particular its total surface area within a growing spherical window. We will study its limit theory, and prove central and non-central limit theorems, depending on the value of lambda and the ambient dimension. Based on joint work with Z. Kabluchko and C. Thäle

This talk deals with a general method to prove final-state observability estimates based on the Lebeau-Robbiano strategy. Complementing previous results obtained in the context of Hilbert spaces, we derive observability estimates of operator semigroups in general Banach spaces by combining an uncertainty principle and a dissipation estimate. Thereby, we allow for generalized growth rates in the assumptions and use an integral expression to estimate the observability constant. We study connections to controllability and stabilization properties and apply our results to linear parabolic equations with thick control sets.

Zunächst erläutern wir ein Eindeutigkeitskriterium für Energieminimierer zu quadratisch gleichmäßig konvexen Funktionalen in einer Klasse von geeigneten maßerhaltenen Abbildungen und bezüglich geeigneter Randbedingungen. Danach diskutieren wir Verallgemeinerungen und mögliche Anwendungen. Besonderen Fokus legen wir dabei auf eine Situation, die einen singulären Energieminimierer erlaubt.

The evolution in time of a quantum system is characterized by the Hamiltonian. The talk will focus on the Hamiltonian related to the N-body problem. After introducing some notations and context for the N-body problem, I will present a result on the decomposition of the essential spectrum of a class of operators that contained the Hamiltonian. This result is a generalization of the well-known HVZ-theorem. This decomposition is obtained using the theory of algebra of operators with, in particular, the use of C*-algebras. In a second step, I will give results on the regularity of the eigenfunctions of the Hamiltonian. This regularity result is obtained using the theory of differentials geometry, with a focus on the blow-up of manifolds with corners. I will end with a result connecting these two approaches: the operators algebra's one and the differential's geometry one's. This connecting result show that these two approach are actually built on the same space. These are joint work with B. Amman, N. Prudhon and V. Nistor.

How can we use topology to analyse a cloud of colored points? With ever improving measuring techniques, data of this type is abundant in modern biology and medical sciences. In this talk, I will explain the basics of persistent homology, a popular tool in a growing field of topological data analysis, and explain our current approach in generalising the methods to chromatic setting.

Precise focusing of nonlinear ultrasonic waves is crucial for the efficiency and safety of various biomedical procedures. Focusing in ultrasound applications is achieved geometrically, with acoustic lenses or spherical arrays of piezoelectric transducers. Such setups can naturally benefit from shape optimization. As ultrasonic waves at high intensities give rise to nonlinear phenomena, these problems are constrained by quasilinear acoustic wave equations. In this talk, we will discuss the mathematical questions arising in this context and present some of the obtained results.

In this talk I will present some recent improvements in the study of homogenisation of compressible viscous fluid in the case of tiny holes and the connection with fluid structure problems. In particular I highlight a situation where the fluid + rigid body problem has more flexibility respect to case of the fluid alone. This flexibility helps us to deduce new ideas to study the case of the fluid alone.

Nemirovski asked in his plenary talk at ICM 2006 whether every convex semialgebraic set is a spectrahedral shadow, i.e. can be defined by linear matrix inequalities. Recently, Scheiderer answered this question to the negative by exhibiting first counter examples. In this talk I will present a joint work with Manuel Bodirsky and Andreas Thom in which we answer this question for certain sets that resisted the efforts of Scheiderer. Our approach relies on a simple model theoretic observation and differs from Scheiderer's methods substantially.

I will present a new approach to the spectral theory of systems of PDEs on closed manifolds, developed in a series of recent papers by Dmitri Vassiliev (UCL) and myself, based on the use of pseudodifferential projections. Emphasis will be placed on ideas and motivation; the talk will include a brief historical overview of the development of the subject area.

We study the homogenisation problem for elliptic operators with high-contrast random coefficients describing a two-phase composite with the heterogeneity scale $\epsilon$ made up of small inclusions randomly distributed in a matrix component. It is assumed that the coefficients of the operator are of order one in the matrix component and of order $\epsilon^2$ in the inclusions (so-called double porosity scaling). High-contrast composites receive special attention in scientific community due to the band-gap structure of their spectrum. In this talk I will give an introduction to the high-contrast setting followed by an overview of our recent results in the topic and some ongoing work.

We determine the asymptotic normalized rank of a random matrix A over an arbitrary field with prescribed numbers of non-zero entries in each row and column. As an application we obtain a formula for the rate of low-density parity check codes. This formula vindicates a conjecture of Lelarge (2013). Moreover we derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. The result covers both matrices over finite fields with independent non-zero entries and {0, 1}-matrices over the rationals.

For any ring R, its Pythagoras number is the smallest number P(R) such that any sum of squares in R can be written as a sum of at most P(R) squares. Lagrange's celebrated four-square theorem can be stated as P(Z)=4. We will look for analogous results in other number fields; the most interesting ones turn out to be totally real fields, i.e. such that the image of all their complex embeddings is in fact a subset of the real numbers. While the Pythagoras number of a number field is easy to determine, the Pythagoras number of its ring of integers is usually unknown. We will discuss the available results and the basic ideas behind them. In particular, to obtain upper bounds for Pythagoras numbers, we introduce the so-called g-invariants, which are similar to the Pythagoras number, but squares of numbers are replaced by squares of linear forms. The study of g-invariants, sometimes called the quadratic Waring's problem, is far from solved even in the case of rational integers; however, we will see some nontrivial results.

A classical inequality of Li and Yau gives a lower bound on the Willmore energy of a surface in terms of its multiplicity at an arbitrary point. We discuss recent extensions to other curvature-dependent bending energies, such as Euler's elastic energy or the Canham-Helfrich functional. Additionally, we illustrate applications to geometric flows and the regularity of minimizers.