Models and Approximations

Research Area C

Alsmeyer, Böhm, Dereich, Engwer, Friedrich (until 2021), Gusakova (since 2021), Hille, Holzegel (since 2020), Huesmann, Jentzen (since 2019), Kabluchko, Lohkamp, Löwe, Mukherjee, Ohlberger, Rave, Schedensack (until 2019), F. Schindler, Schlichting (since 2020), Seis, Simon (since 2021), Stevens, Weber (since 2022), Wilking, Wirth, Wulkenhaar, Zeppieri.

In research area C, we will focus on the development and foundation of mathematical models and their approximations that are relevant in the life sciences, physics, chemistry, and engineering. We will rigorously analyse the dynamics of structures and pattern formation in deterministic and stochastic systems. In particular, we aim at understanding the interplay of macroscopic structures with their driving microscopic mechanisms and their respective topological and geometric properties. We will develop analytical and numerical tools to understand, utilise, and control geometry-driven phenomena, also touching upon dynamics and perturbations of geometries. Structural connections between different mathematical concepts will be investigated, such as between solution manifolds of parameterised PDEs and non-linear interpolation, or between different metric, variational, and multi-scale convergence concepts for geometries. In particular, we aim to characterise distinctive geometric properties of mathematical models and their respective approximations.

• C1. Evolution and asymptotics.

In this unit, we will use generalisations of optimal transport metrics to develop gradient flow descriptions of (cross)-diffusion-reaction systems, rigorously analyse their pattern forming properties, and develop corresponding efficient numerical schemes. Related transport-type- and hyperbolic systems will be compared with respect to their pattern-forming behaviour, especially when mass is conserved. Bifurcations and the effects of noise perturbations will be explored.

Moreover, we aim to understand defect structures, their stability and their interactions. Examples are the evolution of fractures in brittle materials and of vortices in fluids. Our analysis will explore the underlying geometry of defect dynamics such as gradient descents or Hamiltonian structures. Also, we will further develop continuum mechanics and asymptotic descriptions for multiple bodies which deform, divide, move, and dynamically attach to each other in order to better describe the bio-mechanics of growing and dividing soft tissues.

Finally, we are interested in the asymptotic analysis of various random structures as the size or the dimension of the structure goes to infinity. More specifically, we shall consider random polytopes and random trees. For random polytopes we would like to compute the expected number of faces in all dimensions, the expected (intrinsic) volume, and absorption probabilities, as well as higher moments and limit distributions for these quantities.

• C2. Multi-scale phenomena and macroscopic structures.

In multi-scale problems, geometry and dynamics on the micro-scale influence structures on coarser scales. In this research unit we will investigate and analyse such structural interdependence based on topological, geometrical or dynamical properties of the underlying processes.

We are interested in transport-dominated processes, such as in the problem of how efficient a fluid can be stirred to enhance mixing, and in the related analytical questions. A major concern will be the role of molecular diffusion and its interplay with the stirring process. High Péclet number flow in porous media with reaction at the surface of the porous material will be studied. Here, the flow induces pore-scale fluctuations that lead to macroscopic enhanced diffusion and reaction kinetics. We also aim at understanding advection-dominated homogenisation problems in random regimes.

We aim at classifying micro-scale geometry or topology with respect to the macroscopic behaviour of processes considered therein. Examples are meta material modelling and the analysis of processes in biological material. Motivated by network formation and fracture mechanics in random media, we will analyse the effective behaviour of curve and free-discontinuity energies with stochastic inhomogeneity. Furthermore, we are interested in adaptive algorithms that can balance the various design parameters arising in multi-scale methods. The analysis of such algorithms will be the key towards an optimal distribution of computational resources for multi-scale problems.

Finally, we will study multi-scale energy landscapes and analyse asymptotic behaviour of hierarchical patterns occurring in variational models for transportation networks and related optimal transport problems. In particular, we will treat questions of self-similarity, cost distribution, and locality of the fine-scale pattern. We will establish new multilevel stochastic approximation algorithms with the aim of numerical optimisation in high dimensions.

• C3. Interacting particle systems and phase transitions.

The question of whether a system undergoes phase transitions and what the critical parameters are is intrinsically related to the structure and geometry of the underlying space. We will study such phase transitions for variational models, for processes in random environments, for interacting particle systems, and for complex networks. Of special interest are the combined effects of fine-scale randomly distributed heterogeneities and small gradient perturbations.

We aim to connect different existing variational formulations for transportation networks, image segmentation, and fracture mechanics and explore the resulting implications on modelling, analysis, and numerical simulation of such processes.

We will study various aspects of complex networks, i.e. sequences of random graphs (Gn)n∈N, asking for limit theorems as n tends to infinity. A main task will be to broaden the class of networks that can be investigated, in particular, models which include geometry and evolve in time. We will study Ising models on random networks or with random interactions, i.e. spin glasses. Fluctuations of order parameters and free energies will be analysed, especially at the critical values where the system undergoes a phase transition. We will also investigate whether a new class of interacting quantum fields connected with random matrices and non-commutative geometry satisfies the Osterwalder–Schrader axioms.

Further, we will study condensation phenomena, where complex network models combine the preferential attachment paradigm with the concept of fitness. In the condensation regime, a certain fraction of the total mass dynamically accumulates at one point, the condensate. The aim is a qualitative and quantitative analysis of the condensation. We will also explore connections to structured population models. Further, we will study interacting particle systems on graphs that describe social interaction or information exchange. Examples are the averaging process or the Deffuant model.

We will also analyse asymmetric exclusion processes (ASEP) on arbitrary network structures. An interesting aspect will be how these processes are influenced by different distribution mechanisms of the particles at networks nodes. If the graph is given by a lattice, we aim to derive hydrodynamic limits for the ASEP with jumps of different ranges for multiple species, and for stochastic interacting many-particle models of reinforced random walks. Formally, local cross-diffusion systems are obtained as limits of the classical multi-species ASEP and of the many-particle random walk. We will compare the newly resulting limiting equations and are interested in fluctuations, pattern formation, and the long-time behaviour of these models on the microscopic and the macroscopic scale. Further, we will analyse properties of the continuous directed polymer in a random environment.

• C4. Geometry-based modelling, approximation, and reduction.

In mathematical modelling and its application to the sciences, the notion of geometry enters in multiple related but different flavours: the geometry of the underlying space (in which e.g. data may be given), the geometry of patterns (as observed in experiments or solutions of corresponding mathematical models), or the geometry of domains (on which PDEs and their approximations act). We will develop analytical and numerical tools to understand, utilise and control geometry, also touching upon dynamically changing geometries and structural connections between different mathematical concepts, such as PDE solution manifolds, analysis of pattern formation, and geometry.

We will interpret data from different contexts (in particular measurements from the life sciences and physics, shapes from computer graphics applications, and solutions to parameterised PDEs) as elements of an underlying non-linear (infinite-dimensional) geometric space, e.g. a Riemannian manifold. This geometric structure will be exploited for the development of data processing tools. A focus will lie on variational and numerical methods for data fitting and regression via submanifolds, on singular perturbation methods, and on asymptotics and model reduction for parameterised PDEs by decomposing each solution into an element of a linear space and a Lie group element acting on it.

The geometry of spatial patterns often determines the average, effective properties of such structures, e.g. in a material, its effective material properties. Motivated by particular patterns and their defects, as observed in biological organisms or materials, we will examine their macroscopic, homogenised properties and their stability with respect to pattern perturbations. The effective, homogenised structure will again be described in geometric terms: For instance, the homogenised free energy of carbon nanotubes may depend on their bending curvature, the transport efficiency of molecules in strongly layered biological membranes of cell organelles may depend on an effective distance metric, and defects in atomic crystals can be related to specific singularities of two-dimensional surfaces.

Applications like shape optimisation or shape reconstruction problems are concerned with the identification of a geometry. Typically, there is an additional PDE constraint for which the sought geometry serves as the PDE domain. It is a challenge to efficiently approximate this geometry. We will develop concepts that quantify the efficiency of the geometry approximation in terms of the involved computational effort per desired accuracy, and we will investigate numerical schemes that can efficiently deal with complex (time-)varying PDE domains without the need for remeshing.

Further research projects of Research Area C members

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

$\bullet$
Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Christian Seis: Invariante Mannigfaltigkeiten für schnelle Diffusionen nahe Auslöschung (2022-2022) Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82$\bullet$Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82
Christoph Böhm
Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 , Arnulf Jentzen: Dynamical systems and irregular gradient flows (2021-2025) The central goal of this project is to study asymptotic properties for gradient flows (GFs) and related dynamical systems. In particular, we intend to establish a priori bounds and related regularity properties for solutions of GFs, we intend to study the behaviour of GFs near unstable critical regions, we intend to derive lower and upper bounds for attracting regions, and we intend to establish convergence speeds towards global attrators. Special attention will be given to GFs with irregularities (discontinuities) in the gradient and for such GFs we also intend to reveal sufficient conditions for existence, uniqueness, and flow properties in dependence of the given potential. We intend to accomplish the above goals by extending techniques and concepts from differential geometry to describe and study attracting and critical regions, by using tools from convex analysis such as subdifferentials and other generalized derivatives, as well as by employing concepts from real algebraic geometry to describe domains of attraction. In particular, we intend to generalize the center-stable manifold theorem from the theory of dynamical systems to the considered non-smooth setting. Beside finite dimensional GFs, we also study GFs in their associated infinite dimensional limits. The considered irregular GFs and related dynamical systems naturally arise, for example, in the context of molecular dynamics (to model the configuration of atoms along temporal evoluation) and machine learning (to model the training process of artificial neural networks). Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82$\bullet$Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82
Mario Ohlberger
Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 , René Fritze (Milk) Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82
, Stephan Rave
Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 , Christian Himpe: Mathematical Research Data Initiative (2021-2026) Driven by the needs and requirements of mathematical research as well as scientific disciplines using mathematics, the NFDI-consortium MaRDI (Mathematical Research Data Initiative) will develop and establish standards and services for mathematical research data. Mathematical research data ranges from databases of special functions and mathematical objects, aspects of scientific computing such as models and algorithms to statistical analysis of data with uncertainties. It is also widely used in other scientific disciplines due to the cross-disciplinary nature of mathematical methods. Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82$\bullet$Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82
Christian Engwer: Personalised diagnosis and treatment for refractory focal paediatric and adult epilepsy (2021-2024)
Epilepsy is among the most common neurological diseases, affecting between 0.5% and 1% of the general population. Therefore, new diagnosis and treatment methods have a high impact on society. Epilepsy is also among the most frequently diagnosed neurological paediatric disorders, with long-term implications for the quality of life of those affected. Only in two-thirds of cases, seizures can be adequately controlled with anticonvulsant drug treatment. For the remaining drug-refractory patients with focal epilepsy (up to about 2 Mill. in Europe), epilepsy surgery is currently the most effective treatment. However, only 15-20% of those patients are eligible for epilepsy surgery. That is either because the epileptogenic zone in the brain cannot be localized with sufficient accuracy with standard diagnostic means, or because the epileptogenic zone overlaps with eloquent cortical areas, so that it cannot be surgically removed without considerable neurological deficit. PerEpi aims to bring together a group of experts at the European level to improve this situation in two ways, both of which use concepts of non-invasive personalised medicine: The first one focuses on a new individualised multimodal approach to set a new milestone in localization accuracy of the epileptogenic zone in order to offer the most appropriate personalised therapy. The second one focuses on a new individually optimized transcranial electric brain stimulation technique as a new treatment option to reduce seizure frequency and severity. This is particularly attractive for those focal refractory patients where surgery is not an option because of an overlap with eloquent cortical areas. A dedicated ethics work package will ensure that the research in the consortium is designed and conducted following the highest ethical standards. In addition, this work package will study the translational pathways of the new approaches to foster clinical integration that is ethically and socially responsible.

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

$\bullet$
Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Benedikt Wirth: CRC 1450 A05 - Targeting immune cell dynamics by longitudinal whole-body imaging and mathematical modelling (2021-2024) We develop strategies for tracking and quantifying (immune) cell populations or even single cells in long-term (days) whole-body PET studies in mice and humans. This will be achieved through novel acquisition protocols, measured and simulated phantom data, use of prior information from MRI and microscopy, mathematical modelling, and mathematical analysis of image reconstruction with novel regularization paradigms based on optimal transport. Particular applications include imaging and tracking of macrophages and neutrophils following myocardial ischemia-reperfusion or in arthritis and sepsis. Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82$\bullet$Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82
Benedikt Wirth: CRC 1450 A06 - Improving intravital microscopy of inflammatory cell response by active motion compensation using controlled adaptive optics (2021-2024)
We will advance multiphoton fluorescence microscopy by developing a novel optical module comprised of a high-speed deformable mirror that will actively compensate tissue motion during intravital imaging, for instance due to heart beat (8 Hz), breathing (3 Hz, in mm-range) or peristaltic movement of the gut in mice. To control this module in real-time, we will develop mathematical methods that track and predict tissue deformation. This will allow imaging of inflammatory processes at cellular resolution without mechanical tissue fixation.

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

$\bullet$
Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Martin Huesmann: SPP 2265: Random Geometric Systems - Subproject: Optimal transport for stationary point processes (2021-2023) Optimal transport by now has found manifold applications in various areas of mathematics, in particular it has turned into a powerful tool in the analysis of stochastic processes, particle dynamics, and the associated evolution equations, mostly however in a finite-dimensional setting. The goal of this project is to develop a counterpart to this theory in the framework of stationary point processes or more general random (infinite) measures and to employ these novel tools e.g. in the analysis of infinite particle dynamics or to attack questions for particular point process models of interest. Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82$\bullet$Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82
Benedikt Wirth: SPP 2256: Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials - Subproject: Multiscale structure in compliance minimization (2020-2023)
It is a classic engineering problem to identify which geometry of an elastic material best supports a load while consuming a minimum material amount. Typically, the optimal geometry would involve an infinitely fine microstructure with infinitely many, infinitely fine holes. To avoid dealing with the microstructure, the optimization problem is typically relaxed (meaning that microstructured regions are simply replaced by a non-microstructured material with same macroscopic elastic properties, adapting the optimization problem correspondingly) or strongly regularized (meaning that one adds something like production costs to the optimization problem, which will prevent structures with too many or too fine holes). In contrast, in this project we are interested in the case of very weak regularization, in which fine structures and coarsening phenomena over multiple scales will occur (that is, in some regions the structure will be very fine and in others quite coarse). Our aim is to better understand these structures via variational methods. Thus, rather than modelling and understanding the behaviour or response of an existing material, we here consider a material design problem that results in a complex structure.

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

$\bullet$
Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Caterina Zeppieri: SPP 2256: Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials - Subproject: Variational modelling of fracture in high-contrast microstructured materials: mathematical analysis and computational mechanics (2020-2023) After the seminal work of Francfort and Marigo, free-discontinuity functionals of Mumford-Shah type have been established as simplified and yet relevant mathematical models to study fracture in brittle materials. For finite-contrast constituents, the homogenisation of brittle energies is by-now well-understood and provides a rigorous micro-to-macro upscaling for brittle fracture.Only recently, explicit high-contrast brittle microstructures have been provided, which show that, already for simple free-discontinuity energies of Mumford-Shah type, the high-contrast nature of the constituents can induce a complex effective behaviour going beyond that of the single constituents. In particular, macroscopic cohesive-zone models and damage models can be obtained by homogenising purely brittle microscopic energies with high-contrast coefficients. In this framework, the simple-to-complex transition originates from a microscopic bulk-surface energy-coupling which is possible due to the degeneracy of the functionals.Motivated by the need to understand the mathematical foundations of mechanical material-failure and to develop computationally tractable numerical techniques, the main goal of this project is to characterise all possible materials which can be obtained by homogenising simple high-contrast brittle materials. In mathematical terms, this amounts to determine the variational-limit closure of the set of high-contrast free-discontinuity functionals. This problem has a long history in the setting of elasticity, whereas is far less understood if fracture is allowed.For the variational analysis it will be crucial to determine novel homogenisation formulas which “quantify” the microscopic bulk-surface energy-coupling. Moreover, the effect of high-contrast constituents on macroscopic anisotropy will be investigated by providing explicit microstructures realising limit models with preferred crack-directions.The relevant mathematical tools will come from the Calculus of Variations and Geometric Measure Theory. Along the way, new ad hoc extension and approximation results for SBV-functions will be established. The latter will be of mathematical interest in their own right, and appear to be widely applicable in the analysis of scale-dependent free-discontinuity problems.The computational mechanics results will build upon the mathematical theory, and will complement it with relevant insights when the analysis becomes impracticable. High performance fast Fourier transform and adaptive tree-based computational methods will be developed to evaluate the novel cell formulas. The identified damage and cohesive-zone models will be transferred to simulations on component scale.The findings are expected to significantly enhance the understanding of the sources and mechanisms of material-failure and to provide computational tools for identifying anisotropic material-models useful for estimating the strength of industrial components. Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82$\bullet$Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82
Martin Huesmann: SPP 2265: Random Geometric Systems - Subproject: Optimal matching and balancing transport (2020-2023)
The optimal matching problem is one of the classical problems in probability. By now, there is a good understanding of the macroscopic behaviour with some very detailed results, several challenging predictions, and open problems. The goal of this project is to develop a refined analysis of solutions to the optimal matching problem from a macroscopic scale down to a microscopic scale. Based on a recent quantitative linearization result of the Monge-Ampère equation developed in collaboration with Michael Goldman, we will investigate two main directions. On the one hand, we aim at rigorously connecting the solutions to the optimal matching problem to a Gaussian field which scales as the Gaussian free field. On the other hand, we seek to close the gap and show that rescaled solutions to the optimal matching problem converge in the thermo-dynamic limit to invariant allocations to point processes, or more generally to balancing transports between random measures. In the long run, combining these limit results with the Gaussian like behaviour of solutions to the matching problem, we seek to analyse the random tessellations induced by the limiting invariant allocations.

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

$\bullet$
Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Zakhar Kabluchko: SPP 2265: Random Geometric Systems - Subproject: Random polytopes (2020-2023) The aim of the project is to investigate random polytopes. Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82$\bullet$Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82
Christoph Böhm
Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 , Burkhard Wilking: CRC 1442: Geometry: Deformation and Rigidity - Geometric evolution equations (2020-2024) Hamilton’s Ricci flow is a geometric evolution equation on the space of Riemannian metrics of a smooth manifold. In a first subproject we would like to show a differentiable stability result for noncollapsed converging sequences of Riemannian manifolds with nonnegative sectional curvature, generalising Perelman’s topological stability. In a second subproject, next to classifying homogeneous Ricci solitons on non-compact homogeneous spaces, we would like to prove the dynamical Alekseevskii conjecture. Finally, in a third subproject we would like to find new Ricci flow invariant curvature conditions, a starting point for introducing a Ricci flow with surgery in higher dimensions. Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82$\bullet$Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82
Raimar Wulkenhaar: CRC 1442: Geometry: Deformation and Rigidity - D03: Integrability (2020-2024)
The project investigates a novel integrable system which arises from a quantum field theory on noncommutative geometry. It is characterised by a recursive system of equations with conjecturally rational solutions. The goal is to deduce their generating function and to relate the rational coefficients in the generating function to intersection numbers of tautological characteristic classes on some moduli space.

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

$\bullet$
Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Michael Wiemeler Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82
, Burkhard Wilking: CRC 1442: Geometry: Deformation and Rigidity - B01: Curvature and Symmetry (2020-2024)
The question of how far geometric properties of a manifold determine its global topology is a classical problem in global differential geometry. In a first subproject we study the topology of positively curved manifolds with torus symmetry. We think that the methods used in this subproject can also be used to attack the Salamon conjecture for positive quaternionic Kähler manifolds. In a third subproject we study fundamental groups of non-negatively curved manifolds. Two other subprojects are concerned with the classification of manifolds all of whose geodesics are closed and the existence of closed geodesics on Riemannian orbifolds.

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

$\bullet$
Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Mario Ohlberger Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82
, Felix Schindler: ML-MORE: Machine learning and model order reduction to predict the efficiency of catalytic filters. Subproject 1: Model Order Reduction (2020-2023)
Reaktiver Stofftransport in porösen Medien in Verbindung mit katalytischen Reaktionen ist die Grundlage für viele industrielle Prozesse und Anlagen, wie z.B. Brennstoffzellen, Photovoltaikzellen, katalytische Filter für Abgase, etc. Die Modellierung und Simulation der Prozesse auf der Porenskala kann bei der Optimierung des Designs von katalytischen Komponenten und der Prozessführung helfen, ist jedoch derzeit dadurch eingeschränkt, dass solche Simulationen zu grossen Datenmengen führen, zeitaufwändig sind und von einer grossen Anzahl von Parametern abhängen. Außerdem werden auf diese Weise die im Laufe der Jahre gesammelten Versuchsdaten nicht wiederverwendet. Die Entwicklung von Lösungsansätzen für die Vorhersage der chemischen Konversionsrate mittels moderner datenbasierter Methoden des Maschinellen Lernens (ML) ist essenziell, um zu schnellen, zuverlässigen prädiktiven Modellen zu gelangen. Hierzu sind verschiedene Methodenklassen erforderlich. Neben den experimentellen Daten sind voll aufgelöste Simulationen auf der Porenskala notwendig. Diese sind jedoch zu teuer, um einen umfangreichen Satz an Trainingsdaten zu generieren. Daher ist die Modellordnungsreduktion (MOR) zur Beschleunigung entscheidend. Es werden reduzierte Modelle fur den betrachteten instationären reaktiven Transport entwickelt, um große Mengen an Trainingsdaten zu simulieren. Als ML-Methodik werden mehrschichtige Kern-basierte Lernverfahren entwickelt, um die heterogenen Daten zu kalibrieren und nichtlineare prädiktive Modelle zur Effizienzvorhersage zu entwickeln.Hierbei werden große Daten (bzgl. Dimensionalität und Sample-Zahl) zu behandeln sein, was Datenkompression und Parallelisierung des Trainings erfordern wird. Das Hauptziel des Projekts ist es, alle oben genannten Entwicklungen in einem prädiktiven ML-Tool zu integrieren, das die Industrie bei der Entwicklung neuer katalytischer Filter unterstützt und auf viele andere vergleichbare Prozesse übertragbar ist.

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

$\bullet$
Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Christian Seis: Transport Equations, mixing and fluid dynamics (2020-2023) Advection-diffusion equations are of fundamental importance in many areas of science. They describe systems, in which a quantity is simultaneously diffused and advected by a velocity field. In many applications these velocity fields are highly irregular. In this project, several quantitative aspects shall be investigated. One is related to mixing properties in fluids caused by shear flows. The interplay between the transport by the shear flow and the regularizing diffusion leads after a certain time, to the emergence of a dominant length scales which persist during the subsequent evolution and determine mixing rates. A rigorous understanding of these phenomena is desired. In addition, stability estimates for advection-diffusion equations will be derived. These shall give a deep insight into how solutions depend on coefficients and data. The new results shall subsequently be applied to estimate the error generated by numerical finite volume schemes approximating the model equations. Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

$\bullet$
Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Raimar Wulkenhaar: GRK 2149 - Starke und schwache Wechselwirkung - von Hadronen zu Dunkler Materie (2020-2024) The Research Training Group (Graduiertenkolleg) 2149 "Strong and Weak Interactions - from Hadrons to Dark Matter" funded by the Deutsche Forschungsgemeinschaft focuses on the close collaboration of theoretical and experimental nuclear, particle and astroparticle physicists further supported by a mathematician and a computer scientist. This explicit cooperation is of essence for the PhD topics of our Research Training Group.Scientifically this Research Training Group addresses questions at the forefront of our present knowledge of particle physics. In strong interactions we investigate questions of high complexity, such as the parton distributions in nuclear matter, the transition of the hot quark-gluon plasma into hadrons, or features of meson decays and spectroscopy. In weak interactions we pursue questions, which are by definition more speculative and which go beyond the Standard Model of particle physics, particularly with regard to the nature of dark matter. We will confront theoretical predictions with direct searches for cold dark matter particles or for heavy neutrinos as well as with new particle searches at the LHC.The pillars of our qualification programme are individual supervision and mentoring by one senior experimentalist and one senior theorist, topical lectures in physics and related fields (e.g. advanced computation), peer-to-peer training through active participation in two research groups, dedicated training in soft skills, and the promotion of research experience in the international community. We envisage early career steps through a transfer of responsibilities and international visibility with stays at external partner institutions. An important goal of this Research Training Group is to train a new generation of scientists, who are not only successful specialists in their fields, but who have a broader training both in theoretical and experimental nuclear, particle and astroparticle physics. Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

$\bullet$
Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Zakhar Kabluchko: Multidimensional Stochastic Models and Their Applications (2020-2022) The developments of Probability, Statistics, and their applications lead to increasingly complex models, involving a large number of variables as well as hidden or explicit parameters including a variety of random factors as well. The objects of our research like random matrices and random polynomials, Gaussian fields, random polytopes, fractional Laplacians as generators of Levy processes, U-max statistics and kernel density estimators are topics which are currently intensively studied. Research in this area requires the solution of basic open questions in modern Probability and Statistics using techniques relying on the latest mathematical achievements. In particular we shall concentrate on the solution of open problems concerning U-max statistics, the relation of random polytopes and intrinsic volumes to Gaussian random matrices and Gaussian processes, complexity measures for quantization and coding errors and the distribution of fractional processes as well as problems of small deviations. Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82$\bullet$Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82
Benedikt Wirth: Mathematische Rekonstruktion und Modellierung der CAR T-Zell Verteilung in vivo in einem Tumormodell (2019-2023)
Krebstherapien, wie Bestrahlung oder Chemotherapie, liefern häufig nur unzureichende Behandlungserfolge, so dass der Bedarf an neuartigen Therapiestrategien groß ist. Immuntherapien verwenden das körpereigene Immunsystem, um die Krebszellen zu erkennen und zu bekämpfen. Dem Patienten werden hierzu Abwehrzellen (T-Zellen) entnommen und diese werden genetisch verändert, sodass sie in der Lage sind, Krebszellen zu erkennen. Die so modifizierten "CAR T-Zellen" werden angereichert und dem Patienten zurückgegeben (transfundiert).

Für T-Zell-Therapien besteht in zweierlei Hinsicht Forschungsbedarf:

• Spezifizität: Die CAR T-Zellen werden auf bestimmte Erkennungsmerkmale (sogenannte Antigene) der Tumorzellen abgerichtet. Allerdings treten diese Antigene teilweise auch bei gesunden Zellen auf, sodass die CAR T-Zellen auch gesunde Zellen angreifen, was zu unerwünschten Nebenwirkungen führt. Um dies zu verhindern, müssen spezifischere Antigene gefunden bzw. Methoden erforscht werden, eine spezifischere Aktivierung der CAR T-Zellen zu erreichen. Eine Idee besteht hier z.B. in der Kombination mehrerer Antigene.
• Solide Tumoren: Während CAR T-Zelltherapien bei Leukämien (Blutkrebs) schon vielversprechende Erfolge zeigen, ist dies bei soliden Tumoren noch nicht der Fall. Der Grund wird in der Mikroumgebung solider Tumoren vermutet, wo verschiedene Barrieren ein effektives Eindringen der Immunzellen verhindern.
Bis heute ist die Verteilung und die Aktivität der transfundierten Zellen im Körper und im Tumor nur unzureichend bekannt.

Das Ziel dieses Projektes ist es, CAR T-Zellen im Körper mittels nicht-invasiver Bildgebungsverfahren wie PET/SPECT zu beobachten. Hierzu nutzen wir ein Tumormodell in der Maus. CAR T-Zellen werden mit nuklearmedizinischen Tracern markiert und ihre Verteilung und Aktivität wird in der Maus beobachtet.

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82$\bullet$Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82
Arnulf Jentzen
Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 , Benno Kuckuck: Mathematical Theory for Deep Learning (2019-2024) It is the key goal of this project to provide a rigorous mathematical analysis for deep learning algorithms and thereby to establish mathematical theorems which explain the success and the limitations of deep learning algorithms. In particular, this projects aims (i) to provide a mathematical theory for high-dimensional approximation capacities for deep neural networks, (ii) to reveal suitable regular sequences of functions which can be approximated by deep neural networks but not by shallow neural networks without the curse of dimensionality, and (iii) to establish dimension independent convergence rates for stochastic gradient descent optimization algorithms when employed to train deep neural networks with error constants which grow at most polynomially in the dimension. Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82$\bullet$Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82
Arnulf Jentzen: Existence, uniqueness, and regularity properties of solutions of partial differential equations (2019-2024)
The goal of this project is to reveal existence, uniqueness, and regularity properties of solutions of partial differential equations (PDEs). In particular, we intend to study existence, uniqueness, and regularity properties of viscosity solutions of degenerate semilinear Kolmogorov PDEs of the parabolic type. We plan to investigate such PDEs by means of probabilistic representations of the Feynman-Kac type. We also intend to study the connections of such PDEs to optimal control problems.

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

$\bullet$
Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Arnulf Jentzen: Regularity properties and approximations for stochastic ordinary and partial differential equations with non-globally Lipschitz continuous nonlinearities (2019-2024) A number of stochastic ordinary and partial differential equations from the literature (such as, for example, the Heston and the 3/2-model from financial engineering, (overdamped) Langevin-type equations from molecular dynamics, stochastic spatially extended FitzHugh-Nagumo systems from neurobiology, stochastic Navier-Stokes equations, Cahn-Hilliard-Cook equations) contain non-globally Lipschitz continuous nonlinearities in their drift or diffusion coefficients. A central aim of this project is to investigate regularity properties with respect to the initial values of such stochastic differential equations in a systematic way. A further goal of this project is to analyze the regularity of solutions of the deterministic Kolmogorov partial dfferential equations associated to such stochastic differential equations. Another aim of this project is to analyze weak and strong convergence and convergence rates of numerical approximations for such stochastic differential equations. Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82$\bullet$Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82
Arnulf Jentzen: Overcoming the curse of dimensionality: stochastic algorithms for high-dimensional partial differential equations (2019-2024)
Partial differential equations (PDEs) are among the most universal tools used in modeling problems in nature and man-made complex systems. The PDEs appearing in applications are often high dimensional. Such PDEs can typically not be solved explicitly and developing efficient numerical algorithms for high dimensional PDEs is one of the most challenging tasks in applied mathematics. As is well-known, the difficulty lies in the so-called ''curse of dimensionality'' in the sense that the computational effort of standard approximation algorithms grows exponentially in the dimension of the considered PDE. It is the key objective of this research project to overcome this curse of dimensionality and to construct and analyze new approximation algorithms which solve high dimensional PDEs with a computational effffort that grows at most polynomially in both the dimension of the PDE and the reciprocal of the prescribed approximation precision.

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82$\bullet$Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82
Benedikt Wirth: SPP 1962: Non-smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization - SP: Non-smooth and non-convex optimal transport problems (2019-2022)
In recent years a strong interest has developed within mathematics in so-called "branched Transport" models, which allow to describe transportation networks as they occur in road systems, river basins, communication networks, vasculature, and many other natural and artificial contexts. As in classical optimal transport, an amount of material needs to be transported efficiently from a given initial to a final mass distribution. In branched transport, however, the transportation cost is not proportional, but subadditive in the transported mass, modelling an increased transport efficiency if mass is transported in bulk. This automatically favours transportation schemes in which the mass flux concentrates on a complicated, ramified network of one-dimensional lines. The branched transport problem is an intricate nonconvex, nonsmooth variational problem on Radon measures (in fact on normal currents) that describe the mass flux. Various different formulations were developed and analysed (including work by the PIs), however, they all all take the viewpoint of geometric measure theory, working with flat chains, probability measures on the space of Lipschitz curves, or the like. What is completely lacking is an optimization and optimal control perspective (even though some ideas of optimization shimmer through in the existing variational arguments such as regularity analysis via necessary optimality conditions or the concept of calibrations which are related to dual optimization variables). This situation is also reflected in the fact that the field of numerics for branched transport is rather underdeveloped and consists of ad hoc graph optimization methods for special cases and two-dimensional phase field approximations. We will reformulate branched transport in the framework of optimization and optimal control for Radon measures, work out this optimization viewpoint in the variational analysis of branched transport networks, and exploit the results in novel numerical approaches. The new perspective will at the same time help variational analysts, advance the understanding of nonsmooth, nonconvex optimization problems on measures, and provide numerical methods to obtain efficient transport networks.

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82

Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82 Deprecated: substr(): Passing null to parameter #1 ($string) of type string is deprecated in /opt/common/php_includes/mysql-emulation.phps on line 82