# Models and Approximations

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 (G_{n})_{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 syste ms 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.