## Current Projects at the Department

The CRC/TRR “Geometry and Arithmetic of Uniformized Structures (GAUS)” aims to answer structural questions in geometry and arithmetic. The basic idea of uniformization is to replace complicated geometric objects by simpler ones without changing the local structure. Here the original mathematical complexity is encoded in a suitable symmetry group. Uniformization thus enables a translation of complexity into another “language” and opens up new perspectives on the original mathematical objects. The CRC/TRR wants to use these new perspectives to investigate the geometry and arithmetic of algebraic varieties. The researchers are searching for fundamental connections, for example regarding moduli spaces, automorphic forms, Galois representations and cohomological structures.

PIs at the Department of Mathematics: Jan Bruinier, Yingkun Li, Timo Richarz, Nils Scheithauer, Torsten Wedhorn

Funding: 07/2021-06/2025

Webpage: CRC 326

The “turnaround in energy policy” is currently in the main focus of public interest. It concerns social, political and scientific aspects as the dependence on a reliable, efficient and affordable energy supply becomes increasingly dominant. On the other side, the desire for a clean, environmentally consistent and climate-friendly energy production is stronger than ever. To balance these tendencies while making a transition to nuclear-free energy supply, gas becomes more and more important in the decades to come. Natural gas is and will be sufficiently available, is storable and can be traded. On the other side focusing on an efficient handling of gas transportation induces a number of technical and regulatory problems, also in the context of coupling to other energy carriers. As an example, energy transporters are required by law to provide evidence that within the given capacities all contracts defining the market are physically and technically feasible. Given the amount of data and the potential of stochastic effects, this is a formidable task all by itself, regardless from the actual process of distributing the proper amount of gas with the required quality to the customer. It is the goal of the Transregio-CRC to provide certified novel answers to these grand challenges, based on mathematical modeling, simulation and optimization. In order to achieve this goal new paradigms in the integration of these disciplines and in particular in the interplay between integer and nonlinear programming in the context of stochastic data have to be established and brought to bear. Clearly, without a specified underlying structure of the problems to face, such a breakthrough is rather unlikely. Thus, the particular network structure, the given hierarchical hybrid modeling in terms of switching algebraic, ordinary and partial differential-algebraic equations of hyperbolic type that is present in gas network transportation systems gives rise to the confidence that the challenges can be met by the team of the proposed Transregio-CRC. Moreover, the fundamental research conducted here will also be applicable in the context of other energy networks such as fresh- and waste water networks. In this respect the proposed research goes beyond the exemplary problem chosen and will provide, besides a cutting edge in enabling technologies, new mathematics in the emerging area of discrete, respectively, integer and continuous problems.

Besides TU Darmstadt the following institutions are participating in the TRR 154: Friedrich-Alexander-University Erlangen-Nürnberg (speaker university), HU Berlin, TU Berlin, University of Duisburg-Essen, and the Weierstraß-Institute for Applied Analysis and Stochastics (WIAS). Principle investigators in the SFB/Transregio at the mathematics department of TU Darmstadt are Dr. Domschke and the Professors Disser, Egger, Giesselmann, Lang, Pfetsch, and Ulbrich.

duration: 07/2018-06/2022 (second funding period)

Vice spokesperson: Stefan Ulbrich

Multiscale modeling is a central topic in theoretical condensed matter physics and materials science. One prominent class of materials, whose properties can rarely be understood on one length scale and one time scale alone, is soft matter. The properties of soft materials are determined by an intricate interplay of energy and entropy, and minute changes of molecular interactions may lead to massive changes of the system’s macroscopic properties.

In our collaborative research center (CRC TRR 146), we plan to tackle some of the most pressing problems in multiscale modeling in a joint effort of physicists, chemists, applied mathematicians, and computer scientists. The TRR 146 receives funding from the german science foundation (DFG) since October 2014. We address three major challenges:

*Dynamics*: In the past, multiscale coarse-graining approaches have to a large extent focused on static equilibrium properties. However, a thorough understanding of a coarse-grained model’s dynamical properties is necessary if one wants to apply multiscale concepts to the study of transport and nonequilibrium processes.*Coarse-graining and mixed resolution*: In many applications, selected small (e.g., functional) regions of a material must be treated in great detail, whereas the large bulk can be modeled at a coarse-grained level. Simulation schemes are desirable, where fine-grained and coarse-grained regions can dynamically be assigned to the current state of the system. In this context, we will also have to re-analyze fundamental aspects of coarse-graining from a mathematical point of view.*Bridging the particle-continuum gap*: So far, only few successful attempts have been made to combine particle models of soft matter with continuum models in a nontrivial fashion. Multiscale schemes for particle models have mostly been developed in the soft matter community, whereas schemes for treating continuum models with variable resolution are developed in the applied mathematics community. In the CRC-TR, we will bring these two communities together to advance the field as a whole.

The problems addressed in the TRR 146 require a massive interdisciplinary effort at the level of fundamental science and algorithmic development. The TRR 146 brings together scientists with a complementary expertise in a wide range of modeling methods.

Principle investigator in this SFB/Transregio at our department is Herbert Egger.

duration: 07/2018-06/2022 (second funding period)

The Collaborative Research Center (CRC) 1194 involves researchers from the TU Darmstadt and the Max Planck Institute for Polymer Research Mainz. Their common goal is the fundamental analysis of the interaction between transport and wetting processes – particularly when, parallel to momentum transport, also heat and mass transport, complex fluids or complex surfaces are involved.

What happens when surfaces are printed and coated with different liquids? What processes occur when a liquid meets a solid? How do wetting and dewetting depend on the local momentum, heat and mass transport processes? The underlying mechanisms of the interaction between these processes have to a large extent not been understood to date and are the focus of the Collaborative Research Centre (CRC) 1194. Although the physical phenomena take place only in a range of nanometres or micrometres, they often determine the efficiency of the overall process and the resulting product quality.

To date research in this area has focused on the dependence of the wetting process on the local fluid velocity at the contact line, i.e. wetting coupled with momentum transport. However serious deficits exist in our understanding when, parallel to momentum transport, also heat and mass transport are involved. These deficits become even more blatant when complex fluids are involved, e.g. suspensions or mixtures, or when complex surfaces are examined, e.g. rough or porous.

Fundamental processes and phenomena are examined over a wide range of length scales (nano-micro-macro) and the transfer of basic research to applications is an integral part of the research program. Overall the subject demands the use of very diverse methods and expertise from a multitude of disciplines.

The CRC is grouped into three research areas:

(A) Generic Experiments

(B) Modelling and Simulation

(C) New and Improved Applications.

As an overarching and integrating factor two generic configurations have been specified (Immersed Body, Sessile Drop). Furthermore, OpenFOAM has been selected as a common software platform.

Deputy Speaker: Dieter Bothe

The new LOEWE research unit Uniformized Structures in Arithmetic and Geometry aims at joining the broad expertise of TU Darmstadt and GU Frankfurt in the fields of number theory and arithmetic/algebraic geometry.

Our research program focusses on the following three research areas:

A. Special Subvarieties

B. Automorphic Forms

C. Variation of Geometry

In research area A we explore Orthogonal Shimura Varieties and the Kudla Conjecture, in research area B we investigate Borcherds-Products as well as Vertex Algebras, and in research area C we study the Uniformization of Spherical Varieties, the Anabelian Section Conjecture, as well as Tropical Moduli Spaces.

The research areas A, B, and C are mutually interconnected and techniques of uniformization are crucial in our research approaches.

Coordinator: Jan Hendrik Bruinier

Funding: 2018-2022

Iron has enormous potential to boost the energy transition. The project Clean Circles teams up scientists from multiple disciplines to explore how the metal and its oxides can be used in a cycle as carbon-free chemical energy carrier to store wind and solar power.

At the centre of the project is an innovative cycle of energy and substance as important contribution to the energy transition. Electric energy from renewable sources is accumulated in iron with a high energy density and thus becomes storable and transportable. This can be done directly by electrochemical reduction or by using green hydrogen in a thermochemical reduction. The chemical energy is released again with high power densities via high-temperature thermochemical oxidation and reconverted into electricty in thermal power stations. Iron is a carbon-free energy carrier. Thus, the release will not emit carbon dioxide as greenhouse gas. Well packed, iron can be stored for long periods of time. This makes iron a chemical energy carrier to guarantee the supply by controllable power station capacities.

PIs at the Department of Mathematics: Dieter Bothe, Marc Pfetsch, Stefan Ulbrich

Funding: 2021-2025 via the Hessian Cluster Initiative

The Graduate School Computational Engineering – Beyond Traditional Sciences – has been estabilished in 2007 In the context of the German excellence program. It was extended by 5 years in June 2012. After that the graduate school was integrated into the Center of Computational Engineering. Principal Investigators from Mathematics are the professors Aurzada, Bothe, Disser, Egger, Lang, Pfetsch, Ulbrich, and Wollner.

The research topics of the graduate school are the modeling and simulation of coupled multi-physical problems, simulation-based optimization, hierarchical multi-scale modeling and simulation as well as life cycle research with CE methods. The research approaches are connected by cross-sectional areas visualization, simulated reality, high performance computing, validation, software engineering, and life cycle research.

The homepage of the Graduate School is here

The mission of the Darmstadt Graduate School of Excellence Energy Science and Engineering is to educate tomorrow’s leading Energy Engineers in a multidisciplinary field of expertise needed to identify and master the most demanding scientific, engineering, economic and social challenges in an interdisciplinary approach. The main challenge is viewed to be a continuous transition from the carbon-based, non-renewable primary energy sources of today to renewable and environmentally friendly energy resources of tomorrow.

The optimal strategy to meet this challenge is on the one hand to improve conventional energy technologies and render them progressively more efficient, to meet the ever more stringent demands on pollutant emissions, and on the other hand to simultaneously develop innovative, advanced renewable energy technologies, which must be brought to a competitive technological readiness level and provide safe, reliable and cost-effective solutions.

Since October 2021, Anton Freund and his team work on a project with the title “Continuous Order Transformations: A Bridge Between Ordinal Analysis, Reverse Mathematics, and Combinatorics”, funded by the Emmy Noether Programme of DFG – German Research Foundation.

The following question is central for several branches of mathematical logic: Which axiom systems are strong enough to prove a given mathematical theorem? In addition to its intrinsic intellectual interest, an answer to this question does often yield further information about the theorem in question, for example on the quality of approximations or the complexity of algorithmic solutions. Our project aims at new and uniform answers to the given question.

Traditional ordinal analysis tends to focus on individual orders. A key idea of our project is to shift the focus to transformations between orders, including transformations that take other transformations as input. This allows us, for example, to speak about general recursive data types. Indeed, it is a well-known observation in theoretical computer science that these data types can be characterized as the initial fixed points of suitable transformations. Our approach yields new and elegant definitions and results, which are often more transparent than their traditional counterparts. It also builds bridges to other areas of logic, mathematics and computer science, in particular to reverse mathematics and combinatorics.

Emmy Noether Group Leader: Anton Freund

Funding: 2021-2027 from the Emmy Noether Programme of DFG

Website: Emmy Noether Programme 8210

The project studies the representation theory of reductive algebraic groups related to the computation of character formulas for simple and indecomposable tilting modules. We aim to explore new perspectives offered by geometry and categorification going beyond the techniques which have already been developed. The main geometric input will be the development of a modular ramified geometric Satake equivalence. We expect in particular applications in the study of tilting modules, for example, their behaviour under restriction to reductive subgroups, and their multiplicative properties.

Funding: 2021-2026

Project members:

Simon Riche (PI), Université Clermont Auvergne;

Timo Richarz (Partner), TU Darmstadt.

The project EXPRESS is funded within the framework of the DFG program of emphasis “Compressed Sensing in Information Processing” (CoSIP).

In the EXPRESS project we study the compressed sensing (CS) problem in the presence of side information and additional constraints. Side information as well as constraints are due to a specific structure encountered in the system model and may originate from the structure of the measurement system or the sensing matrix (shift-invariance, subarray structure, etc.), the structure of the signal waveforms (integrality, box constraints, constellation constraints such as non-circularity, constant modulus, finite constellation size, etc.), the sparsity structure of the signal (block or group sparsity, rank sparsity, etc.) or the channel, as well as the structure of the measurements (quantization effects, K-bit measures, magnitude-only measurements, etc.). We will investigate in which sense structural information can be incorporated into the CS problem and how it affects existing algorithms and theoretical results. Based on this analysis, we will develop new algorithms and theoretical results particularly suited for these models. It is expected, on the one hand, that exploiting structure in the measurement system, i.e., the sensing matrix, can lead to fast CS algorithms with novel model identifiability conditions and perfect reconstruction/recovery results. In this sense, exploiting structure in the observed signal waveforms and the sparsity structure of the signal representation can lead to reduced complexity CS algorithms with simplified recovery conditions and provably enhanced convergence properties. On the other hand, we expect that quantized measurements, which are of great importance when considering cost efficient hardware and distributed measurement systems, will generally result in a loss of information for which new algorithms and perfect recovery conditions need to be derived.

As an application in this project, we consider collaborative (distributed) multi- dimensional spatial spectrum sensing, i.e., sensing along the frequency-, time-, and space-axes using a network of multi-antenna sensing devices. Depending on the signal model under consideration, the frequency-, time-, and space dependence of the measurements can emerge in several ways. For example, the sensing parameters of interest can include directions-of- arrival, carrier frequencies, and Doppler-shifts. The EXPRESS project aims at exploiting the underlying sparsity properties in the signal model for this application while incorporating the aforementioned various types of side information.

Funding: 2015-2021

Projects with PIs from the department:

EXPRESS: Exploiting structure in compressed sensing using side constraints

The short and mid-term goals of the DFG-PP consist in establishing a theoretical and numerical foundation as well as in developing new algorithmic paradigms for the treatment of non-smooth phenomena and associated parameter influences. Long-term goals involve the realization and further advance of these concepts in the context of robust and hierarchical optimization, partial differential games, and nonlinear partial differential complementarity problems as well as the validation in the context of complex applications. This DFG-PP is motivated by important applications and very recent advances in theory and numerics for non-smooth distributed parameter systems (such as semi-smoothness in function space) and their optimization as well as the beginning of the blending of such non-smooth systems with problems requiring robust solutions. Rooted in applied mathematics, the DFG-PP has an interdisciplinary outreach as its results will benefit computational scientists and engineers, who face challenging applications involving non-smooth components. Structurally, on the one hand, the DFG-PP requires basic research in single projects, while for the synthesis and fostering of ideas and techniques the networking of research groups is needed. Moreover, clustering the research projects around prototypical applications is important to achieve landmark results and to succeed in addressing challenging applications.

Funding: 2016-2023

Projects with PIs from the department:

Optimizing Fracture Propagation using a Phase-Field Approach

Optimization methods for mathematical programs with equilibrium constraints in function spaces based on adaptive error control and reduced order or low rank tensor approximations

The main objective of this Priority Programme is the development of modern non-conventional discretisation methods, based on e.g. mixed (Galerkin or least-squares) finite element or discontinuous Galerkin formulations, including the mathematical analysis for geometrically as well as physically non-linear problems in the fields of e.g. incompressibility, anisotropies and discontinuities (cracks, contact). Our goal is to pool the expertise of mechanics and mathematics in Germany and to create new and strengthen existing networks. In the framework of this cooperation the experiences should be exchanged in between the different working groups to create synergies, save time and costs and raise the efficiency. Furthermore, it is intended to lead this research union to international excellence in the field of non-conventional discretisation techniques.In detail the Priority Programme will drive research towards the following directions concerning non-conventional finite element formulations:

- deep mathematical understanding of the structural requirements of reliable non-conforming finite element method (FEM) approaches for finite deformations,
- mathematically sound variational formulations,
- robust and stiffening-free discretisations at finite deformations for (quasi-)incompressible, isotropic and anisotropic material behaviour as well as for domains with oscillating coefficients,
- accurate approximation of all process variables in the latter mentioned extremal cases,
- insensitive behaviour concerning significant mesh deformation,
- convergence of adaptive mesh refinement,and discontinuities:
- creation of a variational basis as well as suitable discretisation techniques for discontinuities: convergence, stability and approximation properties,
- resolution of discontinuities based on isogeometric formulations,
- novel crack growth and crack branching models,
- contact formulations based on non-conventional discretisation techniques exceeding Mortar-methods.

Funding: 2017-2021

Project with PIs from the department:

The DFG-priority programme 2026 ''Geometry at infinity'' combines research in differential geometry, geometric topology, and global analysis. Crossing and transcending the frontiers of these disciplines it is concerned with convergence and limits in geometric-topological settings and with asymptotic properties of objects of infinite size. The overall theme can roughly be divided into the three cross-sectional topics convergence, compactifications, and rigidity.

Examples of convergence arise in Gromov-Hausdorff limits and geometric evolution equations. The behaviour of geometric, topological and analytic invariants under limits is of fundamental interest. Often limit spaces are non-smooth so that it is desirable to generalize notions like curvature or spectral invariants appropriately. Limits can also be used to construct asymptotic invariants in geometry and topology such as simplicial volume or L^2-invariants. Compactifications reflect asymptotic properties of geometric objects under suitable curvature conditions.

Methods from topology, differential geometry, operator algebras and probability play a role in this study. Important issues are boundary value problems for Laplace or Dirac type operators, both in the Riemannian and Lorentzian setting, as well as spectral geometry and Brownian motion on non-compact manifolds. Besides continuous deformations rigidity is essential for many classification problems in geometry and topology. It appears in geometric contexts, typically in the presence of negative curvature, and in topological and even algebraic settings. Rigidity also underlies isomorphism conjectures relating analytic, geometric and homological invariants of infinite groups and more general coarse spaces. The priority programme supports individual research projects and coordinated research activities.

Funding: 2017-2020 and 2020-2023.

The coordinator of the SPP 2026 is Prof. Dr. Bernhard Hanke from the Universität Augsburg.

Project with PIs from the department:

Asymptotics of singularties and deformations (Elena Mäder-Baumdicker)

## Completed Projects

The DFG Research Unit 1920 “Symmetry, Geometry and Arithmetic” examines current issues in modern arithmetic. An important and key theme is the investigation of absolute Galois groups and their generalisations. These elegantly code arithmetic information which can be extracted through the study of these groups and their representations. The researchers, who are based in Heidelberg and Darmstadt, are hoping that by dovetailing motivic homotopy theory, deformation theory, Iwasawa theory, the theory of automorphic forms and L-functions, they will be able to draw interesting conclusions from new insight into one of these areas which they can apply to the others, in a contemporary vision and modern understanding of basic mathematical research.

As a principal investigator Jan Bruinier is part of this research unit with a project centered around special cycles on the moduli space of abelian surfaces and their connections with L-functions. The spokesperson is Alexander Schmidt from the Universität Heidelberg.

Funding: 2013-2020

The International Research Training Group (IGK 1529) (Internationales Graduiertenkolleg) Mathematical Fluid Dynamics funded by DFG and JSPS and associated with TU Darmstadt, Waseda University in Tokyo and University of Tokyo has started on June 1, 2009.

Our research focuses on analytical, stochastic, geometric and optimization as well as on aerodynamical aspects of Fluid Dynamics. The program mainly seeks to combine methods from several mathematical disciplines, as analysis, stochastics, geometry and optimization to pursue fundamental research in Fluid Dynamics.

The International Research Training Group distinguishes itself through joint teaching and supervision. The core program consists of interdisciplinary lectures and seminars and includes research and study periods in Tokyo.

At the Collaborative Research Center (CRC) 805, about 40 researchers from the fields of mechanical engineering, mathematics and law work closely together. Our common goal is to control uncertainty in load carrying systems. Together, we investigate new methods for product development under uncertainty, enhance procedures in manufacturing, and establish advanced mechatronic and adaptronic technologies to stabilize load-bearing systems during usage.

The social and economic impact of uncertainty is tremendous. This is reflected in the alarmingly high number of product recalls. At times, more than one million vehicles a year are recalled by manufacturers in the automotive industry. By controlling uncertainty, product failures can be limited, safety factors can be minimized, current oversizing can be avoided, and thus resources can be saved.

Focus of our research is a holistic approach in order to control uncertainty along all phases of the product life: from product development to production to usage. A total of 19 subprojects from the three project areas work together:

In the project area product development, special design principles and mathematical optimization methods are developed in order to control uncertainty as early as in the design phase of technical systems.

In the Projects focusing on the production phase, forming and cutting production processes are designed flexibly while maintaining the same production quality, and functional materials for active components are integrated synchronously with the shaping.

In the project area usage, methods as well as passive, semi-active and active technologies are developed and tested in order to control uncertainty during the usage phase of load-carrying structures.

Funding: 2009-2020

In multiphase flows, there are different physicochemical processes that determine the behavior of such fluid systems, e.g. the fluid mechanics of the different fluids and the dynamics of the interfaces, heat and mass transport between the phases, adsorption effects at the interface and transport of species on the interface, variable interface properties, phase changes, etc.

As a rule, these sub-processes are closely coupled with each other, whereby the properties of the interface play a decisive role. A precise and deeper understanding of the behavior of such very complex flow problems must be based on physically sound mathematical models that take into account, in particular, the local processes at the interface.

The goal of the Priority Programme SPP 1506 is to further develop and extend such models, to analyze their mathematical properties and to develop and advance numerical methods for rigorous simulation of these models. This requires strongly interdisciplinary research with expertise from Applied Analysis, Numerical Mathematics, Interface Physics and Chemistry as well as relevant research areas in the Engineering Sciences. Important goals of the Priority Programme are:

- Derive and extend mathematical models that describe relevant physico-chemical interfacial phenomena.
- Improve and deepen the understanding of mechanisms and phenomena occurring at fluidic interfaces through rigorous mathematical analysis of the underlying PDE systems.
- Develop and analyze numerical methods for simulating multiphase flows that resolve the local processes at the interface.
- The validation of these models and the corresponding numerical simulation methods through specially designed experiments.

The long-term vision is that the validated models and simulation tools developed in the Priority Programme will induce significant advances in future high-tech applications such as lab-on-a-chip systems, multiphase reactors in chemical engineering and micro-process engineering.

Coordination: Dieter Bothe (TU Darmstadt) and Arnold Reusken (RWTH Aachen)

Funding: 2010-2016

In this Project we aim at using proof-theoretic methods from logic for the extraction of new data (such as effective bounds, “proof mining”) from prima facie noneffective proofs in convex optimization and related areas. We need to tailor the proof-theoretic methods to the specific domain of applications and will then apply them for the extraction of rates of asymptotic regularity, metastability (in the sense of T. Tao) and convergence of central iterative procedures used in convex optimization. In particular, we will study convergence proofs which make use of facts from the abstract theory of set-valued operators (e.g. maximally monotone operators).

Acyclicity criteria play a role in many areas of algorithmic model theory and logic in computer science. Acyclicity proves to be useful for the complexity of algorithmic problems as well as for model theory analysis. Often, ideal acyclicity conditions can be achieved by unwinding and superposition constructions; typical constructions (such as tree unwindings), however, are usually not available if one has to limit oneself to finite structures due to the nature of the problem. Here, qualitatively and quantitatively constrained approximations become important and it is important to

- (i) isolate suitable approximate acyclicity terms that allow for corresponding finite overlay or unwind constructions, and
- (ii) to obtain methods that make good algorithmic or logical properties available with such controlled acyclicity.

New construction methods, new analysis techniques and new fields of application are to be systematically researched and developed in a broader context.