# Research Areas (RAs)

The research areas are clustered in three thematic sections:

Section A: Deep structures of spaces and invariants

Section B: High dimensionality, singularities, and radomness

Section C: Mathematical modeling, analysis, and algorithms

/Leaders of the respective areas are marked in italics/

# Section A: Deep structures of spaces and invariants

Section A comprises research on 'Deep structures of spaces and invariants' in the arithmetic, algebraic, geometric, and topological core areas of pure mathematics. We will address the formidable questions posed by the Langlands program, with its arithmetic as well as its representation-theoretic aspects. Cohomological, homotopical, and categorical methods will be developed to study fundamental problems in geometry and topology, ranging from integral p-adic Hodge theory, through L2-Invariants of manifolds, to algebraic K-theory.

**Investigators**: Ana Caraiani, Gerd Faltings, *Daniel Huybrechts*, *Peter Scholze*, Don Zagier

Central to modern arithmetic and algebraic geometry are the concepts of Shimura varieties, algebraic cycles, modular forms, and cohomology theories, with many interactions among them and links to other areas. For example, points on Shimura varieties parametrize motives, which can be understood either in terms of cohomology theories or in terms of algebraic cycles; functions on Shimura varieties are modular forms; and the cohomology of Shimura varieties realizes Langlands correspondences. We want to further our understanding, leading to new results on the Tate and Hodge conjectures, diophantine height estimates, the Langland conjectures, algebraic K-theory, and the geometry of K3 surfaces.

**Investigators**: Gustavo Jasso, Wolfgang Lück, Peter Scholze, Jan Schröer, *Stefan Schwede*, *Catharina Stroppel*, Geordie Williamson

This research area focuses on problems where modern topological tools are brought to bear on algebraic and representation-theoretic problems, and where refined algebraic structures and additional symmetries facilitate computations in topology. Cohomological invariants and categorification techniques are essential and connecting concepts. Hecke algebras and group representations appear in different disguises and are subject of our study through algebraic geometry, combinatorial and topological methods.

**Investigators**: Ursula Hamenstädt, Daniel Huybrechts, Albrecht Klemm, *Wolfgang Lück*, Arunima Ray, Karl-Theodor Sturm, *Peter Teichner*

This research area aims for deep insights about manifolds, their automorphism groups and moduli spaces, and generalizations to metric measure spaces. This requires tools from various fields such as surgery theory, algebraic K-theory, measured and geometric group theory, harmonic analysis, and L2-invariants. We will combine forces to transfer techniques from one area to another. Among our main goals are the extension of the Farrell-Jones conjecture to reductive p-adic groups and new classification results for four-dimensional manifolds.

# Section B: High dimensionality, singularities, and randomness

Research in Section B will be devoted to the challenges of 'High dimensionality, singularities, and randomness'. We will deal with the demanding problems of universality beyond integrable models, convergence towards singular limit models, and understanding singular stochastic partial differential equations. Analyzing structures and evolutions of singular spaces will be among our challenging aims. Furthermore, we will pursue research on high-dimensional approximation in data spaces and on statistical analysis of large data sets.

**Investigators**: *Anton Bovier*, Andreas Eberle, Patrick Ferrari, *Massimiliano Gubinelli*, Lorens Imhof, Sven Rady, Karl-Theodor Sturm

The understanding of large-scale observables of probabilistic models is a central objective of probability theory. Leveraging our expertise on exact computations, asymptotic analysis, potential theory, coupling methods, control theory, multiscale and infinite-dimensional analysis, we will advance the understanding of basic mechanisms that drive large-scale observables. These involve extremal events ind highly correlated and high-dimensional random fields and their influence on complex and singular stochastic dynamics. The crucial task here is to identify universal asymptotic structures and to show convergence to these in a variety of particular situations, ranging from spatial branching processes, log-correlated Gaussian fields, and models of interface growth to singular stochastic partial differential equations and, on the applied side, to models of strategic acquisition and transmission of information and Markov Monte Carlo methods.

**Investigators**: Christian Bayer, Jochen Garcke, *Michael Griebel*, *Alois Kneip*, Michael Vogt

The efficient treatment of large sets of high-dimensional data by machine learning methods is a major challenge in big data applications. This project focuses on exploiting the low intrinsic dimensionality of nominally high-dimensional data sets to develop numerical and statistical techniques for the transformation and subsequent analysis of these sets. Our research will be driven by applications in econometrics, macroecronomics, biostatistics, and engineering.

**Investigators**: Werner Ballmann, Matthias Erbar, Leif Kobbelt, Matthias Lesch, Benny Moldovanu, Joe Neeman, Martin Rumpf, *Karl-Theodor Sturm*

Concepts of transport have led to fundamental new insights in the geometry of metric measure spaces and to powerful tools in mechanism design, computer vision, and geometry processing. There is a deep connection between these developments which is a leitmotiv of this research area. We will push forward the geometric calculus on metric measure spaces with synthetic Ricci bounds and extend it to discrete and non-commutative geometries, quantum systems, and kinetic evolution equations. We will study the many facets of the heat trace and optimal design problems associated with small eigenvalues. Tools based on transport models will be developed further and applied to problems of mechanism design, model uncertainty, and image processing. We aim to expand our Riemannian calculus on shape spaces and to combine it with shape statistics, machine learning, and shape semantic.

# Section C: Mathematical modeling, analysis, and algorithms

Research in Section C will deal with 'Mathematical modeling, analysis, and algorithms'. Our agenda includes the ambitious goal of understanding the relation between atomistic and continuum theories of solids through statistical mechanics and rigorous renormalization. We will advance our understanding of the role of information in economics and of fundamental algorithmic problems such as the traveling salesman problem. Moreover, we will derive sharp regularity results in analysis and optimal numerical discretization. This involves fundamental questions concerning frames and multiscale expansions.

**Investigators**: Sergio Conti, Margherita Disertori, *Stefan Müller*, Ira Neitzel, *Barbara Niethammer*, Alessia Nota, Michael Ortiz, Marc Alexander Schweitzer, Juan J. L. Velázquez

Mathematical methods to analyze and connect descriptions of matter at different scales are crucial to understanding and designing advanced materials. Through strategic hiring, Bonn has built a powerful team to address key new challenges of the next decade in a close interaction of modeling, analysis, simulation, and optimization: from toy models in statistical mechanics with discrete symmetries to more realistic models with continuous symmetries and phase transitions; from equilibrium, asymptotic self-similarity and local interactions to non-equilibrium, complex space-time behavior and long-range effects in kinetic equations; from energy based methods in statistics to the dynamics of defects and microstructure; from the simulation and analysis of single physics models to more realistic coupled real-world models, involving the full modeling-analysis-simulation-optimization cycle.

**Investigators**: Francesc Dilmé, Florian Hoffmann, Daniel Krähmer, Stephan Lauermann, *Benny Moldovanu*, Martin Pollrich, *Sven Rady*, Dezsö Szalay, Ngoc Tran

Information shapes the way individuals and groups make decisions. Modern economic theory recognizes information as a scarce good that is dispersed among many agents: time and resources have to be spent to acquire information, and individuals typically differ in the information generating technology they can deploy. The performance of institutions that govern collective decision-making crucially depends on the incentives they provide for individuals to acquire information, on their ability to aggregate information, and on the information to which they themselves give access. The research area thus focuses on three themes: information acquisition, information aggregation, and information design.

**Investigators**: Jens Franke, Stephan Held, Stephan Hougardy, Thomas Kesselheim, *Heiko Röglin*, *Jens Vygen*

The impact of combinatorial optimization to real-world applications can hardly be overestimated. We study fundamental combinatorial optimization problems (such as the traveling salesman problem) and their complexity, and we devise and analyze efficient algorithms. We also design and implement algorithms for real-world applications and benefit from the interplay between theory and practice. We have industrial cooperations with IBM and DHL, leading companies in two key application areas: chip design and logistics. On the one hand, we show that mathematics makes the difference in such applications, leading to better solutions. On the other hand, practical needs lead to new theoretical questions and to a refocusing of fundamental research. The theoretically best algorithms are often not the same as those that work best in practice; we aim to explain this phenomenon rigorously in order to narrow the gap between theory and practice.

**Investigators**: Carsten Burstedde, Michael Griebel, Herbert Koch, *Martin Rumpf*, *Christoph Thiele*, Pavel Zorin-Kranich

This research area focuses on multiscale phenomena, which are the core of many open problems in pure and applied mathematics. Harmonic analysis, in particular the analysis of function spaces and frames as well as optimal approximation therein, plays a fundamental role in the study of these phenomena. The theory of Riesz bases and multiscale expansions is used in time-frequency analysis and in modern approaches to numerical discretization. We develop and use sharp estimates in suitable choices of function spaces to establish well-posedness of PDEs and to derive efficient a posteriori estimates for nonconvex variational problems. We study dispersive equations from the rough-path perspective, and we investigate the interplay between multiscale frame representations of random fields and the approximation of PDEs with stochastic coefficients.