Find below, as a small sample of the variety of mathematical research represented in Bonn, four of the ten research areas in the Hausdorff Center for Mathematics (HCM). A full list is found on HCM's webpages.
- HCM Research Area F*: ‘Structures and invariants in algebra and topology’
This is a new research area formed by researchers, half of whom came to Bonn after the cluster started. The research area also reﬂects a new emphasis in topology and representation theory.
The activities of this research area aim at the interplay of geometry, topology, algebra and group theory. The topics range over
- the classiﬁcation of manifolds,
- algebraic K- and L-theory of group rings,
- the geometry and homology of mapping class groups,
- equivariant and global homotopy theory,
- and categoriﬁcation of knot invariants and group algebras.
They have in common that they lead to explicit invariants in geometry and topology, designed to answer speciﬁc questions and solve speciﬁc problems, and to a better and deeper understanding of important general structures, which are of basic fundamental interest and will open the door to new projects and proofs. The investigators are experts in different ﬁelds; their respective backgrounds and expertise will lead to a fruitful cooperation and an exchange of knowledge and techniques.
- HCM Research Area B: ‘Shape, pattern and partial differential equations’
The interplay of the concepts of shape (interfaces in materials or geometric contours in images) and pattern (microstructures in materials or textures in images) characterises mathematical models both in the natural sciences and in computer vision and imaging. This Research Area capitalises on the similarity of the mathematical tools involved: differential geometry, the calculus of variations, and nonlinear partial differential equations. Examples of this fruitful interplay are the rigorous understanding of
- lower dimensional elasticity theories,
- multiscale models bridging between statistical physics and continuum mechanics,
- pattern formation and interface dynamics in biological models, or
- the combination of Riemannian geometry and continuum mechanics in shape space theory.
Research in this area emphasises the understanding of concrete phenomenons in connection to challenging applications over the development of abstract theory. Furthermore, we aim at developing fast and reliable numerical algorithms in a close interplay with modeling and analysis.
- HCM Research Area J: ‘High-Dimensional problems and multi-scale methods’
Mathematical modelling of physical phenomena often leads to high-dimensional partial differential equations. Examples are the many particle Schrödinger equation in quantum physics, the description of queueing networks, reaction mechanisms in molecular biology, visco-elasticity in polymer fluids, or models for the pricing of financial derivatives. Also, homogenisation and stochastic modelling usually result in high-dimensional PDEs. Typically, besides their high dimension, these problems involve multiple scales in space and time. In this Research Area we deal with high-dimensional problems and multiscale methods from the perspective of modelling, analysis, and numerical simulation. In the numerical treatment, the so-called curse of dimension is encountered. The computational cost required for an approximate solution scales exponentially with the dimension of the problem, and thus renders classical numerical approaches useless in practise. Therefore, the Research Area focuses on:
- Dimension-independent discretisation and solution methods
- Simplified effective models and their macroscopic behaviour of large high-dimensional systems.
- HCM Research Area KL: ‘Algorithms, combinatorics, and complexity’
The interaction between mathematics and computation, and more specifically between optimization and complexity, is central in our research area. Mathematical methods are employed to devise and analyze algorithms, and algorithmic ideas are applied in mathematical domains. On the one hand, efficient algorithms for various key problems are designed in this Research Area; for example in combinatorial optimisation and number theory, some of which had been considered hitherto to be computationally intractable. On the other hand, recent years have witnessed dramatic progress in our understanding of the phenomena of efficient computation: new and deep techniques for establishing intrinsic intractability barriers, and new means for surmounting them. In most cases, some combinatorial structure needs to be explored to gain insight. Notions of efficient computation and complexity are also evolving into central paradigms in mathematics and its foundations. In a worldwide unique industrial cooperation, this Research Area develops mathematical foundations and algorithms for designing next-generation computer chips, the most complex structures that mankind has ever developed.