The section carries out fundamental mathematics research. We form the mathematical knowledge base for emerging new areas in the technical sciences, and we investigate new applications of pure mathematics. We cooperate with colleagues from industry, mathematical research groups as well as engineering and natural science research groups in Denmark and abroad.
The section is active in the following four mathematical fields:
Algebra
Algebra describes the essence underlying countless structures in mathematics: prime numbers, polynomials, symmetries, vector spaces, geometrical objects, and on the lighter side also games, puzzles and even origami. So it is not surprising that algebra crops up in many real-life applications.
We teach, supervise projects and research in both the abstract nature and the applications of algebra. Our main focus is in Discrete Mathematics, notably Algebraic Coding Theory and Computer Algebra. Algebraic Coding Theory is used to represent data in such a way that it becomes resilient to noise and tampering. To use these solutions, we should know how to program computers so they operate fast on the algebraic structures: this is the aim of Computer Algebra.
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Contact persons: Peter Beelen, Johan Rosenkilde
Dynamical Systems
A dynamical system is a system that changes over time in accordance with certain rules. Dynamical systems is a central topic in mathematics, science and engineering.
In the Dynamical Systems Group we study differential equations, as well as difference equations, from a dynamical systems point of view. We develop new theory and new methods, primarily in the area of singular perturbation/slow-fast theory and the theory of complex dynamics. We also apply dynamical system theory to a range of problems in science and engineering, including fluid dynamics, chemistry, ecology, and mechanics. We collaborate with researchers from these fields and contribute to a mathematical understanding of their models. The methods we use include bifurcation theory, center- and slow manifold theory and blowup methods.
Contact persons: Morten Brøns, Christian Henriksen, Kristian U. Kristiansen
Functional Analysis
Functional analysis is a branch of mathematical analysis, dealing with infinite-dimensional vector spaces and operators hereon. Functional analysis is a research topic by itself, but it is also a toolbox that provides insight into the underlying mathematical structure of problems in dynamical systems, geometry, optimization and other areas of applied mathematics. The Functional analysis group focuses on applications within harmonic analysis, where a key issue is how to decompose complicated signals in terms of elementary building blocks.A central role is played by the theory of frames in general Hilbert spaces. The work in the group deals withthe abstract theory of frames and its concrete manifestations in terms of structured frames in function spaces. Among the considered frames are the Gabor systems (or more generally shift-invariant systems) arising from time-frequency analysis,wavelet/shearlet systems, and other generalized shift-invariant systems.
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Contact persons: Ole Christensen,Jakob Lemvig
Geometry
The geometry group develops novel theories, concepts, and applications related to the construction, analysis, and optimization of shapes in the most general sense of this very broad category, but also in its low dimensional concrete ramifications and approximations.
The research strategy of the group is focused on this - two-way - bridge between the general and the concrete. It is exemplified by recent new results:
- on capillary and CMC surfaces (from integrable geometry)
- on flowmeters (from isogeometric analysis)
- on sweep forms (from robotic geometry)
- on wildfire frontals (from Finsler and Lagrange geometry)
- on membranes and foams (from Laplace geometry).
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Contact persons: Steen Markvorsen, Jens Gravesen, David Brander