Goals of Research Area B

Variational problems in geometry and elasticity.

In the context of rigidity and elasticity of thin sheets there are many open problems on the boundary between the calculus of variations and geometry. We are in particular working on establishing rigorous scaling laws for the energy of elastic sheets with different types of boundary conditions. In some cases a logarithmic scaling is expected in analogy to energy concentration in dislocations in crystals. Furthermore, the optimality of energy scaling in folding patterns with thickness to the power 5/3 is expected to result from an interplay of the local elastic energy with large scale rigidity of almost isometric embeddings. A very much related problem, the optimal regularity of the Nash-Kuiper construction, has been highlighted as an outstanding open problem in geometry by S. T. Yau and is the subject of continued investigation.
Rank-one convexity and quasiconvexity lead to different families of constitutive relations in nonlinear elasticity where in essence one asks for a characterisation of possible  microstructures. Associated relaxation methods are investigated numerically in Research Area J. On the theoretical side a fundamental open problem is whether rank-one convexity implies quasi-convexity in two dimensions. Recent progress was made by Székelyhidi in his work on compensated compactness and Tartar’s conjecture [B:FS08]. A very different perspective builds on a deeply rooted connection to harmonic analysis, singular integral operators and stochastic optimal control and thus to Research Area A. A winter school on the calculus of variation and stochastic control in 2009 with Iwaniec, Struwe and Volberg as lecturers underpinned current investigations. Volberg will return to Bonn for 12 months in 2012–14 with a Humboldt Research Award.

Pattern formation in materials science and biology.

A major outstanding problem is to develop a statistical mechanics of solids. In statistical mechanics a key property is the convexity of the free energy, whereas in the continuum theory of elasticity the much weaker  condition of quasiconvexity is the natural condition. A route we plan to pursue (in collaboration with Research Area J and Research Area G) is rigorous renormalisation theory. Even in a deterministic setting there are still many challenges in the modeling of materials, in particular in the context of plasticity. We intend to address the relation of atomistic models of dislocations to continuum plasticity and to study formation of dislocation patterns at the mesoscopic scale, also in connection to relaxation theory. A close cooperation with the division on virtual material design of Fraunhofer SCAI on multiscale modeling and efficient parallel simulation is planned within the HCM also interconnecting Research Area B and Research Area J. The numerical simulation of materials governed by effects on different length scales will benefit from tailored schemes based on hierarchical matrices co-developed by Bebendorf [B:BK09] and parallel algorithms able to treat complicated domains as those studied by Beuchler. The recent appointment of Velázquez further strengthened the Research Area’s profile on free boundary problems, mean field models and singularities in geometric PDE. He in particular brings in expertise on pattern formation and interface dynamics in biological models. For instance, a deeper understanding of cell aggregates with local cell to cell interaction beyond approaches based on diffusion of chemicals requires models combining properties of parabolic and hyperbolic equations. Often they can be understood as limits of stochastic processes. This forms a link to Research Area G. Furthermore, the study of cell sorting induced by differential chemotactic aggregation – conjectured to play a crucial role in morphogenesis – in which chemotaxis propagates by means of chemical spiral waves poses many analytical challenges. Biology as a rich field of mathematical modeling and deep questions in analysis and stochastics were the topic of a workshop in cooperation with the Research Area G in 2009.

Variational methods in imaging and vision.

Computing geodesics paths is significantly more expensive than evaluating an elastic dissimilarity measure between shapes. Even though the latter at first does not induce a metric in shape space, we expect to establish a metric structure building upon the paradigm of diffusion maps. Furthermore, we aim at developing shape space analysis for shapes considered as thin shells, and study least action functionals associated with geodesic paths measuring tangential and bending stresses. Effective variational time discretisations will be based on discrete geometry and discrete exterior calculus. In addition, we intend to investigate the G-convergence of the arising time discrete functionals on sequences of shapes and incremental deformations to time continuous BV-type least action functionals on shapes and motion fields in space time. A particular challenge in image analysis, reconstruction and fusion is the proper combination of texture and geometry processing tools. Jointly with Rauhut (Bonn Junior Fellow engaged in Research Area J) we will combine sparse approximation via learned dictionaries to tackle applications in shape analysis and tissue classification in medical imaging.