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Differentialgeometrie im Grossen (hybrid meeting)
The field of classical differential geometry has expanded enormously over the last several decades, helped by the development of tools from neighboring fields such as partial differential equations, complex analysis and geometric topology. In the spirit of the previous meetings in the series, this meeting will bring together researchers from apparently separate subfields of differential geometry, but whose work is linked by common themes. In particular, this meeting will emphasize intrinsic geometric questions motivated by the classification and rigidity of global geometric structures and the interaction of curvature with the underlying geometry and topology.
Mechanics of Materials: Mechanics of Interfaces and Evolving Microstructure
Emphasis in modern day efforts in mechanics of materials is increasingly directed towards integration with computational materials science, which itself rests on solid physical and mathematical foundations in thermodynamics and kinetics of processes. Practical applications demand attention to length and time scales which are sufficiently large to preclude direct application of quantum mechanics approaches; accordingly, there are numerous pathways to mathematical modelling of the complexity of material structure during processing and in service. The conventional mathematical machinery of energy minimization provides guidance but has limited direct applicability to material systems evolving away from equilibrium. Material response depends on driving forces, whether arising from mechanical, electromagnetic, or thermal fields. When microstructures evolve, as during plastic deformation, progressive damage and fracture, corrosion, stress-assisted diffusion, migration or chemical/thermal aging, the associated classical mathematical frameworks are often ad hoc and heuristic. Advancing new and improved methods is a major focus of 21st century mechanics of materials of interfaces and evolving microstructure.
Applications of Asymptotic Analysis
This workshop focused on asymptotic analysis and its fundamental role in the derivation and understanding of the nonlinear structure of mathematical models in various fields of applications, its impact on the development of new numerical methods and on other fields of applied mathematics such as shape optimization. This was complemented by a review as well as the presentation of some of the latest developments of singular perturbation methods.
Arbeitsgemeinschaft: Percolation
Abstract. Percolation as a mathematical theory is more than fifty years old. During its life, it has attracted the attention of both physicists and mathematicians. This is due in large part to the fact that it represents one of the simplest examples of a statistical mechanical model undergoing a phase transition, and that several interesting results can be obtained rigorously. In recent years the interest in percolation has spread even further, following the introduction by Oded Schramm of the Schramm-Loewner Evolution (SLE) and a theorem by Stanislav Smirnov showing the conformal invariance of the continuum scaling limit of two-dimensional critical percolation. These results establish a new, powerful and mathematically rigorous, link between lattice-based statistical mechanical models and conformally invariant models in the plane, studied by physicists under the name of Conformal Field Theory (CFT). The Arbeitsgemeinschaft on percolation has attracted more than thirty participants, most of them young researchers, from several countries in Europe, North America, and Brazil. The main focus has been on recent developments, but several classical results have also been presented.
Multivariate Splines and Algebraic Geometry
Multivariate splines are effective tools in numerical analysis and approximation theory. Despite an extensive literature on the subject, there remain open questions in finding their dimension, constructing local bases, and determining their approximation power. Much of what is currently known was developed by numerical analysts, using classical methods, in particular the so-called Bernstein-B´ezier techniques. Due to their many interesting structural properties, splines have become of keen interest to researchers in commutative and homological algebra and algebraic geometry. Unfortunately, these communities have not collaborated much. The purpose of the half-size workshop is to intensify the interaction between the different groups by bringing them together. This could lead to essential breakthroughs on several of the above problems.
Adaptive Numerical Methods for PDEs
This collection contains the extended abstracts of the talks given at the Oberwolfach Conference on “Adaptive Numerical Methods for PDEs”, June 10th - June 16th, 2007. These talks covered various aspects of a posteriori error estimation and mesh as well as model adaptation in solving partial differential equations. The topics ranged from the theoretical convergence analysis of self-adaptive methods, over the derivation of a posteriori error estimates for the finite element Galerkin discretization of various types of problems to the practical implementation and application of adaptive methods.