Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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2063 research outputs found
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Mini-Workshop: Zero-Range and Point-Like Singular Perturbations: For a Spillover to Analysis, PDE and Differential Geometry
No full textThe field of contact interactions and perturbations of differential operators supported on subsets with non-trivial co-dimension is an increasingly active mainstream of mathematical physics (in particular, operator and spectral theory and quantum mechanics), with intimately related applications and mathematical challenges in partial differential equations, and neighbouring sectors of analysis, PDEs, and differential geometry. This Mini-Workshop fostered intense and prolific discussions on recent advances and trends in the field
Non-Commutative Geometry and Cyclic Homology
No full textThe meeting displayed the cyclic theory as a fundamental mathematical tool with applications in diverse domains such as analysis, algebraic K-theory, algebraic geometry, arithmetic geometry, solid state physics and quantum field theory
Representations and degenerations
No full textIn this snapshot, we explain two important mathematical concepts (representation and degeneration) in elementary terms. We will focus on the simplest meaningful examples, and motivate both concepts by study of symmetry
Quasi-Equilibria and Click Times for a Variant of Muller's Ratchet
No full textConsider a population of individuals, each of them carrying a type in . The population evolves according to a Moran dynamics with selection and mutation, where an individual of type has the same selective advantage over all individuals with type , and type mutates to type
at a constant rate. This model is thus a variation of the classical Muller's ratchet: there the selective advantage is proportional to . For a regime of selection strength and mutation rates which is between the regimes of weak and strong selection/mutation, we obtain the asymptotic rate of the click times of the ratchet (i.e. the times at which the hitherto minimal ('best') type in the population is lost), and reveal the quasi-stationary type frequency profile between clicks. The large population limit of this profile is characterized as the normalized attractor of a "dual" hierarchical multitype logistic system, and also via the distribution of the final minimal
displacement in a branching random walk with one-sided steps. An important role in the proofs is played by a graphical representation of the model, both forward and backward in time, and a central tool is the ancestral selection graph decorated by mutations
At the Interface between Semiclassical Analysis and Numerical Analysis of Wave Scattering Problems
No full textIn this context of wave scattering, both semiclassical analysis and numerical analysis share the same goal - that of understanding the behaviour of the scattered wave - but these two fields operate largely in isolation, mainly because the tools and techniques of the two fields are largely disjoint.
In recent years there have been promising examples of successful collaboration at the interface of semiclassical analysis and numerical analysis, to the mutual benefit of both fields. This workshop sought to capitalise on these successes by bringing together members of the semiclassical-analysis and numerical-analysis communities and catalysing activity at this interface
Embedding Spaces of Split Links
No full textWe study the homotopy type of the space of unparametrised embeddings of a split link in . Inspired by work of Brendle and Hatcher, we introduce a semi-simplicial space
of separating systems and show that this is homotopy equivalent to . This combinatorial object provides a gateway to studying the homotopy type of via the homotopy type of the spaces . We apply this tool to find a simple description of the fundamental group, or motion group, of , and extend this to a description of the motion group of embeddings in
Root Cycles in Coxeter Groups
No full textFor an element of a Coxeter group there are two important attributes, namely its length, and its expression as a product of disjoint cycles in its action on , the root system of . This paper investigates the interaction between these two features of , introducing the notion of the crossing number of , . Writing as a product of disjoint cycles we associate to each cycle a `crossing number' , which is the number of positive roots in for which is negative. Let Seq be the sequence of written in increasing order, and let = max Seq. The length of can be retrieved from this sequence, but Seq provides much more information. For a conjugacy class of let and let be the maximum value of across all conjugacy classes of . We call and , respectively, the crossing numbers of and . Here we determine the crossing numbers of all finite Coxeter groups and of all universal Coxeter groups. We also show, among other things, that for finite irreducible Coxeter groups if and are two elements of minimal length in the same conjugacy class , then Seq = Seq and . Also it is shown that the crossing number of an arbitrary Coxeter group is bounded below by the crossing number of a standard parabolic subgroup. Finally, examples are given to show that crossing numbers can be arbitrarily large for finite and infinite irreducible Coxeter groups
Convergence and Error Analysis of Compressible Fluid Flows with Random Data: Monte Carlo Method
No full textThe goal of this paper is to study convergence and error estimates of the Monte Carlo method for the Navier-Stokes equations with random data. To discretize in space and time, the Monte Carlo method is combined with a suitable deterministic discretization scheme, such as a fnite volume method. We assume that the initial data, force and the viscosity coefficients are random variables and study both, the statistical convergence rates as well as the approximation errors. Since the compressible Navier-Stokes equations are not known to be uniquely solvable in the class of global weak solutions, we cannot apply pathwise arguments to analyze the random Navier-Stokes equations. Instead we have to apply intrinsic stochastic compactness arguments via the Skorokhod representation theorem and the Gyöngy-Krylov method. Assuming that the numerical solutions are bounded in probability, we prove that the Monte Carlo fnite volume method converges to a statistical strong solution. The convergence rates are discussed as well. Numerical experiments illustrate theoretical results
Algebraic Structures in Statistical Methodology
No full textAlgebraic structures arise naturally in a broad variety of statistical problems, and
numerous fruitful connections have been made between algebra and discrete mathematics and research on statistical methodology. The workshop took up this theme with a particular focus on algebraic approaches to graphical models, causality, axiomatic systems for independence and non-parametric models
Mathematical Imaging and Surface Processing
No full textThis workshop was gathering applied mathematicians and computer scientists
interested in image and geometry processing. These topics have developed
tremendously in the past few years with the rise of artificial intelligence,
parallel hardware and strong needs for real-world applications (3D
scene reconstruction, architecture, medical imaging and data analysis, etc).
These research fields are at the intersection of many mathematical
disciplines, from geometry, calculus of variations and optimization to the
analysis and numerical analysis of PDEs. The almost 50 participants
to this workshop, including many young researchers, had many
fruitful exchanges, interested in common issues and
speaking a common language, yet often coming from
different backgrounds and with different knowledge.
This volume collects the abstracts for all the presentations covering this wide spectrum of tools and application domains