Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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Oberwolfach Preprints Vol. 2017,04
No full text2010 Mathematics Subject Classification: Primary 06B35, 06F05, 15A69, 46L05. Secondary
06B30, 06F25, 13J25, 16W80, 16Y60, 18B35, 18D20, 19K14, 46L06, 46M15, 54F05Research in Pairs 2015We show that abstract Cuntz semigroups form a closed symmetric monoidal category. Thus, given Cuntz semigroups and , there is another Cuntz semigroup playing the role of morphisms from to . Applied to C*-algebras and , the semigroup should be considered as the target in analogues of the UCT for bivariant theories of Cuntz semigroups.
Abstract bivariant Cuntz semigroups are computable in a number of interesting cases. We explore its behaviour under the tensor product with the Cuntz semigroup of strongly self-absorbing C*-algebras and the Jacelon-Razak algebra. We also show that order-zero maps between C*-algebras naturally define elements
in the respective bivariant Cuntz semigroup
K-Theory for Group C*-Algebras and Semigroup C*-Algebras
No full textThis book gives an account of the necessary background for group algebras and crossed products for actions of a group or a semigroup on a space and reports on some very recently developed techniques with applications to particular examples. Much of the material is available here for the first time in book form. The topics discussed are among the most classical and intensely studied C*-algebras. They are important for applications in fields as diverse as the theory of unitary group representations, index theory, the topology of manifolds or ergodic theory of group actions
Mini-Workshop: PDE Models of Motility and Invasion in Active Biosystems
No full textCell migration is crucial for the development and functioning of multicellular organisms; it plays an essential role in, e.g., morphogenesis, immune system dynamics, wound healing, angiogenesis, bacterial motion and biofilm formation, tumor growth and metastasis. Cell motility is a highly complex phenomenon involving a plethora of biophysical and biochemical events occuring on several time and space scales. The associated dynamics range from the subcellular level over individual cell behavior and up to the macroscopic level of cell populations; all these scales are tightly interrelated.
For decades, partial differential equations have been used to model the motility of single cells as well as the collective motion of cell assemblies like tumors. Mathematical models for both individual motile cells and invading tumors have major features in common. The active nature of cells leads to very similar nonlinear systems of coupled equations, the solutions of which often determine the position and shape of the objects of interest. Recently, several types of models attracted particular attention in the description of these systems: free boundary problems, phase field models, reaction-diffusion-taxis and kinetic transport equations. Both tumor growth/invasion and cell motility can be described by parabolic, hyperbolic, or elliptic equations; in case of free boundary problems, the boundary conditions are very similar. Thereby, the involved free boundaries can describe cell membranes, tumor margins, or interfaces between different tissues.
In this mini-workshop applied mathematicians and biophysicists using these model classes to describe different but related biological systems came together, presented their recent work and identified commonalities and differences in their approaches. Moreover, they discussed possible model extensions and their application to different, but related problems, along with the innovative utilization of certain mathematical tools to the analysis of the resulting systems
Mathematics of Quantitative Finance
No full textThe workshop on Mathematics of Quantitative Finance, organised at the Mathematisches Forschungsinstitut Oberwolfach from 26 February to 4 March 2017, focused on cutting edge areas of mathematical finance, with an emphasis on the applicability of the new techniques and models presented by the participants
Mini-Workshop: Spaces and Moduli Spaces of Riemannian Metrics
No full textThe mini-workshop focused on central questions and new results concerning spaces and moduli spaces of Riemannian metrics with lower or upper curvature bounds on open and closed manifolds and, moreover, related themes from Anosov geometry. These are all described in detail below. The event brought together young and senior researchers working about (moduli) spaces of negative and nonnegative sectional, nonnegative Ricci and positive scalar curvature as well as Anosov metrics, and the talks and discussions brought about many new and inspiring research problems to pursue
Real Algebraic Geometry With a View Toward Moment Problems and Optimization
No full textContinuing the tradition initiated in MFO workshop held in 2014, the aim of this workshop was to foster the interaction between real algebraic geometry, operator theory, optimization, and algorithms for systems control. A particular emphasis was given to moment problems through an interesting dialogue between researchers working on these problems in finite and infinite dimensional settings, from which emerged new challenges and interdisciplinary applications
The Colored Jones Polynomial and Kontsevich-Zagier Series for Double Twist Knots
No full textResearch in Pairs 2016Using a result of Takata, we prove a formula for the colored Jones polynomial of the double twist knots and where and are positive integers. In the case, this leads to new families of -hypergeometric series generalizing the Kontsevich-Zagier series. Comparing with the cyclotomic expansion of the colored Jones polynomials of gives a generalization of a duality at roots of unity between the Kontsevich-Zagier function and the generating function for strongly unimodal sequences
Mini-Workshop: Interactions between Low-dimensional Topology and Complex Algebraic Geometry
No full textRecent developments exhibit a strong connection between low-dimensional topology and complex algebraic geometry. A common theme is provided by the Alexander polynomial and its many avatars. The mini-Workshop brought together at Oberwolfach groups of researchers working in mostly separate areas, but sharing common interests in a vibrant, emerging field at the crossroads of Topology, Group Theory, and Geometry
Review of the Methods of Reflections
No full textResearch in Pairs 2017The methods of reflections were invented to obtain approximate solutions of the motion of more than one particle in a given environment, provided that one can represent the solution for one particle rather easily. This motivation is quite similar to the motivation of the Schwarz domain decomposition method, which was invented to prove existence and uniqueness of solutions of the Laplace equation on complicated domains, which are composed of simpler ones, for which existence and uniqueness of solutions was known. Like for Schwarz methods, there is also an alternating and a parallel method of reflections, but interestingly, the parallel method is not always convergent. We carefully trace in this paper the historical development of these methods of reflections, give several precise mathematical formulations, an equivalence result with the alternating Schwarz method for two particles, and also an analysis for a one dimensional model problem with three particles of the alternating, parallel, and a recent averaged parallel method of reflections
Detecting Ineffective Features for Pattern Recognition
No full textResearch in Pairs 2017For a binary classification problem, the hypothesis testing is studied, that a component of the observation vector is not effective, i.e., that component carries no information for the classification. We introduce nearest neighbor and partitioning estimates of the Bayes error probability, which result in a strongly consistent test