Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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2063 research outputs found
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The Index of Singular Zeros of Harmonic Mappings of Anti-Analytic Degree One
No full textMathematics Subject Classification (2010): 31A05, 30C55
Keywords: Harmonic mappings; Poincaré index; singular zero; multiplicity; critical setResearch in Pairs 2017We study harmonic mappings of the form , where is an analytic function. In particular we are interested in the index (a generalized multiplicity) of the zeros of such functions. Outside the critical set of , where the Jacobian of is non-vanishing, it is known that this index has similar properties as the classical multiplicity of zeros of analytic functions. Little is known about the index of zeros on the critical set, where the Jacobian vanishes; such zeros are called singular zeros. Our main result is a characterization of the index of singular zeros, which enables one to determine the index directly from the power series of
Bredon Cohomology and Robot Motion Planning
No full textResearch in Pairs 2017In this paper we study the topological invariant reflecting the complexity of algorithms for autonomous robot motion. Here, stands for the configuration space of a system and is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in . We focus on the case when the space is aspherical; then the number depends only on the fundamental group and we denote it . We prove that can be characterised as the smallest integer such that the canonical -equivariant map of classifying spaces can be equivariantly deformed into the -dimensional skeleton of . The symbol denotes the classifying space for free actions and denotes the classifying space for actions with isotropy in a certain family of subgroups of . Using this result we show how one can estimate in terms of the equivariant Bredon cohomology theory. We prove that where denotes the cohomological dimension of with respect to the family of subgroups . We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion free hyperbolic groups as well as all torsion free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher are exactly the classes having Bredon cohomology extensions with respect to the family
Emerging Developments in Interfaces and Free Boundaries
No full textThe field of the mathematical and numerical analysis of systems of nonlinear partial differential equations involving interfaces and free boundaries is a well established and flourishing area of research. This workshop focused on recent developments and emerging new themes. By bringing together experts in these fields we achieved progress in open questions and developed novel research directions in mathematics related to interfaces and free boundaries. This interdisciplinary workshop brought together researchers from distinct mathematical fields such as analysis, computation, optimisation and modelling to discuss emerging challenges
Differentialgeometrie im Großen
No full textThe topics discussed at the meeting were Kähler geometry, geometric evolution equations, manifolds of nonnegative curvature, metric geometry and geometric representations of groups. The choice of topics reflects current trends in the development of differential geometry
Computational Optimal Transport
No full textOptimal transport is the mathematical discipline of
matching supply to demand while minimizing shipping
costs. This matching problem becomes extremely
challenging as the quantity of supply and demand
points increases; modern applications must cope with
thousands or millions of these at a time. Here, we
introduce the computational optimal transport problem
and summarize recent ideas for achieving new
heights in efficiency and scalability
Numerical Invariants and Moduli Spaces for Line Arrangements
No full text2010 Mathematics Subject Classification: Primary 32S22; Secondary 14H50, 14B05, 13D02.
Key words and phrases: plane curves; line arrangement; free curves; syzygy; Terao's conjecture;
intersection lattice, Castelnuovo-Mumford regularity.Research in Pairs 2016Using several numerical invariants, we study a partition of the space of line arrangements in the complex projective plane, given by the intersection lattice types. We offer also a new characterization of the free plane curves using the Castelnuovo-Mumford regularity of the associated Milnor/Jacobian algebra
Mini-Workshop: Perspectives in High-dimensional Probability and Convexity
No full textUnderstanding the geometric structure of systems involving a huge amount of parameters is a central problem in mathematics and applied sciences today. Here, geometric and analytical ideas meet in a non-trivial way and powerful probabilistic tools play a key role in many discoveries. Two essentially independent areas of mathematics concerned with high-dimensional problems are asymptotic geometric analysis and information-based complexity. In this Mini-Workshop we brought together researchers from both fields to explore the connections and form synergies to develop new perspectives
Nonlinear Waves and Dispersive Equations
No full textNonlinear dispersive equations are models for nonlinear waves in a wide range of physical contexts. Mathematically they display an interplay between linear dispersion and nonlinear interactions, which can result in a wide range of outcomes from finite time blow-up to solitons and scattering. They are linked to many areas of mathematics and physics, ranging from integrable systems and harmonic analysis to fluid dynamics, geometry, general relativity and probability
The Minimal Resolution Conjecture on a general quartic surface in
No full textMSC: 13D02; 13C40; 13D40; 13E10; 14M06Research in Pairs 2017Mustaţă has given a conjecture for the graded Betti numbers in the minimal free resolution of the ideal of a general set of points on an irreducible projective algebraic variety. For surfaces in this conjecture has been proven for points on quadric surfaces and on general cubic surfaces. In the latter case, Gorenstein liaison was the main tool. Here we prove the conjecture for general quartic surfaces. Gorenstein liaison continues to be a central tool, but to prove the existence of our links we make use of certain dimension computations. We also discuss the higher degree case, but now the dimension count does not force the existence of our links
Arbeitsgemeinschaft: Additive Combinatorics, Entropy, and Fractal Geometry
No full textThe aim of the workshop was to survey recent developments in fractal geometry, specifically those related to projections and slices of planar self-similar sets, and dimension and absolute continuity of self-similar measures on the line, in particular Bernoulli convolutions. The methods combine ergodic theory, additive combinatorics, and algebraic number theory. Talks were high-level descriptions of the results, aimed at a mixed audience with minimal background in real analysis, ergodic theory and dimension theory