Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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2063 research outputs found
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Topological Complexity, Robotics and Social Choice
No full textTopological complexity is a number that measures
how hard it is to plan motions (for robots, say) in
terms of a particular space associated to the kind of
motion to be planned. This is a burgeoning subject
within the wider area of Applied Algebraic Topology.
Surprisingly, the same mathematics gives insight into
the question of creating social choice functions, which
may be viewed as algorithms for making decisions by
artificial intelligences
Convex Geometry and its Applications
No full textThe geometry of convex domains in Euclidean space plays a central role in several branches of mathematics: functional and harmonic analysis, the theory of PDE, linear programming and, increasingly, in the study of algorithms in computer science. The purpose of this meeting was to bring together researchers from the analytic, geometric and probabilistic groups who have contributed to these developments
Real Analyticity is Concentrated in Dimension 2
No full textWe prove that a real-valued function on a real analytic manifold is analytic whenever all its restrictions to -dimensional analytic submanifolds are analytic functions. We also obtain analogous results in the framework of Nash manifolds and nonsingular real algebraic sets. These results can be regarded as substitutes in the real case for the classical theorem of Hartogs, asserting that a complex-valued function defined on an open subset of is holomorphic if it is holomorphic with respect to each variable separately. In the proofs we use methods of real algebraic geometry even though the initial problem is purely analytic
Subgroups of Cremona Groups
No full textThere have been recent breakthroughs in two related questions: the study of Cremona groups, and the problem of rationality of algebraic varieties. Our workshop brought together algebraic geometers, who discussed and tried to solve problems that are relevant to the classification of finite subgroups of Cremona groups. Priority was given to the following four related areas: automorphisms of algebraic varieties, birational geometry of Mori fibre spaces, and rationality problems
Computability Theory
No full textComputability and computable enumerability are two of the fundamental notions of mathematics. Interest in effectiveness is already apparent in the famous Hilbert problems, in particular the second and tenth, and in early 20th century work of Dehn, initiating the study of word problems in group theory. The last decade has seen both completely new subareas develop as well as remarkable growth in two-way interactions between classical computability theory and areas of applications. There is also a great deal of work on algorithmic randomness, reverse mathematics, computable analysis, and in computable structure theory/computable model theory. The goal of this workshop is to bring together researchers representing different aspects of computability theory to discuss recent advances, and to stimulate future work
Enumerative Combinatorics
No full textEnumerative Combinatorics focusses on the exact and asymptotic counting of combinatorial objects. It is strongly connected to the probabilistic analysis of large combinatorial structures and has fruitful connections to several disciplines, including statistical physics, algebraic combinatorics, graph theory and computer science. This workshop brought together experts from all these various fields, including also computer algebra, with the goal of promoting cooperation and interaction among researchers with largely varying backgrounds
Non-commutative Geometry, Index Theory and Mathematical Physics
No full textNon-commutative geometry today is a new but mature branch of mathematics shedding light on many other areas from number theory to operator algebras. In the 2018 meeting two of these connections were highlighted. For once, the applications to mathematical physics, in particular quantum field theory. Indeed, it was quantum theory which told us first that the world on small scales inherently is non-commutative. The second connection was to index theory with its applications in differential geometry. Here, non-commutative geometry provides the fine tools to obtain higher information
Statistics for Data with Geometric Structure
No full textStatistics for data with geometric structure is an active and diverse topic of research. Applications include manifold spaces in directional data or symmetric positive definite matrices and some shape representations. But in some cases, more involved metric spaces like stratified spaces play a crucial role in different ways. On the one hand, phylogenetic trees are represented as points in a stratified data space, whereas branching trees, for example of veins, are data objects, whose stratified structure is of essential importance. For the latter case, one important tool is persistent homology, which is currently a very active area of research. As data sets become not only larger but also more complex, the need for theoretical and methodological progress in dealing with data on non-Euclidean spaces or data objects with nontrivial geometric structure is growing. A number of fundamental results have been achieved recently and the development of new methods for refined, more informative data representation is ongoing. Two complimentary approaches are pursued: on the one hand developing sophisticated new parameters to describe the data, like persistent homology, and on the other hand achieving simpler representations in terms of given parameters, like dimension reduction. Some foundational works in stochastic process theory on manifolds open the doors to this field and stochastic analysis on manifolds, thus enabling a well-founded treatment of non-Euclidean dynamic data. The results presented in the workshop by leading experts in the field are great accomplishments of collaboration between mathematicians from statistics, geometry and topology and the open problems which were discussed show the need for an expansion of this interdisciplinary effort, which could also tie in more closely with computer science
Applied Harmonic Analysis and Data Processing
No full textMassive data sets have their own architecture. Each data source has an inherent structure, which we should attempt to detect in order to utilize it for applications, such as denoising, clustering, anomaly detection, knowledge extraction, or classification. Harmonic analysis revolves around creating new structures for decomposition, rearrangement and reconstruction of operators and functions—in other words inventing and exploring new architectures for information and inference. Two previous very successful workshops on applied harmonic analysis and sparse approximation have taken place in 2012 and in 2015. This workshop was the an evolution and continuation of these workshops and intended to bring together world leading experts in applied harmonic analysis, data analysis, optimization, statistics, and machine learning to report on recent developments, and to foster new developments and collaborations
Fast Solvers for Highly Oscillatory Problems
No full textWaves of diverse types surround us. Sound, light
and other waves, such as microwaves, are crucial for
speech, mobile phones, and other communication technologies.
Elastic waves propagating through the Earth
bounce through the Earth’s crust and enable us to
“see” thousands of kilometres in depth. These propagating
waves are highly oscillatory in time and space,
and may scatter off obstacles or get “trapped” in
cavities. Simulating these phenomena on computers
is extremely important. However, the achievable
speeds for accurate numerical modelling are low even
on large modern computers. Our snapshot will introduce
the reader to recent progress in designing
algorithms that allow for much more rapid solutions