http://publications.mfo.de/handle/mfo/2813 Oberwolfach Reports Volume 9 (2012) Thu, 09 Apr 2026 20:37:15 GMT 2026-04-09T20:37:15Z http://publications.mfo.de/handle/mfo/3331 Dynamics of Patterns Patterns and nonlinear waves arise in many applications. Mathematical descriptions and analyses draw from a variety of fields such as partial differential equations of various types, differential and difference equations on networks and lattices, multi-particle systems, time-delayed systems, and numerical analysis. This workshop brought together researchers from these diverse areas to bridge existing gaps and to facilitate interaction. Sun, 01 Jan 2012 00:00:00 GMT http://publications.mfo.de/handle/mfo/3331 2012-01-01T00:00:00Z Patterns and nonlinear waves arise in many applications. Mathematical descriptions and analyses draw from a variety of fields such as partial differential equations of various types, differential and difference equations on networks and lattices, multi-particle systems, time-delayed systems, and numerical analysis. This workshop brought together researchers from these diverse areas to bridge existing gaps and to facilitate interaction. http://publications.mfo.de/handle/mfo/3330 Convex Geometry and its Applications The 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 other algorithms in computer science. High-dimensional geometry is an extremely active area of research: the participation of a considerable number of talented young mathematicians at this meeting is testament to the fact that the field is flourishing. Sun, 01 Jan 2012 00:00:00 GMT http://publications.mfo.de/handle/mfo/3330 2012-01-01T00:00:00Z The 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 other algorithms in computer science. High-dimensional geometry is an extremely active area of research: the participation of a considerable number of talented young mathematicians at this meeting is testament to the fact that the field is flourishing. http://publications.mfo.de/handle/mfo/3329 Mathematical and Algorithmic Aspects of Atmosphere-Ocean Data Assimilation The nomenclature “data assimilation” arises from applications in the geosciences where complex mathematical models are interfaced with observational data in order to improve model forecasts. Mathematically, data assimilation is closely related to filtering and smoothing on the one hand and inverse problems and statistical inference on the other. Key challenges of data assimilation arise from the high-dimensionality of the underlying models, combined with systematic spatio-temporal model errors, pure model uncertainty quantifications and relatively sparse observation networks. Advances in the field of data assimilation will require combination of a broad range of mathematical techniques from differential equations, statistics, probability, scientific computing and mathematical modelling, together with insights from practitioners in the field. The workshop brought together a collection of scientists representing this broad spectrum of research strands. Sun, 01 Jan 2012 00:00:00 GMT http://publications.mfo.de/handle/mfo/3329 2012-01-01T00:00:00Z The nomenclature “data assimilation” arises from applications in the geosciences where complex mathematical models are interfaced with observational data in order to improve model forecasts. Mathematically, data assimilation is closely related to filtering and smoothing on the one hand and inverse problems and statistical inference on the other. Key challenges of data assimilation arise from the high-dimensionality of the underlying models, combined with systematic spatio-temporal model errors, pure model uncertainty quantifications and relatively sparse observation networks. Advances in the field of data assimilation will require combination of a broad range of mathematical techniques from differential equations, statistics, probability, scientific computing and mathematical modelling, together with insights from practitioners in the field. The workshop brought together a collection of scientists representing this broad spectrum of research strands. http://publications.mfo.de/handle/mfo/3328 Mini-Workshop: Frontiers in Quantile Regression Quantiles play an essential role in modern statistics, as emphasized by the fundamental work of Parzen (1978) and Tukey (1977). Quantile regression was introduced by Koenker and Bassett (1978) as a complement to least squares estimation (LSE) or maximum likelihood estimation (MLE) and leads to far-reaching extensions of ”classical” regression analysis by estimating families of conditional quantile surfaces, which describe the relation between a one-dimensional response y and a high dimensional predictor x. Since its introduction quantile regression has found great attraction in mathematical and applied statistics because of its natural interpretability and robustness, which yields attractive applications in such important areas as medicine, economics, engineering and environmental modeling. Although classical quantile regression theory is very well developed, the implicit definition of quantile regression still yields many new mathematical challenges such as multivariate, censored and longitudinal data, which were discussed during the workshop. Sun, 01 Jan 2012 00:00:00 GMT http://publications.mfo.de/handle/mfo/3328 2012-01-01T00:00:00Z Quantiles play an essential role in modern statistics, as emphasized by the fundamental work of Parzen (1978) and Tukey (1977). Quantile regression was introduced by Koenker and Bassett (1978) as a complement to least squares estimation (LSE) or maximum likelihood estimation (MLE) and leads to far-reaching extensions of ”classical” regression analysis by estimating families of conditional quantile surfaces, which describe the relation between a one-dimensional response y and a high dimensional predictor x. Since its introduction quantile regression has found great attraction in mathematical and applied statistics because of its natural interpretability and robustness, which yields attractive applications in such important areas as medicine, economics, engineering and environmental modeling. Although classical quantile regression theory is very well developed, the implicit definition of quantile regression still yields many new mathematical challenges such as multivariate, censored and longitudinal data, which were discussed during the workshop. http://publications.mfo.de/handle/mfo/3327 Mini-Workshop: Efficient and Robust Approximation of the Helmholtz Equation The accurate and efficient treatment of wave propogation phenomena is still a challenging problem. A prototypical equation is the Helmholtz equation at high wavenumbers. For this equation, Babuška & Sauter showed in 2000 in their seminal SIAM Review paper that standard discretizations must fail in the sense that the ratio of true error and best approximation error has to grow with the frequency. This has spurred the development of alternative, non-standard discretization techniques. This workshop focused on evaluating and comparing these different approaches also with a view to their applicability to more general wave propagation problems. Sun, 01 Jan 2012 00:00:00 GMT http://publications.mfo.de/handle/mfo/3327 2012-01-01T00:00:00Z The accurate and efficient treatment of wave propogation phenomena is still a challenging problem. A prototypical equation is the Helmholtz equation at high wavenumbers. For this equation, Babuška & Sauter showed in 2000 in their seminal SIAM Review paper that standard discretizations must fail in the sense that the ratio of true error and best approximation error has to grow with the frequency. This has spurred the development of alternative, non-standard discretization techniques. This workshop focused on evaluating and comparing these different approaches also with a view to their applicability to more general wave propagation problems. http://publications.mfo.de/handle/mfo/3326 Mini-Workshop: Geometries, Shapes and Topologies in PDE-based Applications The aim of the workshop was to study geometrical objects and their sensitivities in applications based on partial differential equations or differential variational inequalities. Focus topics comprised analytical investigations, numerical developments, issues in applications as well as new and future directions. Particular emphasis was put on: (i) combined shape and topological sensitivity; (ii) extended topological expansions and their numerical realization; (iii) level set based shape and topology optimization. Sun, 01 Jan 2012 00:00:00 GMT http://publications.mfo.de/handle/mfo/3326 2012-01-01T00:00:00Z The aim of the workshop was to study geometrical objects and their sensitivities in applications based on partial differential equations or differential variational inequalities. Focus topics comprised analytical investigations, numerical developments, issues in applications as well as new and future directions. Particular emphasis was put on: (i) combined shape and topological sensitivity; (ii) extended topological expansions and their numerical realization; (iii) level set based shape and topology optimization. http://publications.mfo.de/handle/mfo/3325 Complexity Theory Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, and pseudorandomness. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes. Sun, 01 Jan 2012 00:00:00 GMT http://publications.mfo.de/handle/mfo/3325 2012-01-01T00:00:00Z Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, and pseudorandomness. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes. http://publications.mfo.de/handle/mfo/3324 Non-Archimedean Analytic Geometry The workshop focused on recent developments in non-Archimedean analytic geometry with various applications to arithmetic and algebraic geometry. These applications include questions in Arakelov theory, p-adic differential equations, p-adic Hodge theory and the geometry of moduli spaces. Various methods were used in combination with analytic geometry, in particular perfectoid spaces, model theory, skeleta, formal geometry and tropical geometry. Sun, 01 Jan 2012 00:00:00 GMT http://publications.mfo.de/handle/mfo/3324 2012-01-01T00:00:00Z The workshop focused on recent developments in non-Archimedean analytic geometry with various applications to arithmetic and algebraic geometry. These applications include questions in Arakelov theory, p-adic differential equations, p-adic Hodge theory and the geometry of moduli spaces. Various methods were used in combination with analytic geometry, in particular perfectoid spaces, model theory, skeleta, formal geometry and tropical geometry. http://publications.mfo.de/handle/mfo/3323 C*-Algebras, Dynamics, and Classification Classification is a central theme in mathematics, and a particularly rich one in the theory of operator algebras. Indeed, one of the first major results in the theory is Murray and von Neumann’s type classification of factors (weakly closed self-adjoint algebras of operators on Hilbert space with trivial center), and one of its modern touchstones is the mid-1970s Connes-Haagerup classification of amenable factors with separable predual. Several significant themes in the classification theory of norm-separable C -algebras have emerged since the work of Connes-Haagerup, and these were the focus of our workshop. They include Elliott’s program to classify separable nuclear C -algebras via K-theoretic invariants, the role of C -algebras in the classification of orbit equivalence relations of discrete countable group actions, and the more recent contact between descriptive set theorists and operator algebraists which seeks to quantify the Borel complexity of the isomorphism relation for various natural classes of algebras. Sun, 01 Jan 2012 00:00:00 GMT http://publications.mfo.de/handle/mfo/3323 2012-01-01T00:00:00Z Classification is a central theme in mathematics, and a particularly rich one in the theory of operator algebras. Indeed, one of the first major results in the theory is Murray and von Neumann’s type classification of factors (weakly closed self-adjoint algebras of operators on Hilbert space with trivial center), and one of its modern touchstones is the mid-1970s Connes-Haagerup classification of amenable factors with separable predual. Several significant themes in the classification theory of norm-separable C -algebras have emerged since the work of Connes-Haagerup, and these were the focus of our workshop. They include Elliott’s program to classify separable nuclear C -algebras via K-theoretic invariants, the role of C -algebras in the classification of orbit equivalence relations of discrete countable group actions, and the more recent contact between descriptive set theorists and operator algebraists which seeks to quantify the Borel complexity of the isomorphism relation for various natural classes of algebras. http://publications.mfo.de/handle/mfo/3322 Computational Inverse Problems Inverse problem typically deal with the identification of unknown quantities from indirect measurements and appear in many areas in technology, medicine, biology, finance, and econometrics. The computational solution of such problems is a very active, interdisciplinary field with close connections to optimization, control theory, differential equations, asymptotic analysis, statistics, and probability. The focus of this workshop was on hybrid methods, model reduction, regularization in Banach spaces, and statistical approaches. Sun, 01 Jan 2012 00:00:00 GMT http://publications.mfo.de/handle/mfo/3322 2012-01-01T00:00:00Z Inverse problem typically deal with the identification of unknown quantities from indirect measurements and appear in many areas in technology, medicine, biology, finance, and econometrics. The computational solution of such problems is a very active, interdisciplinary field with close connections to optimization, control theory, differential equations, asymptotic analysis, statistics, and probability. The focus of this workshop was on hybrid methods, model reduction, regularization in Banach spaces, and statistical approaches.