Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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    2063 research outputs found

    Innovative Approaches to the Numerical Approximation of PDEs

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      This workshop was about the numerical solution of PDEs for which classical approaches, such as the finite element method, are not well suited or need further (theoretical) underpinnings. A prominent example of PDEs for which classical methods are not well suited are PDEs posed in high space dimensions. New results on low rank tensor approximation for those problems were presented. Other presentations dealt with regularity of PDEs, the numerical solution of PDEs on surfaces, PDEs of fractional order, numerical solvers for PDEs that converge with exponential rates, and the application of deep neural networks for solving PDEs

      Mathematical Theory of Water Waves

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        Water waves, that is waves on the surface of a fluid (or the interface between different fluids) are omnipresent phenomena. However, as Feynman wrote in his lecture, water waves that are easily seen by everyone, and which are usually used as an example of waves in elementary courses, are the worst possible example; they have all the complications that waves can have. These complications make mathematical investigations particularly challenging and the physics particularly rich. Indeed, expertise gained in modelling, mathematical analysis and numerical simulation of water waves can be expected to lead to progress in issues of high societal impact (renewable energies in marine environments, vorticity generation and wave breaking, macro-vortices and coastal erosion, ocean shipping and near-shore navigation, tsunamis and hurricane-generated waves, floating airports, ice-sea interactions, ferrofluids in high-technology applications, ...). The workshop was mostly devoted to rigorous mathematical theory for the exact hydrodynamic equations; numerical simulations, modelling and experimental issues were included insofar as they had an evident synergy effect

        Mini-Workshop: Degeneration Techniques in Representation Theory

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          Modern Representation Theory has numerous applications in many mathematical areas such as algebraic geometry, combinatorics, convex geometry, mathematical physics, probability. Many of the object and problems of interest show up in a family. Degeneration techniques allow to study the properties of the whole family instead of concentrating on a single member. This idea has many incarnations in modern mathematics, including Newton-Okounkov bodies, tropical geometry, PBW degenerations, Hessenberg varieties. During the mini-workshop Degeneration Techniques in Representation Theory various sides of the existing applications of the degenerations techniques were discussed and several new possible directions were reported

          Analytic Number Theory

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            Analytic number theory is a subject which is central to modern mathematics. There are many important unsolved problems which have stimulated a large amount of activity by many talented researchers. At least two of the Millennium Problems can be considered to be in this area. Moreover in recent years there has been very substantial progress on a number of these questions

            Group Algebras of Compact Groups. A New Way of Producing Group Hopf Algebras over Real and Complex Fields: Weakly Complete Topological Vector Spaces

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            Mathematisches Forschungsinstitut Oberwolfach
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            Weakly complete real or complex associative algebras AA are necessarily projective limits of finite dimensional algebras. Their group of units A−1A^{-1} is a pro-Lie group with the associated topological Lie algebra ALieA_{\rm Lie} of AA as Lie algebra and the globally defined exponential function exp⁡ ⁣:A→A−1\exp\colon A\to A^{-1} as the exponential function of A−1A^{-1}. With each topological group GG, a weakly complete group algebra K[G]\mathbb K[G] is associated functorially so that the functor G↦K[G]G\mapsto \mathbb K[G] is left adjoint to A↦A−1A\mapsto A^{-1}. The group algebra K[G]\mathbb K[G] is a weakly complete Hopf algebra. If GG is compact, then R[G]\mathbb R[G] contains GG as the set of grouplike elements. The category of all real weakly complete Hopf algebras AA with a compact group of grouplike elements whose linear span is dense in AA is equivalent to the category of compact groups. The group algebra A=R[G]A=\mathbb R[G] of a compact group GG contains a copy of the Lie algebra L(G)\mathfrak L(G) in ALieA_{\rm Lie}; it also contains all probability measures on GG. The dual of the group algebra R[G]\mathbb R[G] is the Hopf algebra R(G,R){\cal R}(G,\mathbb R) of representative functions of GG. The rather straightforward duality between vector spaces and weakly complete vector spaces thus becomes the basis of a duality R(G,R)↔R[G]{\cal R}(G,\mathbb R)\leftrightarrow \mathbb R[G] and thus yields a new aspect of Tannaka duality. In the case of a compact abelian GG, an alternative concrete construction of K[G]\mathbb K[G] is given both for K=C\mathbb K=\mathbb C and K=R\mathbb K=\mathbb R. Because of the presence of L(G)\mathfrak L(G), the enveloping algebra of weakly complete Lie algebras are introduced and placed into relation with K[G]\mathbb K[G]

            The Fourier Transform on Harmonic Manifolds of Purely Exponential Volume Growth

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            Mathematisches Forschungsinstitut Oberwolfach
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            Let XX be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of harmonic manifolds except for the flat spaces. Denote by h>0h > 0 the mean curvature of horospheres in XX, and set ρ=h/2\rho = h/2. Fixing a basepoint o∈Xo \in X, for ξ∈∂X\xi \in \partial X, denote by BξB_{\xi} the Busemann function at ξ\xi such that Bξ(o)=0B_{\xi}(o) = 0. then for λ∈C\lambda \in \mathbb{C} the function e(iλ−ρ)Bξe^{(i\lambda - \rho)B_{\xi}} is an eigenfunction of the Laplace-Beltrami operator with eigenvalue −(λ2+ρ2)-(\lambda^2 + \rho^2). For a function ff on XX, we define the Fourier transform of ff by f~(λ,ξ):=∫Xf(x)e(−iλ−ρ)Bξ(x)dvol(x)\tilde{f}(\lambda, \xi) := \int_X f(x) e^{(-i\lambda - \rho)B_{\xi}(x)} dvol(x) for all λ∈C,ξ∈∂X\lambda \in \mathbb{C}, \xi \in \partial X for which the integral converges. We prove a Fourier inversion formula f(x)=C0∫0∞∫∂Xf~(λ,ξ)e(iλ−ρ)Bξ(x)dλo(ξ)∣c(λ)∣−2dλf(x) = C_0 \int_{0}^{\infty} \int_{\partial X} \tilde{f}(\lambda, \xi) e^{(i\lambda - \rho)B_{\xi}(x)} d\lambda_o(\xi) |c(\lambda)|^{-2} d\lambda for f∈Cc∞(X)f \in C^{\infty}_c(X), where cc is a certain function on R−{0}\mathbb{R} - \{0\}, λo\lambda_o is the visibility measure on ∂X\partial X with respect to the basepoint o∈Xo \in X and C0>0C_0 > 0 is a constant. We also prove a Plancherel theorem, and a version of the Kunze-Stein phenomenon

            Cataland: Why the Fuß?

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            Mathematisches Forschungsinstitut Oberwolfach
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            The three main objects in noncrossing Catalan combinatorics associated to a finite Coxeter system are noncrossing partitions, clusters, and sortable elements. The first two of these have known Fuß-Catalan generalizations. We provide new viewpoints for both and introduce the missing generalization of sortable elements by lifting the theory from the Coxeter system to the associated positive Artin monoid. We show how this new perspective ties together all three generalizations, providing a uniform framework for noncrossing Fuß-Catalan combinatorics. Having developed the combinatorial theory, we provide an interpretation of our generalizations in the language of the representation theory of hereditary Artin algebras

            Tropical Geometry: new directions

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              The workshop "Tropical Geometry: New Directions" was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject, notably, to new phenomena that have opened themselves in the course of the last 4 years. This includes, in particular, refined enumerative geometry (using positive integer q-numbers instead of positive integer numbers), unexpected appearance of tropical curves in scaling limits of Abelian sandpile models, as well as a significant progress in more traditional areas of tropical research, such as tropical moduli spaces, tropical homology and tropical correspondence theorems

              Mini-Workshop: Reflection Groups in Negative Curvature

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                Discrete groups generated by reflections constitute an important source of examples of lattices in simple Lie groups of real rank 11 (whose associated symmetric spaces are negatively curved). Yet a classification for them is far from being achieved, even in the case of hyperbolic geometry. The goal of this mini-workshop was to stimulate the research by bringing together specialists of different aspects of the theory

                Graph Theory

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                  Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem

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                  Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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