Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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2063 research outputs found
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Bounded Weight Modules for Basic Classical Lie Superalgebras at Infinity
No full textWe classify simple bounded weight modules over the complex simple Lie superalgebras and , when at least one of and equals . For such modules are of spinor-oscillator type, i.e., they combine into one the known classes of spinor -modules and oscillator-type -modules. In addition, we characterize the category of bounded weight modules over (under the assumption ) by reducing its study to already
known categories of representations of , where possibly
equals . When classifying simple bounded weight -modules, we prove that every such module is integrable over one of the two infinite-dimensional ideals of the Lie algebra . We finish the paper by establishing some first facts about the category of bounded weight -modules
Discretization of Inherent ODEs and the Geometric Integration of DAEs with Symmetries
No full textDiscretization methods for differential-algebraic equations (DAEs) are considered that are based on the integration of an associated inherent ordinary differential equation (ODE). This allows to make use of any discretization scheme suitable for the numerical integration of ODEs. For DAEs with symmetries it is shown that the inherent ODE can be constructed in such a way that it inherits the symmetry properties of the given DAE and geometric properties of its flow. This in particular allows the use of geometric integration schemes with a numerical flow that has analogous geometric properties
Moduli spaces and Modular forms (hybrid meeting)
No full textThe relation between moduli spaces and modular forms goes back
to the theory of elliptic curves. On the one hand both topics
experience their own growth and development, but from time to
time new unexpected links show up and usually these lead to progress on both
sides. One subject where there has been a lot of progress concerns
the moduli of abelian varieties and K3 surfaces and especially
on the Kodaira dimension of these spaces. The idea of the workshop
was to bring together the experts of the two areas in the hope that
discussion, interaction and lectures would spur the development
of new ideas. The lectures of the workshop gave ample evidence
of the interaction and provided opportunities for further interaction.
Besides the lectures participants interacted via zoom in smaller groups
On the Computational Content of the Theory of Borel Equivalence Relations
No full textThis preprint offers computational insights into the theory of Borel equivalence relations. Specifically, we classify equivalence relations on the Cantor space up to computable reductions, i.e., reductions induced by Turing functionals. The presented results correspond to three main research focuses: (i) the poset of degrees of equivalence relations on reals under computable reducibility; (ii) the complexity of the equivalence relations generated by computability-theoretic reducibilities , (iii) the effectivization of the notion of hyperfiniteness
Reflections on hyperbolic space
No full textIn school, we learn that the interior angles of any triangle sum up to pi. However, there exist spaces different from the usual Euclidean space in which this is not true. One of these spaces is the ''hyperbolic space'', which has another geometry than the classical Euclidean geometry. In this snapshot, we consider the geometry of hyperbolic polytopes, for example polygons, how they tile hyperbolic space, and how reflections along the faces of polytopes give rise to important mathematical structures. The classification of these structures is an open area of research
Ultrafilter methods in combinatorics
No full textGiven a set X, ultrafilters determine which subsets
of X should be considered as large. We illustrate
the use of ultrafilter methods in combinatorics by
discussing two cornerstone results in Ramsey theory,
namely Ramsey’s theorem itself and Hindman’s theorem.
We then present a recent result in combinatorial
number theory that verifies a conjecture of Erdos
known as the “B + C conjecture”
Challenges in Optimization with Complex PDE-Systems (hybrid meeting)
No full textThe workshop concentrated on various aspects of optimization problems with systems of nonlinear partial differential equations (PDEs) or variational inequalities (VIs) as constraints. In particular, discussions around several keynote presentations in the areas of optimal control of nonlinear or non-smooth systems, optimization problems with functional and discrete or switching variables leading to mixed integer nonlinear PDE constrained optimization, shape and topology optimization, feedback control and stabilization, multi-criteria problems and multiple optimization problems with equilibrium constraints as well as versions of these problems under uncertainty or stochastic influences, and the respectively associated numerical analysis as well as design and analysis of solution algorithms were promoted. Moreover, aspects of optimal control of data-driven PDE constraints (e.g. related to machine learning) were addressed
Deep Learning for Inverse Problems (hybrid meeting)
No full textMachine learning and in particular deep learning offer several data-driven methods to amend the typical shortcomings of purely analytical approaches. The mathematical research on these combined models is presently exploding on the experimental side but still lacking on the theoretical point of view. This workshop addresses the challenge of developing a solid mathematical theory for analyzing deep neural networks for inverse problems