Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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The ternary Goldbach problem
No full textLeonhard Euler (1707–1783) war einer der besten Mathematiker des 18. Jahrhunderts und wohl auch aller Zeiten. Er korrespondierte oft mit seinem Freund, Christoph Goldbach (1690–1764), einem Universalgelehrten, der auch Mathematik betrieb und ebenso wie Euler in Russland lebte. In einem dieser Briefe im Juni 1782, machte Goldbach die folgende Vermutung zu Primzahlen: "Es scheinet wenigstens, dass eine jede Zahl, die größer ist als 2, ein aggregatum trium numerorum primorum sey." Goldbach behauptete also, dass sich jede Zahl, die größer ist als 2, als Summe dreier Primzahlen schreiben lässt. In diesem Schnappschuss moderner Mathematik soll es darum gehen, in welchem Ausmaß Mathematiker Goldbachs Vermutung bewiesen haben. Wir wollen dabei insbesondere aktuellen Fortschritt betrachten
Positive Line Bundles Over the Irreducible Quantum Flag Manifolds
No full textNoncommutative Kähler structures were recently introduced by the third author as a framework for studying noncommutative Kähler geometry on quantum homogeneous spaces. It was subsequently observed that the notion of a positive vector bundle directly generalises to this setting, as does the Kodaira vanishing theorem. In this paper, by restricting to covariant Kähler structures of irreducible type (those having an irreducible space of holomorphic -forms) we provide simple cohomological criteria for positivity, offering a means to avoid explicit curvature calculations. These general results are applied to our motivating family of examples, the irreducible quantum flag manifolds . Building on the recently established noncommutative Borel-Weil theorem, every covariant line bundle over can be identified as positive, negative, or flat, and hence we can conclude that each Kähler structure is of Fano type. Moreover, it proves possible to extend the Borel-Weil theorem for to a direct noncommutative generalisation of the classical Bott-Borel-Weil theorem for positive line bundles
Variational Methods for Evolution (hybrid meeting)
No full textVariational principles for evolutionary systems take advantage of the rich toolbox provided by the theory of the calculus of variations. Such principles are available for Hamiltonian systems in classical mechanics, gradient flows for dissipative systems, but also time-incremental minimization techniques for more general evolutionary problems. The new challenges arise via the interplay of two or more functionals (e.g. a free energy and a dissipation potential), new structures (systems with nonlocal transport, gradient flows on graphs, kinetic equations, systems of equations)
thus encompassing a large variety of applications in the modeling of materials and fluids, in biology, in multi-agent systems, and in data science.
This workshop brought together a broad spectrum of researchers from
calculus of variations, partial differential equations, metric
geometry, and stochastics, as well as applied and computational
scientists to discuss and exchange ideas. It focused on variational
tools such as minimizing movement schemes,
optimal transport, gradient flows, and large-deviation principles for
time-continuous Markov processes, -convergence and homogenization
Manifolds and Groups
No full textThe workshop concentrated on the interplay of advances in the understanding
of manifolds and geometric group theory. In particular, we discussed mapping
class groups and moduli spaces of manifolds (also of high dimension) and
aspects
of homological stability related to them; cobordism categories and their
applications;
-invariants, simplicial volume, and their applications; and in
general, the phenomena of rigidity versus flexibility in geometry and
topology
Mini-Workshop: Dimers, Ising and Spanning Trees beyond the Critical Isoradial Case (online meeting)
No full textThe goal of this mini-workshop is to gather specialists of the dimer, Ising and spanning tree models around recent and ongoing progress in two directions. One is understanding the connection to the spectral curve of these models in the cases when the curve has positive genus. The other is the introduction of universal embeddings associated to these models. We aim to use these new tools to progress in the study of scaling limits
Theoretical Analysis and Simulation Methods for Hawkes Processes and their Diffusion Approximation
No full textOscillatory systems of interacting Hawkes processes with Erlang memory kernels were introduced in Ditlevsen and Löcherbach (2017). They are piecewise deterministic Markov processes (PDMP) and can be approximated by a stochastic diffusion. First, a strong error bound between the PDMP and the diffusion is proved. Second, moment bounds for the resulting diffusion are derived. Third, approximation schemes for the diffusion, based on the numerical splitting approach, are proposed. These schemes are proved to converge with meansquare order 1 and to preserve the properties of the diffusion, in particular the hypoellipticity, the ergodicity and the moment bounds. Finally, the PDMP and the diffusion are compared through numerical experiments, where the PDMP is simulated with an adapted thinning procedure
Shape space – a paradigm for character animation in computer graphics
No full textNowadays 3D computer animation is increasingly realistic
as the models used for the characters become
more and more complex. These models are typically
represented by meshes of hundreds of thousands or
even millions of triangles. The mathematical notion
of a shape space allows us to effectively model, manipulate,
and animate such meshes. Once an appropriate
notion of dissimilarity measure between different
triangular meshes is defined, various useful tools
in character modeling and animation turn out to coincide
with basic geometric operations derived from
this definition
Topologie (hybrid meeting)
No full textThe Oberwolfach conference "Topologie" is one of only a few opportunities for
researchers from many different areas in algebraic and geometric topology to meet and
exchange ideas. On this occasion, because of the Corona pandemic, only about 20 participants attended in person, but another 25 attended online. Speakers were selected from both groups. A topic of special interest emphasized at the workshop was the rational
homotopy theory of embedding spaces and relations to graph complexes and formality. Two 50 minute lectures
on this theme were given by Thomas Willwacher, and one by Victor Turchin.
The rest of the program covered a wide range of topics, among them: homotopy properties of diffeomorphism groups
of high dimensional manifolds, advances in the classification of high-dimensional highly connected smooth manifolds,
parametrized algebraic surgery in relation to hermitian algebraic K-theory, other advances in
and geometric applications of algebraic K-theory, stable homotopy interpretation of link invariants,
geometry of surface bundles and cohomology of mapping class groups, boundary concepts in
geometric group theory, and Koszul duality for operads
Singularities and Bifurcations of Pseudospherical Surfaces
No full textWe study singularities and bifurcations of constant negative curvature surfaces in Euclidean 3-space via their association with Lorentzian harmonic maps. This preprint presents the basic results on this, the full proofs of which will appear in an article under preparation. We show that the generic bifurcations in 1-parameter families of such surfaces are the Cuspidal Butterfly, Cuspidal Lips, Cuspidal Beaks, 2/5 Cuspidal edge and Shcherbak bifurcations
Real Analysis, Harmonic Analysis and Applications (hybrid meeting)
No full textThe workshop hosted various research groups on topics in Real Analysis, Harmonic Analysis and Applications