Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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Combinatorics
No full textCombinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their
properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic
and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization,
Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions.
This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session
Fibrés de Higgs sans géométrie
No full textHiggs bundles appeared a few decades ago as solutions to certain equations from physics and have attracted much attention in geometry as well as other areas of mathematics and physics. Here, we take a very informal stroll through some aspects of linear algebra that anticipate the deeper structure in the moduli space of Higgs bundles.[Also available in French
Topological and Smooth Dynamics on Surfaces (individual research only)
No full textBecause of the pandemia, the workshop on "Topological and Smooth Dynamics on Surfaces"
could not be realized in the usual format or in the new hybrid format. Instead, a subgroup
of 3 participants used the week mostly for informal discussions, collaborations and research
in the topic of the workshop. Namely, a study of a parametric family
of planar homeomorphisms was carried out by Boroński and Štimac.
Recently a novel approach to the study of parametric families of
disiaptive diffeomorphisms on surfaces was introduced by Crovisier
and Pujals in their seminal paper [1], in which they initiate the study
of the class of strongly dissipative diffeomorphisms. These maps are
shown in [1] to be very close in a certain sense to 1-dimensional maps.
In particular, Crovisier and Pujals showed that for any of them there
exists a reduction to a 1-dimensional model, that consists of a metric
tree and a continuous map on it that is semi-conjugate to the original
diffeomorphism. The authors also showed that the class of strongly
dissipative diffeomorphisms contains maps in the Hénon family, for an
open set of parameters.
The study carried out by Boroński and Štimac at MFO attempted
on generalizing the aforementioned results to a new setting. Namely, a
Crovisier-Pujals-like 1-dimensional model was constructed for a Lozi
family of maps [3], within the Misiurewicz parameter set [4]. This
model conjugates each Lozi map in the studied parameter set to the
natural extension of a continuous map on a metric tree, thus extending
the earlier result to the case, but also improving semi-conjugation
to a conjugacy. Further results in this direction are expected to come
in a timely fashion.
The Ph.D. student Kilassa Kvaternik studied and presented the
paper [2]
Maximal Quaternion Orders in Quadratic Extensions - in Hurwitz’s Diaries
No full textWe present and comment on some unpublished work of Adolf Hurwitz on quaternion arithmetic from his diaries
Lefschetz Properties in Algebra, Geometry and Combinatorics (hybrid meeting)
No full textThe themes of the workshop are the Weak Lefschetz Property - WLP - and the Strong Lefschetz
Property - SLP. The name of these properties, referring to Artinian algebras, is motivated by the
Lefschetz theory for projective manifolds, initiated by S. Lefschetz, and well established by the late
1950's. In fact, Lefschetz properties of Artinian algebras are algebraic generalizations of the Hard
Lefschetz property of the cohomology ring of a smooth projective complex variety. The
investigation of the Lefschetz properties of Artinian algebras was started in the mid 1980's and nowadays is a very active area of research.
Although there were limited developments on this topic in the 20th century, in the last years this
topic has attracted increasing attention from mathematicians of different areas, such as commutative
algebra, algebraic geometry, combinatorics, algebraic topology and representation theory. One of
the main features of the WLP and the SLP is their ubiquity and the quite surprising and still not completely
understood relations with other themes, including linear configurations, interpolation problems, vector bundle
theory, plane partitions, splines, -webs, differential geometry, coding theory, digital image processing, physics and the theory of
statistical designs, etc. among others
Space-Time Euler Discretization Schemes for the Stochastic 2D Navier-Stokes Equations
No full textWe prove that the implicit time Euler scheme coupled with finite elements space discretization for the 2D Navier-Stokes equations on the torus subject to a random perturbation converges in , and describe the rate of convergence for an -valued initial condition. This refines previous results which only established the convergence in probability of these numerical approximations. Using exponential moment estimates of the solution of the stochastic Navier-Stokes equations and convergence of a localized scheme, we can prove strong convergence of this space-time approximation. The speed of the -convergence depends on the diffusion coefficient and on the viscosity parameter. In case of Scott-Vogelius mixed elements and for an additive noise, the convergence is polynomial
Dynamics of Gravitational Collapse in the Axisymmetric Einstein-Vlasov System
No full textWe numerically investigate the dynamcis near black hole formation of solutions to the Einstein-Vlasov system in axisymmetry. Our results are obtained using a particle-in-cell and finite difference code based on the (2+1)+1 formulation of the Einstein field equations in axisymmetry. Solutions are launched from generic type initial data and exhibit type I critical behaviour. In particular we find lifetime scaling in solutions containing black holes, and support that the critical solutions are stationary. Our results contain examples of solutions that form black holes, perform damped oscillations, and appear to disperse. We prove that complete dispersal of the solution implies that it has nonpositive binding energy
Vertex-to-self trajectories on the platonic solids
No full textWe consider the problem of walking in a straight line
on the surface of a Platonic solid. While the tetrahedron,
octahedron, cube, and icosahedron all exhibit
the same behavior, we find a remarkable difference
with the dodecahedron
(Revised Edition)
No full textDemailly's conjecture, which is a consequence of the Green-Griffths-Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly's conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly's definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraically closed fields. Furthermore, we show that the set of (not necessarily surjective) morphisms from a projective variety to a projective algebraically hyperbolic variety that map a fixed closed subvariety of onto a fixed closed subvariety of is finite. As an application, we obtain that Aut() is finite and that every surjective endomorphism of is an automorphism. Finally, we explore "weaker" notions of hyperbolicity related to boundedness of moduli spaces of maps, and verify similar predictions made by the Green-Griffths-Lang conjecture on hyperbolic projective varieties