Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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Winkeltreue zahlt sich aus
No full textNicht nur Seefahrerinnen, auch Computergrafikerinnen
und Physikerinnen wissen Winkeltreue zu schätzen.
Doch beschränkte Rechenkapazitäten und Vereinfachungen
in theoretischen Modellen erfordern es,
winkeltreue Abbildungen nur mit einer überschaubaren
Datenmenge zu beschreiben. Entsprechende Theorien
werden in der diskreten Mathematik untersucht.
Im Folgenden lade ich Sie auf eine Reise in die faszinierende
Welt der winkeltreuen Abbildungen ein
Classical and Quantum Mechanical Models of Many-Particle Systems
No full textThis workshop was dedicated to the presentation of recent results in the field of the mathematical study of kinetic theory and its naturalextensions (statistical physics and fluid mechanics). The main models are the Vlasov(-Poisson) equation and the Boltzmann equation, which are obtainedas limits of many-body equations (Newton’s equations in the classical case and Schrödinger’s equation in the quantum case) thanks to the mean-field and Boltzmann-Grad scalings. Numerical aspects and applications to mechanics, physics, engineering and biology were also discussed
Computing the long term evolution of the solar system with geometric numerical integrators
No full textSimulating the dynamics of the Sun–Earth–Moon system
with a standard algorithm yields a dramatically
wrong solution, predicting that the Moon is ejected
from its orbit. In contrast, a well chosen algorithm
with the same initial data yields the correct behavior.
We explain the main ideas of how the evolution of
the solar system can be computed over long times
by taking advantage of so-called geometric numerical
methods. Short sample codes are provided for the
Sun–Earth–Moon system
Gradient Canyons, Concentration of Curvature, and Lipschitz Invariants
No full textResearch in Pairs 2017We find new bi-Lipschitz invariants of holomorphic functions of two variables by using the gradient canyons and by combining analytic and geometric viewpoints on the concentration of curvature
Arbeitsgemeinschaft: Higher Gross Zagier Formulas
No full textThe aim of this Arbeitsgemeinschaft is to go over the proof of the higher Gross–Zagier formula established in the paper [YZ15]. The formula relates arbitrary order central derivative of the base change -function of an unramifed automorphic representation of PGL over a function field to the self-intersection number of a certain algebraic cycle on the moduli stack of Shtukas
Cocycle Superrigidity and Group Actions on Stably Finite C*-Algebras
No full text2000 Mathematics Subject Classification. Primary 46L55, 54H05; Secondary 03E15, 37A55OWLF 2016Let be a countably infinite property (T) group, and let be
UHF-algebra of infinite type. We prove that there exists a continuum of
pairwise non (weakly) cocycle conjugate, strongly outer actions of
on . The proof consists in assigning, to any second countable abelian
pro- group , a strongly outer action of on whose (weak)
cocycle conjugacy class completely remembers the group . The group is
reconstructed from the action via its (weak) 1-cohomology set endowed with a
canonical pairing function. The key ingredient in this computation is Popa's
cocycle superrigidity theorem for Bernoulli shifts on the hyperfinite II
factor .
Our construction also shows the following stronger statement: the relations
of conjugacy, cocycle conjugacy, and weak cocycle conjugacy of strongly outer
actions of on are complete analytic sets, and in particular not
Borel. The same conclusions hold more generally when is only assumed
to contain an infinite subgroup with relative property (T), and for actions on
(not necessarily simple) separable, nuclear, UHF-absorbing, self-absorbing
C*-algebras with at least one trace.
Finally, we use the techniques of this paper to construct outer actions on
with prescribed cohomology. Precisely, for every infinite property (T)
group , and for every countable abelian group , we construct
an outer action of on whose 1-cohomology is isomorphic to
Solving quadratic equations in many variables
No full textFields are number systems in which every linear equation
has a solution, such as the set of all rational
numbers or the set of all real numbers . All fields
have the same properties in relation with systems of
linear equations, but quadratic equations behave differently
from field to field. Is there a field in which
every quadratic equation in five variables has a solution,
but some quadratic equation in four variables
has no solution? The answer is in this snapshot
Nonlinear Partial Differential Equations on Graphs
No full textOne-dimensional metric graphs in two and three-dimensional spaces play an important role in emerging areas of modern science such as nano-technology, quantum physics, and biological networks. The workshop focused on the analysis of nonlinear partial differential equations on metric graphs, especially on the bifurcation and stability of nonlinear waves on complex graphs, on the justification of Kirchhoff boundary conditions, on spectral properties and the validity of amplitude equations for periodic graphs, and the existence of ground states for the NLS equation with and without potential
Reflection Positivity
No full textThe main theme of the workshop was reflection positivity and its occurences in various areas of mathematics and physics, such as Representation Theory, Quantum Field Theory, Noncommutative Geometry, Dynamical Systems, Analysis and Statistical Mechanics. Accordingly, the program was intrinsically interdisciplinary and included talks covering different aspects of reflection positivity
Mini-Workshop: Adaptive Methods for Control Problems Constrained by Time-Dependent PDEs
No full textOptimization problems constrained by time-dependent PDEs (Partial Differential Equations) are challenging from a computational point of view: even in the simplest case, one needs to solve a system of PDEs coupled globally in time and space for the unknown solutions (the state, the costate and the control of the system). Typical and practically relevant examples are the control of nonlinear heat equations as they appear in laser hardening or the thermic control of flow problems (Boussinesq equations). Specifically for PDEs with a long time horizon, conventional time-stepping methods require an enormous storage of the respective other variables. In contrast, adaptive methods aim at distributing the available degrees of freedom in an a-posteriori-fashion to capture singularities and are, therefore, most promising