Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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2063 research outputs found
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Random matrix theory: Dyson Brownian motion
No full textThe theory of random matrices was introduced by
John Wishart (1898–1956) in 1928. The theory was
then developed within the field of nuclear physics
from 1955 by Eugene Paul Wigner (1902–1995) and
later by Freeman John Dyson, who were both concerned
with the statistical description of heavy atoms
and their electromagnetic properties. In this snapshot,
we show how mathematical properties can have
unexpected links to physical phenomenena. In particular,
we show that the eigenvalues of some particular
random matrices can mimic the electrostatic repulsion
of the particles in a gas
Splitting Necklaces, with Constraints
No full textWe prove several versions of Alon's "necklace-splitting theorem", subject to additional constraints, as illustrated by the following results.
(1) The "almost equicardinal necklace-splitting theorem" claims that, without increasing the number of cuts, one guarantees the existence of a fair splitting such that each thief is allocated (approximately) one and the same number of pieces of the necklace, provided the number of thieves is a prime power.
(2) The "binary splitting theorem" claims that if and the thieves are associated with the vertices of a d-cube then, without increasing the number of cuts, one can guarantee the existence of a fair splitting such that
adjacent pieces are allocated to thieves that share an edge of the cube. This result provides a positive answer to the "binary splitting necklace conjecture" of Asada at al. (Conjecture 2.11 in [5]) in the case
l-Torsion Bounds for the Class Group of Number Fields with an l -Group as Galois Group
No full textWe describe the relations among the -torsion conjecture for -extensions, the discriminant multiplicity conjecture for nilpotent extensions and a conjecture of Malle giving an upper bound for the number of nilpotent extensions. We then prove all of these conjectures in these cases
Low-Dimensional Topology and Number Theory (individual research only)
No full textBecause of the pandemia, the workshop on "Low-Dimensional Topology and Number Theory" could not be realized in the usual format or in the new hybrid format. Instead, a subgroup consisting of 6 participants used the week at the MFO mostly for informal discussions, collaborations and research in the topic of the workshop, including some online contacts to other participants who were not able to come to the MFO. At the institute talks were given by Campbell Wheeler on "WRT invariants & quantum modularity", Gregor Masbaum on "Generic skein modules", Gaëtan Borot on "Counting multicurves on surfaces with respect to hyperbolic vs. combinatorial geometry", Roland van der Veen on "1-cocycle invariants of knots", and by Michael Ontiveros on "From a triangulated lens space to a modular form"
Generating Finite Coxeter Groups with Elements of the Same Order
No full textSupposing is a group and a natural number, is defined to be the minimal number of elements of of order which generate (setting if has no such generating sets). This paper investigates when is a finite Coxeter group either of type or or of exceptional type. Together with Garzoni [3] and Yu [10], this determines for all finite irreducible Coxeter groups when 2rank (rank when is of type )
Rotating needles, vibrating strings, and Fourier summation
No full textWe give a brief survey of the connection between seemingly unrelated problems such as sets in the plane containing lines pointing in many directions, vibrating strings and drum heads, and a classical problem from Fourier analysis
Mathematical Logic: Proof Theory, Constructive Mathematics (hybrid meeting)
No full textThe Workshop "Mathematical Logic: Proof Theory,
Constructive Mathematics" focused on
proofs both as formal derivations in deductive systems as well as on
the extraction of explicit computational content from
given proofs in core areas of ordinary mathematics using proof-theoretic
methods. The workshop contributed to the following research strands: interactions between foundations and applications; proof mining; constructivity in classical logic; modal logic and provability logic; proof theory and theoretical computer science; structural proof theory
Boundary Element Methods
No full textThe field of boundary element methods (BEM) relies on recasting boundary value
problems for (mostly linear) partial differential equations as (usually singular) integral
equations on boundaries of domains or interfaces. Its main goal is the design and analysis
of methods and algorithms for the stable and accurate discretization of these integral
equations, the data-sparse representation of the resulting systems of equations, and their
efficient direct or iterative solution.
Boundary element methods play a key role in important areas of computational engineering
and physics addressing simulations in acoustics, electromagnetics, and elasticity. Thus
progress in boundary element method, both theoretical and algorithmic, is definitely
relevant beyond mathematics. Boundary element methods had been developed for many decades,
but during the past two decades the field has seen a surge in research activity, spurred
by algorithmic and theoretical breakthroughs concerning BEM for electromagnetics,
time-domain methods, new approaches to eigenvalue problems, adaptivity, local low-rank
matrix compression, and frequency-explicit analysis, to name only a few.
The contributions in this report give an impressive panorama of the many and diverse
current research activities in BEM. They range profound mathematical analyses with
striking results to new algorithmic developments. On the one hand, the results are based
on a large variety of tools from many areas of mathematics. On the other hand, research in
BEM blazes the trail for progress in the numerical treatment of non-local operators, a
field that is rapidly gaining importance
Rational Functions with Small Value Set
No full textIn connection with Galois Theory and Algebraic Curves, this paper investigates rational
functions for which the value set is relatively small. In particular, under certain circumstances, it proves that having a small value set is equivalent to the field extension being Galois
How Quantum Information Can Improve Social Welfare
No full textIn [2, 18, 5, 19, 4] it has been shown that quantum resources can allow us
to achieve a family of equilibria that can have sometimes a better social welfare,
while guaranteeing privacy. We use graph games to propose a way to build non-
cooperative games from graph states, and we show how to achieve an unlimited
improvement with quantum advice compared to classical advice