2 - Snapshots of Modern Mathematics from Oberwolfach
http://publications.mfo.de/handle/mfo/20
The snapshot project is designed to promote the understanding and appreciation of modern mathematics and mathematical research in the general public world-wide. It is part of the project "Oberwolfach meets IMAGINARY“, supported by the Klaus Tschira Foundation.Thu, 09 Jul 2020 01:07:43 GMT2020-07-09T01:07:43ZQuantum symmetry
http://publications.mfo.de/handle/mfo/3747
Quantum symmetry
Weber, Moritz
In mathematics, symmetry is usually captured using
the formalism of groups. However, the developments
of the past few decades revealed the need to go beyond
groups: to “quantum groups”. We explain the
passage from spaces to quantum spaces, from groups
to quantum groups, and from symmetry to quantum
symmetry, following an analytical approach.
Thu, 04 Jun 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37472020-06-04T00:00:00ZWeber, MoritzIn mathematics, symmetry is usually captured using
the formalism of groups. However, the developments
of the past few decades revealed the need to go beyond
groups: to “quantum groups”. We explain the
passage from spaces to quantum spaces, from groups
to quantum groups, and from symmetry to quantum
symmetry, following an analytical approach.Determinacy versus indeterminacy
http://publications.mfo.de/handle/mfo/3739
Determinacy versus indeterminacy
Berg, Christian
Can a continuous function on an interval be uniquely
determined if we know all the integrals of the function
against the natural powers of the variable? Following
Weierstrass and Stieltjes, we show that the answer is
yes if the interval is finite, and no if the interval is
infinite.
Wed, 22 Apr 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37392020-04-22T00:00:00ZBerg, ChristianCan a continuous function on an interval be uniquely
determined if we know all the integrals of the function
against the natural powers of the variable? Following
Weierstrass and Stieltjes, we show that the answer is
yes if the interval is finite, and no if the interval is
infinite.Vertex-to-Self Trajectories on the Platonic Solids
http://publications.mfo.de/handle/mfo/3737
Vertex-to-Self Trajectories on the Platonic Solids
Athreya, Jayadev S.; Aulicino, David
We 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.
Wed, 15 Apr 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37372020-04-15T00:00:00ZAthreya, Jayadev S.Aulicino, DavidWe 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.Random matrix theory: Dyson Brownian motion
http://publications.mfo.de/handle/mfo/3736
Random matrix theory: Dyson Brownian motion
Finocchio, Gianluca
The 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.
Wed, 15 Apr 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37362020-04-15T00:00:00ZFinocchio, GianlucaThe 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.From Betti numbers to ℓ²-Betti numbers
http://publications.mfo.de/handle/mfo/3735
From Betti numbers to ℓ²-Betti numbers
Kammeyer, Holger; Sauer, Roman
We provide a leisurely introduction to ℓ²-Betti numbers,
which are topological invariants, by relating
them to their much older cousins, Betti numbers. In
the end we present an open research problem about
ℓ²-Betti numbers.
Wed, 15 Apr 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37352020-04-15T00:00:00ZKammeyer, HolgerSauer, RomanWe provide a leisurely introduction to ℓ²-Betti numbers,
which are topological invariants, by relating
them to their much older cousins, Betti numbers. In
the end we present an open research problem about
ℓ²-Betti numbers.Is it possible to predict the far future before the near future is known accurately?
http://publications.mfo.de/handle/mfo/3693
Is it possible to predict the far future before the near future is known accurately?
Gander, Martin J.
It has always been the dream of mankind to predict
the future. If the future is governed by laws
of physics, like in the case of the weather, one can
try to make a model, solve the associated equations,
and thus predict the future. However, to make accurate
predictions can require extremely large amounts
of computation. If we need seven days to compute
a prediction for the weather tomorrow and the day
after tomorrow, the prediction arrives too late and
is thus not a prediction any more. Although it may
seem improbable, with the advent of powerful computers
with many parallel processors, it is possible to
compute a prediction for tomorrow and the day after
tomorrow simultaneously. We describe a mathematical
algorithm which is designed to achieve this.
Wed, 18 Dec 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36932019-12-18T00:00:00ZGander, Martin J.It has always been the dream of mankind to predict
the future. If the future is governed by laws
of physics, like in the case of the weather, one can
try to make a model, solve the associated equations,
and thus predict the future. However, to make accurate
predictions can require extremely large amounts
of computation. If we need seven days to compute
a prediction for the weather tomorrow and the day
after tomorrow, the prediction arrives too late and
is thus not a prediction any more. Although it may
seem improbable, with the advent of powerful computers
with many parallel processors, it is possible to
compute a prediction for tomorrow and the day after
tomorrow simultaneously. We describe a mathematical
algorithm which is designed to achieve this.The Interaction of Curvature and Topology
http://publications.mfo.de/handle/mfo/3692
The Interaction of Curvature and Topology
Kordaß, Jan-Bernhard
In this snapshot we will outline the mathematical
notion of curvature by means of comparison geometry.
We will then try to address questions as the ways in
which curvature might influence the topology of a
space, and vice versa.
Wed, 18 Dec 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36922019-12-18T00:00:00ZKordaß, Jan-BernhardIn this snapshot we will outline the mathematical
notion of curvature by means of comparison geometry.
We will then try to address questions as the ways in
which curvature might influence the topology of a
space, and vice versa.The Mathematics of Fluids and Solids
http://publications.mfo.de/handle/mfo/3691
The Mathematics of Fluids and Solids
Kaltenbacher, Barbara; Kukavica, Igor; Lasiecka, Irena; Triggiani, Roberto; Tuffaha, Amjad; Webster, Justin
Fluid-structure interaction is a rich and active field
of mathematics that studies the interaction between
fluids and solid objects. In this short article, we give
a glimpse into this exciting field, as well as a sample
of the most significant questions that mathematicians
try to answer.
Wed, 18 Dec 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36912019-12-18T00:00:00ZKaltenbacher, BarbaraKukavica, IgorLasiecka, IrenaTriggiani, RobertoTuffaha, AmjadWebster, JustinFluid-structure interaction is a rich and active field
of mathematics that studies the interaction between
fluids and solid objects. In this short article, we give
a glimpse into this exciting field, as well as a sample
of the most significant questions that mathematicians
try to answer.A surprising connection between quantum mechanics and shallow water waves
http://publications.mfo.de/handle/mfo/3690
A surprising connection between quantum mechanics and shallow water waves
Fillman, Jake; VandenBoom, Tom
We describe a connection between quantum mechanics
and nonlinear wave equations and highlight a few
problems at the forefront of modern research in the
intersection of these areas.
Wed, 11 Dec 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36902019-12-11T00:00:00ZFillman, JakeVandenBoom, TomWe describe a connection between quantum mechanics
and nonlinear wave equations and highlight a few
problems at the forefront of modern research in the
intersection of these areas.Formation Control and Rigidity Theory
http://publications.mfo.de/handle/mfo/3689
Formation Control and Rigidity Theory
Zelazo, Daniel; Zhao, Shiyu
Formation control is one of the fundamental coordination
tasks for teams of autonomous vehicles. Autonomous
formations are used in applications ranging
from search-and-rescue operations to deep space
exploration, with benefits including increased robustness
to failures and risk mitigation for human operators.
The challenge of formation control is to develop
distributed control strategies using vehicle onboard
sensing that ensures the desired formation is
obtained. This snapshot describes how the mathematical
theory of rigidity has emerged as an important
tool in the study of formation control problems.
Wed, 11 Dec 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36892019-12-11T00:00:00ZZelazo, DanielZhao, ShiyuFormation control is one of the fundamental coordination
tasks for teams of autonomous vehicles. Autonomous
formations are used in applications ranging
from search-and-rescue operations to deep space
exploration, with benefits including increased robustness
to failures and risk mitigation for human operators.
The challenge of formation control is to develop
distributed control strategies using vehicle onboard
sensing that ensures the desired formation is
obtained. This snapshot describes how the mathematical
theory of rigidity has emerged as an important
tool in the study of formation control problems.