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.Fri, 13 Dec 2019 21:23:59 GMT2019-12-13T21:23:59ZA 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.Expander graphs and where to find them
http://publications.mfo.de/handle/mfo/3687
Expander graphs and where to find them
Khukhro, Ana
Graphs are mathematical objects composed of a collection
of “dots” called vertices, some of which are
joined by lines called edges. Graphs are ideal for visually
representing relations between things, and mathematical
properties of graphs can provide an insight
into real-life phenomena. One interesting property is
how connected a graph is, in the sense of how easy it
is to move between the vertices along the edges. The
topic dealt with here is the construction of particularly
well-connected graphs, and whether or not such
graphs can happily exist in worlds similar to ours.
Fri, 22 Nov 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36872019-11-22T00:00:00ZKhukhro, AnaGraphs are mathematical objects composed of a collection
of “dots” called vertices, some of which are
joined by lines called edges. Graphs are ideal for visually
representing relations between things, and mathematical
properties of graphs can provide an insight
into real-life phenomena. One interesting property is
how connected a graph is, in the sense of how easy it
is to move between the vertices along the edges. The
topic dealt with here is the construction of particularly
well-connected graphs, and whether or not such
graphs can happily exist in worlds similar to ours.Deep Learning and Inverse Problems
http://publications.mfo.de/handle/mfo/3686
Deep Learning and Inverse Problems
Arridge, Simon; de Hoop, Maarten; Maass, Peter; Öktem, Ozan; Schönlieb, Carola; Unser, Michael
Big data and deep learning are modern buzz words
which presently infiltrate all fields of science and technology.
These new concepts are impressive in terms
of the stunning results they achieve for a large variety
of applications. However, the theoretical justification
for their success is still very limited. In this snapshot,
we highlight some of the very recent mathematical
results that are the beginnings of a solid theoretical
foundation for the subject.
Thu, 21 Nov 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36862019-11-21T00:00:00ZArridge, Simonde Hoop, MaartenMaass, PeterÖktem, OzanSchönlieb, CarolaUnser, MichaelBig data and deep learning are modern buzz words
which presently infiltrate all fields of science and technology.
These new concepts are impressive in terms
of the stunning results they achieve for a large variety
of applications. However, the theoretical justification
for their success is still very limited. In this snapshot,
we highlight some of the very recent mathematical
results that are the beginnings of a solid theoretical
foundation for the subject.Mixed-dimensional models for real-world applications
http://publications.mfo.de/handle/mfo/3688
Mixed-dimensional models for real-world applications
Nordbotten, Jan Martin
We explore mathematical models for physical problems
in which it is necessary to simultaneously consider
equations in different dimensions; these are called
mixed-dimensional models. We first give several examples,
and then an overview of recent progress made
towards finding a general method of solution of such
problems.
Thu, 21 Nov 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36882019-11-21T00:00:00ZNordbotten, Jan MartinWe explore mathematical models for physical problems
in which it is necessary to simultaneously consider
equations in different dimensions; these are called
mixed-dimensional models. We first give several examples,
and then an overview of recent progress made
towards finding a general method of solution of such
problems.Touching the transcendentals: tractional motion from the bir th of calculus to future perspectives
http://publications.mfo.de/handle/mfo/3685
Touching the transcendentals: tractional motion from the bir th of calculus to future perspectives
Milici, Pietro
When the rigorous foundation of calculus was developed,
it marked an epochal change in the approach
of mathematicians to geometry. Tools from geometry
had been one of the foundations of mathematics
until the 17th century but today, mainstream conception
relegates geometry to be merely a tool of visualization.
In this snapshot, however, we consider
geometric and constructive components of calculus.
We reinterpret “tractional motion”, a late 17th century
method to draw transcendental curves, in order
to reintroduce “ideal machines” in math foundation
for a constructive approach to calculus that avoids
the concept of infinity.
Thu, 21 Nov 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36852019-11-21T00:00:00ZMilici, PietroWhen the rigorous foundation of calculus was developed,
it marked an epochal change in the approach
of mathematicians to geometry. Tools from geometry
had been one of the foundations of mathematics
until the 17th century but today, mainstream conception
relegates geometry to be merely a tool of visualization.
In this snapshot, however, we consider
geometric and constructive components of calculus.
We reinterpret “tractional motion”, a late 17th century
method to draw transcendental curves, in order
to reintroduce “ideal machines” in math foundation
for a constructive approach to calculus that avoids
the concept of infinity.Analogue mathematical instruments: Examples from the “theoretical dynamics” group (France, 1948–1964)
http://publications.mfo.de/handle/mfo/3684
Analogue mathematical instruments: Examples from the “theoretical dynamics” group (France, 1948–1964)
Petitgirard, Loïc
Throughout the history of dynamical systems, instruments
have been used to calculate and visualize (approximate)
solutions of differential equations. Here
we describe the approach of a group of physicists and
engineers in the period 1948–1964, and we give examples
of the specific (analogue) mathematical instruments
they conceived and used. These examples
also illustrate how their analogue culture and practices
faced the advent of the digital computer, which
appeared at that time as a new instrument, full of
promises.
Thu, 21 Nov 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36842019-11-21T00:00:00ZPetitgirard, LoïcThroughout the history of dynamical systems, instruments
have been used to calculate and visualize (approximate)
solutions of differential equations. Here
we describe the approach of a group of physicists and
engineers in the period 1948–1964, and we give examples
of the specific (analogue) mathematical instruments
they conceived and used. These examples
also illustrate how their analogue culture and practices
faced the advent of the digital computer, which
appeared at that time as a new instrument, full of
promises.Configuration spaces and braid groups
http://publications.mfo.de/handle/mfo/2519
Configuration spaces and braid groups
Jiménez Rolland, Rita; Xicoténcatl, Miguel A.
In this snapshot we introduce configuration spaces
and explain how a mathematician studies their ‘shape’.
This will lead us to consider paths of configurations
and braid groups, and to explore how algebraic properties
of these groups determine features of the spaces.
Tue, 08 Oct 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25192019-10-08T00:00:00ZJiménez Rolland, RitaXicoténcatl, Miguel A.In this snapshot we introduce configuration spaces
and explain how a mathematician studies their ‘shape’.
This will lead us to consider paths of configurations
and braid groups, and to explore how algebraic properties
of these groups determine features of the spaces.Limits of graph sequences
http://publications.mfo.de/handle/mfo/2516
Limits of graph sequences
Klimošová, Tereza
Graphs are simple mathematical structures used to
model a wide variety of real-life objects. With the
rise of computers, the size of the graphs used for
these models has grown enormously. The need to efficiently
represent and study properties of extremely
large graphs led to the development of the theory of
graph limits.
Wed, 04 Sep 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25162019-09-04T00:00:00ZKlimošová, TerezaGraphs are simple mathematical structures used to
model a wide variety of real-life objects. With the
rise of computers, the size of the graphs used for
these models has grown enormously. The need to efficiently
represent and study properties of extremely
large graphs led to the development of the theory of
graph limits.On Logic, Choices and Games
http://publications.mfo.de/handle/mfo/2515
On Logic, Choices and Games
Oliva, Paulo
Can we always mathematically formalise our taste
and preferences? We discuss how this has been done
historically in the field of game theory, and how recent
ideas from logic and computer science have brought
an interesting twist to this beautiful theory.
Wed, 04 Sep 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25152019-09-04T00:00:00ZOliva, PauloCan we always mathematically formalise our taste
and preferences? We discuss how this has been done
historically in the field of game theory, and how recent
ideas from logic and computer science have brought
an interesting twist to this beautiful theory.