Oberwolfach Publications
http://publications.mfo.de:80
The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Fri, 21 Feb 2020 22:32:25 GMT2020-02-21T22:32:25ZOberwolfach Publicationshttp://publications.mfo.de/themes/Mirage2/images/apple-touch-icon.png
http://publications.mfo.de:80
Demailly’s Notion of Algebraic Hyperbolicity: Geometricity, Boundedness, Moduli of Maps (Revised Edition)
http://publications.mfo.de/handle/mfo/3694
Demailly’s Notion of Algebraic Hyperbolicity: Geometricity, Boundedness, Moduli of Maps (Revised Edition)
Javanpeykar, Ariyan; Kamenova, Ljudmila
Demailly'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 $Y$ to a projective algebraically hyperbolic variety $X$ that map a fixed closed subvariety of $Y$ onto a fixed closed subvariety of $X$ is finite. As an application, we obtain that Aut($X$) is finite and that every surjective endomorphism of $X$ 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.
Thu, 23 Jan 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36942020-01-23T00:00:00ZJavanpeykar, AriyanKamenova, LjudmilaDemailly'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 $Y$ to a projective algebraically hyperbolic variety $X$ that map a fixed closed subvariety of $Y$ onto a fixed closed subvariety of $X$ is finite. As an application, we obtain that Aut($X$) is finite and that every surjective endomorphism of $X$ 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.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.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.