MFO Repository
http://publications.mfo.de:8080
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http://publications.mfo.de:8080
Demailly’s Notion of Algebraic Hyperbolicity: Geometricity, Boundedness, Moduli of Maps
http://publications.mfo.de/handle/mfo/1387
Demailly’s Notion of Algebraic Hyperbolicity: Geometricity, Boundedness, Moduli of Maps
Javanpeykar, Ariyan; Kamenova, Ljudmila
Demailly's conjecture, which is a consequence of the Green-Griffiths-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-Griffiths-Lang conjecture on hyperbolic projective varieties.
Mon, 08 Oct 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13872018-10-08T00:00:00ZJavanpeykar, AriyanKamenova, LjudmilaDemailly's conjecture, which is a consequence of the Green-Griffiths-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-Griffiths-Lang conjecture on hyperbolic projective varieties.Affine Space Fibrations
http://publications.mfo.de/handle/mfo/1386
Affine Space Fibrations
Gurjar, Rajendra V.; Masuda, Kayo; Miyanishi, Masayoshi
We discuss various aspects of affine space fibrations. Our interest will be focused in the singular fibers, the generic fiber and the propagation of properties of a given smooth special fiber to nearby fibers.
Wed, 05 Sep 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13862018-09-05T00:00:00ZGurjar, Rajendra V.Masuda, KayoMiyanishi, MasayoshiWe discuss various aspects of affine space fibrations. Our interest will be focused in the singular fibers, the generic fiber and the propagation of properties of a given smooth special fiber to nearby fibers.Some Results Related to Schiffer's Problem
http://publications.mfo.de/handle/mfo/1385
Some Results Related to Schiffer's Problem
Kawohl, Bernd; Lucia, Marcello
We consider the following semilinear overdetermined problem on a two dimensional bounded or unbounded domain $\Omega$ with analytic boundary $\partial\Omega$ having at least one bounded connected component \begin{eqnarray*} \left\{ \begin{array}{l} - \Delta u = g(u) \quad \hbox{in } \Omega,\\ \frac{\partial u}{\partial \nu} =0 \, \hbox{ and } \, u = c \hbox{ on } \partial \Omega, \end{array} \right. \end{eqnarray*} where $c$ is a constant. When $g(c) =0$ the constant solution $u \equiv c$ is the unique solution. For $g(c) \not =0$, we show that the boundary is a circle if and only if the problem admits a solution that has constant third or fourth normal derivative along the boundary. A similar result involving the fifth normal derivative is proved.
Thu, 16 Aug 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13852018-08-16T00:00:00ZKawohl, BerndLucia, MarcelloWe consider the following semilinear overdetermined problem on a two dimensional bounded or unbounded domain $\Omega$ with analytic boundary $\partial\Omega$ having at least one bounded connected component \begin{eqnarray*} \left\{ \begin{array}{l} - \Delta u = g(u) \quad \hbox{in } \Omega,\\ \frac{\partial u}{\partial \nu} =0 \, \hbox{ and } \, u = c \hbox{ on } \partial \Omega, \end{array} \right. \end{eqnarray*} where $c$ is a constant. When $g(c) =0$ the constant solution $u \equiv c$ is the unique solution. For $g(c) \not =0$, we show that the boundary is a circle if and only if the problem admits a solution that has constant third or fourth normal derivative along the boundary. A similar result involving the fifth normal derivative is proved.Topological Complexity, Robotics and Social Choice
http://publications.mfo.de/handle/mfo/1384
Topological Complexity, Robotics and Social Choice
Carrasquel, José; Lupton, Gregory; Oprea, John
Topological complexity is a number that measures
how hard it is to plan motions (for robots, say) in
terms of a particular space associated to the kind of
motion to be planned. This is a burgeoning subject
within the wider area of Applied Algebraic Topology.
Surprisingly, the same mathematics gives insight into
the question of creating social choice functions, which
may be viewed as algorithms for making decisions by
artificial intelligences.
Fri, 10 Aug 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13842018-08-10T00:00:00ZCarrasquel, JoséLupton, GregoryOprea, JohnTopological complexity is a number that measures
how hard it is to plan motions (for robots, say) in
terms of a particular space associated to the kind of
motion to be planned. This is a burgeoning subject
within the wider area of Applied Algebraic Topology.
Surprisingly, the same mathematics gives insight into
the question of creating social choice functions, which
may be viewed as algorithms for making decisions by
artificial intelligences.A short story on optimal transport and its many applications
http://publications.mfo.de/handle/mfo/1381
A short story on optimal transport and its many applications
Santambrogio, Filippo
We present some examples of optimal transport problems
and of applications to different sciences (logistics,
economics, image processing, and a little bit of
evolution equations) through the crazy story of an
industrial dynasty regularly asking advice from an
exotic mathematician.
Wed, 08 Aug 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13812018-08-08T00:00:00ZSantambrogio, FilippoWe present some examples of optimal transport problems
and of applications to different sciences (logistics,
economics, image processing, and a little bit of
evolution equations) through the crazy story of an
industrial dynasty regularly asking advice from an
exotic mathematician.Number theory in quantum computing
http://publications.mfo.de/handle/mfo/1380
Number theory in quantum computing
Schönnenbeck, Sebastian
Algorithms are mathematical procedures developed
to solve a problem. When encoded on a computer,
algorithms must be "translated" to a series of simple
steps, each of which the computer knows how
to do. This task is relatively easy to do on a classical
computer and we witness the benefits of this
success in our everyday life. Quantum mechanics,
the physical theory of the very small, promises to enable
completely novel architectures of our machines,
which will provide specific tasks with higher computing
power. Translating and implementing algorithms
on quantum computers is hard. However, we will
show that solutions to this problem can be found and
yield surprising applications to number theory.
Tue, 07 Aug 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13802018-08-07T00:00:00ZSchönnenbeck, SebastianAlgorithms are mathematical procedures developed
to solve a problem. When encoded on a computer,
algorithms must be "translated" to a series of simple
steps, each of which the computer knows how
to do. This task is relatively easy to do on a classical
computer and we witness the benefits of this
success in our everyday life. Quantum mechanics,
the physical theory of the very small, promises to enable
completely novel architectures of our machines,
which will provide specific tasks with higher computing
power. Translating and implementing algorithms
on quantum computers is hard. However, we will
show that solutions to this problem can be found and
yield surprising applications to number theory.Tropical geometry
http://publications.mfo.de/handle/mfo/1378
Tropical geometry
Brugallé, Erwan; Itenberg, Ilia; Shaw, Kristin; Viro, Oleg
What kind of strange spaces hide behind the enigmatic
name of tropical geometry? In the tropics, just
as in other geometries, one of the simplest objects is
a line. Therefore, we begin our exploration by considering
tropical lines. Afterwards, we take a look at
tropical arithmetic and algebra, and describe how to
define tropical curves using tropical polynomials.
Thu, 19 Jul 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13782018-07-19T00:00:00ZBrugallé, ErwanItenberg, IliaShaw, KristinViro, OlegWhat kind of strange spaces hide behind the enigmatic
name of tropical geometry? In the tropics, just
as in other geometries, one of the simplest objects is
a line. Therefore, we begin our exploration by considering
tropical lines. Afterwards, we take a look at
tropical arithmetic and algebra, and describe how to
define tropical curves using tropical polynomials.Generalized Vector Cross Products and Killing Forms on Negatively Curved Manifolds
http://publications.mfo.de/handle/mfo/1377
Generalized Vector Cross Products and Killing Forms on Negatively Curved Manifolds
Barberis, María Laura; Moroianu, Andrei; Semmelmann, Uwe
Motivated by the study of Killing forms on compact Riemannian manifolds of negative sectional curvature, we introduce the notion of generalized vector cross products on $\mathbb{R}^n$ and give their classification. Using previous results about Killing tensors on negatively curved manifolds and a new characterization of $\mathrm{SU}(3)$-structures in dimension $6$ whose associated $3$-form is Killing, we then show that every Killing $3$-form on a compact $n$-dimensional Riemannian manifold with negative sectional curvature vanishes if $n\ge 4$.
Tue, 17 Jul 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13772018-07-17T00:00:00ZBarberis, María LauraMoroianu, AndreiSemmelmann, UweMotivated by the study of Killing forms on compact Riemannian manifolds of negative sectional curvature, we introduce the notion of generalized vector cross products on $\mathbb{R}^n$ and give their classification. Using previous results about Killing tensors on negatively curved manifolds and a new characterization of $\mathrm{SU}(3)$-structures in dimension $6$ whose associated $3$-form is Killing, we then show that every Killing $3$-form on a compact $n$-dimensional Riemannian manifold with negative sectional curvature vanishes if $n\ge 4$.Metric Connections with Parallel Skew-Symmetric Torsion
http://publications.mfo.de/handle/mfo/1376
Metric Connections with Parallel Skew-Symmetric Torsion
Cleyton, Richard; Moroianu, Andrei; Semmelmann, Uwe
A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel skew-symmetric torsion arise naturally in several geometric contexts, e.g. on naturally reductive homogeneous spaces, nearly Kähler or nearly parallel G2-manifolds, Sasakian and 3-Sasakian manifolds, or twistor spaces over quaternion-Kähler manifolds with positive scalar curvature. In this paper we study the local structure of Riemannian manifolds carrying a metric connection with parallel skew-symmetric torsion. On every such manifold one can define a natural splitting of the tangent bundle which gives rise to a Riemannian submersion over a geometry with parallel skew-symmetric torsion of smaller dimension endowed with some extra structure. We show how previously known examples of geometries with parallel skew-symmetric torsion fit into this pattern, and construct several new examples. In the particular case where the above Riemannian submersion has the structure of a principal bundle, we give the complete local classification of the corresponding geometries with parallel skew-symmetric torsion.
Mon, 16 Jul 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13762018-07-16T00:00:00ZCleyton, RichardMoroianu, AndreiSemmelmann, UweA geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel skew-symmetric torsion arise naturally in several geometric contexts, e.g. on naturally reductive homogeneous spaces, nearly Kähler or nearly parallel G2-manifolds, Sasakian and 3-Sasakian manifolds, or twistor spaces over quaternion-Kähler manifolds with positive scalar curvature. In this paper we study the local structure of Riemannian manifolds carrying a metric connection with parallel skew-symmetric torsion. On every such manifold one can define a natural splitting of the tangent bundle which gives rise to a Riemannian submersion over a geometry with parallel skew-symmetric torsion of smaller dimension endowed with some extra structure. We show how previously known examples of geometries with parallel skew-symmetric torsion fit into this pattern, and construct several new examples. In the particular case where the above Riemannian submersion has the structure of a principal bundle, we give the complete local classification of the corresponding geometries with parallel skew-symmetric torsion.Data assimilation: mathematics for merging models and data
http://publications.mfo.de/handle/mfo/1375
Data assimilation: mathematics for merging models and data
Morzfeld, Matthias; Reich, Sebastian
When you describe a physical process, for example,
the weather on Earth, or an engineered system, such
as a self-driving car, you typically have two sources of
information. The first is a mathematical model, and
the second is information obtained by collecting data.
To make the best predictions for the weather, or most
effectively operate the self-driving car, you want to
use both sources of information. Data assimilation
describes the mathematical, numerical and computational
framework for doing just that.
Tue, 10 Jul 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13752018-07-10T00:00:00ZMorzfeld, MatthiasReich, SebastianWhen you describe a physical process, for example,
the weather on Earth, or an engineered system, such
as a self-driving car, you typically have two sources of
information. The first is a mathematical model, and
the second is information obtained by collecting data.
To make the best predictions for the weather, or most
effectively operate the self-driving car, you want to
use both sources of information. Data assimilation
describes the mathematical, numerical and computational
framework for doing just that.