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http://publications.mfo.de:8080
The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Wed, 22 May 2019 10:39:13 GMT2019-05-22T10:39:13ZMFO Repositoryhttp://publications.mfo.de/themes/Mirage2/images/apple-touch-icon.png
http://publications.mfo.de:8080
Chirality of Real Non-Singular Cubic Fourfolds and Their Pure Deformation Classification
http://publications.mfo.de/handle/mfo/1419
Chirality of Real Non-Singular Cubic Fourfolds and Their Pure Deformation Classification
Finashin, Sergey; Kharlamov, Viatcheslav
In our previous works we have classified real non-singular cubic hypersurfaces in the 5-dimensional projective space up to equivalence that includes both real projective transformations and continuous variations of co-efficients preserving the hypersurface non-singular. Here, we perform a finer classification giving a full answer to the chirality problem: which of real non-singular cubic hypersurfaces can not be continuously deformed to their mirror reflection.
Wed, 15 May 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14192019-05-15T00:00:00ZFinashin, SergeyKharlamov, ViatcheslavIn our previous works we have classified real non-singular cubic hypersurfaces in the 5-dimensional projective space up to equivalence that includes both real projective transformations and continuous variations of co-efficients preserving the hypersurface non-singular. Here, we perform a finer classification giving a full answer to the chirality problem: which of real non-singular cubic hypersurfaces can not be continuously deformed to their mirror reflection.The Becker-Gottlieb Transfer: a Geometric Description
http://publications.mfo.de/handle/mfo/1418
The Becker-Gottlieb Transfer: a Geometric Description
Wang, Yi-Sheng
In this note, we examine geometric aspects of the Becker-Gottlieb transfer in terms of the Umkehr and index maps, and rework some classic index theorems, using the cohomological formulae of the Becker-Gottlieb transfer. The results are natural from the homotopy-theoretic point of view; they reveal subtle geometric information in the Umkehr map, and demonstrate the beauty of the Atiyah-Singer index theorem for families and its generalizations.
Tue, 14 May 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14182019-05-14T00:00:00ZWang, Yi-ShengIn this note, we examine geometric aspects of the Becker-Gottlieb transfer in terms of the Umkehr and index maps, and rework some classic index theorems, using the cohomological formulae of the Becker-Gottlieb transfer. The results are natural from the homotopy-theoretic point of view; they reveal subtle geometric information in the Umkehr map, and demonstrate the beauty of the Atiyah-Singer index theorem for families and its generalizations.The Fourier Transform on Harmonic Manifolds of Purely Exponential Volume Growth
http://publications.mfo.de/handle/mfo/1417
The Fourier Transform on Harmonic Manifolds of Purely Exponential Volume Growth
Biswas, Kingshook; Knieper, Gerhard; Peyerimhoff, Norbert
Let $X$ be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of harmonic manifolds except for the flat spaces. Denote by $h > 0$ the mean curvature of horospheres in $X$, and set $\rho = h/2$. Fixing a basepoint $o \in X$, for $\xi \in \partial X$, denote by $B_{\xi}$ the Busemann function at $\xi$ such that $B_{\xi}(o) = 0$. then for $\lambda \in \mathbb{C}$ the function $e^{(i\lambda - \rho)B_{\xi}}$ is an eigenfunction of the Laplace-Beltrami operator with eigenvalue $-(\lambda^2 + \rho^2)$. For a function $f$ on $X$, we define the Fourier transform of $f$ by $$\tilde{f}(\lambda, \xi) := \int_X f(x) e^{(-i\lambda - \rho)B_{\xi}(x)} dvol(x)$$ for all $\lambda \in \mathbb{C}, \xi \in \partial X$ for which the integral converges. We prove a Fourier inversion formula $$f(x) = C_0 \int_{0}^{\infty} \int_{\partial X}
\tilde{f}(\lambda, \xi) e^{(i\lambda - \rho)B_{\xi}(x)} d\lambda_o(\xi) |c(\lambda)|^{-2} d\lambda$$ for $f \in C^{\infty}_c(X)$, where $c$ is a certain function on $\mathbb{R} - \{0\}$, $\lambda_o$ is the visibility measure on $\partial X$ with respect to the basepoint $o \in X$ and $C_0 > 0$ is a constant. We also prove a Plancherel theorem, and a version of the Kunze-Stein phenomenon.
Wed, 08 May 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14172019-05-08T00:00:00ZBiswas, KingshookKnieper, GerhardPeyerimhoff, NorbertLet $X$ be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of harmonic manifolds except for the flat spaces. Denote by $h > 0$ the mean curvature of horospheres in $X$, and set $\rho = h/2$. Fixing a basepoint $o \in X$, for $\xi \in \partial X$, denote by $B_{\xi}$ the Busemann function at $\xi$ such that $B_{\xi}(o) = 0$. then for $\lambda \in \mathbb{C}$ the function $e^{(i\lambda - \rho)B_{\xi}}$ is an eigenfunction of the Laplace-Beltrami operator with eigenvalue $-(\lambda^2 + \rho^2)$. For a function $f$ on $X$, we define the Fourier transform of $f$ by $$\tilde{f}(\lambda, \xi) := \int_X f(x) e^{(-i\lambda - \rho)B_{\xi}(x)} dvol(x)$$ for all $\lambda \in \mathbb{C}, \xi \in \partial X$ for which the integral converges. We prove a Fourier inversion formula $$f(x) = C_0 \int_{0}^{\infty} \int_{\partial X}
\tilde{f}(\lambda, \xi) e^{(i\lambda - \rho)B_{\xi}(x)} d\lambda_o(\xi) |c(\lambda)|^{-2} d\lambda$$ for $f \in C^{\infty}_c(X)$, where $c$ is a certain function on $\mathbb{R} - \{0\}$, $\lambda_o$ is the visibility measure on $\partial X$ with respect to the basepoint $o \in X$ and $C_0 > 0$ is a constant. We also prove a Plancherel theorem, and a version of the Kunze-Stein phenomenon.Minimal Codimension One Foliation of a Symmetric Space by Damek-Ricci Spaces
http://publications.mfo.de/handle/mfo/1416
Minimal Codimension One Foliation of a Symmetric Space by Damek-Ricci Spaces
Knieper, Gerhard; Parker, John R.; Peyerimhoff, Norbert
In this article we consider solvable hypersurfaces of the form $N \exp(\mathbb{R} H)$ with induced metrics in the symmetric space $M = SL(3,\mathbb{C})/SU(3)$, where $H$ a suitable unit length vector in the subgroup $A$ of the Iwasawa decomposition $SL(3,\mathbb{C}) = NAK$. Since $M$ is rank $2$, $A$ is $2$-dimensional and we can parametrize these hypersurfaces via an angle $\alpha \in [0,\pi/2]$ determining the direction of $H$. We show that one of the hypersurfaces (corresponding to $\alpha = 0$) is minimally embedded and isometric to the non-symmetric $7$-dimensional Damek-Ricci space. We also provide an explicit formula for the
Ricci curvature of these hypersurfaces and show that all hypersurfaces for $\alpha \in (0,\frac{\pi}{2}]$ admit planes of both negative and positive sectional curvature. Moreover, the symmetric space $M$ admits a minimal foliation with all leaves isometric to the non-symmetric $7$-dimensional Damek-Ricci space.
Tue, 07 May 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14162019-05-07T00:00:00ZKnieper, GerhardParker, John R.Peyerimhoff, NorbertIn this article we consider solvable hypersurfaces of the form $N \exp(\mathbb{R} H)$ with induced metrics in the symmetric space $M = SL(3,\mathbb{C})/SU(3)$, where $H$ a suitable unit length vector in the subgroup $A$ of the Iwasawa decomposition $SL(3,\mathbb{C}) = NAK$. Since $M$ is rank $2$, $A$ is $2$-dimensional and we can parametrize these hypersurfaces via an angle $\alpha \in [0,\pi/2]$ determining the direction of $H$. We show that one of the hypersurfaces (corresponding to $\alpha = 0$) is minimally embedded and isometric to the non-symmetric $7$-dimensional Damek-Ricci space. We also provide an explicit formula for the
Ricci curvature of these hypersurfaces and show that all hypersurfaces for $\alpha \in (0,\frac{\pi}{2}]$ admit planes of both negative and positive sectional curvature. Moreover, the symmetric space $M$ admits a minimal foliation with all leaves isometric to the non-symmetric $7$-dimensional Damek-Ricci space.Algebra, matrices, and computers
http://publications.mfo.de/handle/mfo/1415
Algebra, matrices, and computers
Detinko, Alla; Flannery, Dane; Hulpke, Alexander
What part does algebra play in representing the real
world abstractly? How can algebra be used to solve
hard mathematical problems with the aid of modern
computing technology? We provide answers to these
questions that rely on the theory of matrix groups
and new methods for handling matrix groups in a
computer.
Fri, 03 May 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14152019-05-03T00:00:00ZDetinko, AllaFlannery, DaneHulpke, AlexanderWhat part does algebra play in representing the real
world abstractly? How can algebra be used to solve
hard mathematical problems with the aid of modern
computing technology? We provide answers to these
questions that rely on the theory of matrix groups
and new methods for handling matrix groups in a
computer.Positive Scalar Curvature and Applications
http://publications.mfo.de/handle/mfo/1414
Positive Scalar Curvature and Applications
Rosenberg, Jonathan; Wraith, David
We introduce the idea of curvature, including how it
developed historically, and focus on the scalar curvature
of a manifold. A major current research topic
involves understanding positive scalar curvature. We
discuss why this is interesting and how it relates to
general relativity.
Thu, 25 Apr 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14142019-04-25T00:00:00ZRosenberg, JonathanWraith, DavidWe introduce the idea of curvature, including how it
developed historically, and focus on the scalar curvature
of a manifold. A major current research topic
involves understanding positive scalar curvature. We
discuss why this is interesting and how it relates to
general relativity.Diophantine equations and why they are hard
http://publications.mfo.de/handle/mfo/1413
Diophantine equations and why they are hard
Pasten, Hector
Diophantine equations are polynomial equations whose
solutions are required to be integer numbers. They
have captured the attention of mathematicians during
millennia and are at the center of much of contemporary
research. Some Diophantine equations are easy,
while some others are truly difficult. After some time
spent with these equations, it might seem that no
matter what powerful methods we learn or develop,
there will always be a Diophantine equation immune
to them, which requires a new trick, a better idea, or
a refined technique. In this snapshot we explain why.
Wed, 24 Apr 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14132019-04-24T00:00:00ZPasten, HectorDiophantine equations are polynomial equations whose
solutions are required to be integer numbers. They
have captured the attention of mathematicians during
millennia and are at the center of much of contemporary
research. Some Diophantine equations are easy,
while some others are truly difficult. After some time
spent with these equations, it might seem that no
matter what powerful methods we learn or develop,
there will always be a Diophantine equation immune
to them, which requires a new trick, a better idea, or
a refined technique. In this snapshot we explain why.On the Lie Algebra Structure of $HH^1(A)$ of a Finite-Dimensional Algebra A
http://publications.mfo.de/handle/mfo/1412
On the Lie Algebra Structure of $HH^1(A)$ of a Finite-Dimensional Algebra A
Linckelmann, Markus; Rubio y Degrassi, Lleonard
Let $A$ be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext-quiver of $A$ is a simple directed graph, then $HH^1(A)$ is a solvable Lie algebra. The second main result shows that if the Ext-quiver of $A$ has no loops and at most two parallel arrows in any direction, and if $HH^1(A)$ is a simple Lie algebra, then char(k) is not equal to $2$ and $HH^1(A)\cong$ $sl_2(k)$. The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.
Wed, 17 Apr 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14122019-04-17T00:00:00ZLinckelmann, MarkusRubio y Degrassi, LleonardLet $A$ be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext-quiver of $A$ is a simple directed graph, then $HH^1(A)$ is a solvable Lie algebra. The second main result shows that if the Ext-quiver of $A$ has no loops and at most two parallel arrows in any direction, and if $HH^1(A)$ is a simple Lie algebra, then char(k) is not equal to $2$ and $HH^1(A)\cong$ $sl_2(k)$. The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.The First Hochschild Cohomology as a Lie Algebra
http://publications.mfo.de/handle/mfo/1411
The First Hochschild Cohomology as a Lie Algebra
Rubio y Degrassi, Lleonard; Schroll, Sibylle; Solotar, Andrea
In this paper we study sufficient conditions for the solvability of the first Hochschild cohomology of a finite dimensional algebra as a Lie algebra in terms of its Ext-quiver in arbitrary characteristic. In particular, we show that if the quiver has no parallel arrows and no loops then the first Hochschild cohomology is solvable. For quivers containing loops, we determine easily verifiable sufficient conditions for the solvability of the first Hochschild cohomology. We apply these criteria to show the solvabilty of the first Hochschild cohomology space for large families of algebras, namely, several families of self-injective tame algebras including all tame blocks of finite groups and some wild algebras including quantum complete intersections.
Tue, 16 Apr 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14112019-04-16T00:00:00ZRubio y Degrassi, LleonardSchroll, SibylleSolotar, AndreaIn this paper we study sufficient conditions for the solvability of the first Hochschild cohomology of a finite dimensional algebra as a Lie algebra in terms of its Ext-quiver in arbitrary characteristic. In particular, we show that if the quiver has no parallel arrows and no loops then the first Hochschild cohomology is solvable. For quivers containing loops, we determine easily verifiable sufficient conditions for the solvability of the first Hochschild cohomology. We apply these criteria to show the solvabilty of the first Hochschild cohomology space for large families of algebras, namely, several families of self-injective tame algebras including all tame blocks of finite groups and some wild algebras including quantum complete intersections.On radial basis functions
http://publications.mfo.de/handle/mfo/1410
On radial basis functions
Buhmann, Martin; Jäger, Janin
Many sciences and other areas of research and applications
from engineering to economics require the approximation
of functions that depend on many variables.
This can be for a variety of reasons. Sometimes
we have a discrete set of data points and we
want to find an approximating function that completes
this data; another possibility is that precise
functions are either not known or it would take too
long to compute them explicitly. In this snapshot
we want to introduce a particular method of approximation
which uses functions called radial basis functions.
This method is particularly useful when approximating
functions that depend on very many variables.
We describe the basic approach to approximation
with radial basis functions, including their computation,
give several examples of such functions and
show some applications.
Wed, 13 Mar 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14102019-03-13T00:00:00ZBuhmann, MartinJäger, JaninMany sciences and other areas of research and applications
from engineering to economics require the approximation
of functions that depend on many variables.
This can be for a variety of reasons. Sometimes
we have a discrete set of data points and we
want to find an approximating function that completes
this data; another possibility is that precise
functions are either not known or it would take too
long to compute them explicitly. In this snapshot
we want to introduce a particular method of approximation
which uses functions called radial basis functions.
This method is particularly useful when approximating
functions that depend on very many variables.
We describe the basic approach to approximation
with radial basis functions, including their computation,
give several examples of such functions and
show some applications.