1 - Oberwolfach Preprints (OWP)
http://publications.mfo.de/handle/mfo/19
The Oberwolfach Preprints (OWP) mainly contain research results related to a longer stay in Oberwolfach. In particular, this concerns the Research in Pairs program and the Oberwolfach Leibniz Fellows, but this can also include an Oberwolfach Lecture, for example.Wed, 21 Aug 2019 03:15:42 GMT2019-08-21T03:15:42ZA Cheeger Type Inequality in Finite Cayley Sum Graphs
http://publications.mfo.de/handle/mfo/2512
A Cheeger Type Inequality in Finite Cayley Sum Graphs
Biswas, Arindam; Saha, Jyoti Prakash
Let $G$ be a finite group and $S$ be a symmetric generating set of $G$ with $|S| = d$. We show that if the undirected Cayley sum graph $C_{\Sigma}(G,S)$ is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from $-1$. We also establish an explicit lower bound for the spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval $\left(-1+\frac{h(G)^{4}}{\eta}, 1-\frac{h(G)^{2}}{2d^{2}}\right]$, where $h(G)$ denotes the (vertex) Cheeger constant of the $d$-regular graph $C_{\Sigma}(G,S)$ and $\eta = 2^{9}d^{8}$. Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the non-bipartite Cayley graph $C(G,S)$.
Wed, 31 Jul 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25122019-07-31T00:00:00ZBiswas, ArindamSaha, Jyoti PrakashLet $G$ be a finite group and $S$ be a symmetric generating set of $G$ with $|S| = d$. We show that if the undirected Cayley sum graph $C_{\Sigma}(G,S)$ is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from $-1$. We also establish an explicit lower bound for the spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval $\left(-1+\frac{h(G)^{4}}{\eta}, 1-\frac{h(G)^{2}}{2d^{2}}\right]$, where $h(G)$ denotes the (vertex) Cheeger constant of the $d$-regular graph $C_{\Sigma}(G,S)$ and $\eta = 2^{9}d^{8}$. Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the non-bipartite Cayley graph $C(G,S)$.On a Cheeger Type Inequality in Cayley Graphs of Finite Groups
http://publications.mfo.de/handle/mfo/2511
On a Cheeger Type Inequality in Cayley Graphs of Finite Groups
Biswas, Arindam
Let $G$ be a finite group. It was remarked by Breuillard-Green-Guralnick-Tao that if the Cayley graph $C(G,S)$ is an expander graph and is non-bipartite then the spectrum of the adjacency operator $T$ is bounded away from $-1$. In this article we are interested in explicit bounds for the spectrum of these graphs. Specifically, we show that the non-trivial spectrum of the adjacency operator lies in the interval $\left[-1+\frac{h(\mathbb{G})^{4}}{\gamma}, 1-\frac{h(\mathbb{G})^{2}}{2d^{2}}\right]$, where $h(\mathbb{G})$ denotes the (vertex) Cheeger constant of the $d$ regular graph $C(G,S)$ with respect to a symmetric set $S$ of generators and $\gamma = 2^{9}d^{6}(d+1)^{2}$.
Mon, 22 Jul 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25112019-07-22T00:00:00ZBiswas, ArindamLet $G$ be a finite group. It was remarked by Breuillard-Green-Guralnick-Tao that if the Cayley graph $C(G,S)$ is an expander graph and is non-bipartite then the spectrum of the adjacency operator $T$ is bounded away from $-1$. In this article we are interested in explicit bounds for the spectrum of these graphs. Specifically, we show that the non-trivial spectrum of the adjacency operator lies in the interval $\left[-1+\frac{h(\mathbb{G})^{4}}{\gamma}, 1-\frac{h(\mathbb{G})^{2}}{2d^{2}}\right]$, where $h(\mathbb{G})$ denotes the (vertex) Cheeger constant of the $d$ regular graph $C(G,S)$ with respect to a symmetric set $S$ of generators and $\gamma = 2^{9}d^{6}(d+1)^{2}$.On Co-Minimal Pairs in Abelian Groups
http://publications.mfo.de/handle/mfo/2509
On Co-Minimal Pairs in Abelian Groups
Biswas, Arindam; Saha, Jyoti Prakash
A pair of non-empty subsets $(W,W')$ in an abelian group $G$ is a complement pair if $W+W'=G$. $W'$ is said to be minimal to $W$ if $W+(W'\setminus \{w'\}) \neq G, \forall \,w'\in W'$. In general, given an arbitrary subset in a group, the existence of minimal complement(s) depends on its structure. The dual problem asks that given such a set, if it is a minimal complement to some subset. We study tightness property of complement pairs $(W,W')$ such that both $W$ and $W'$ are minimal to each other. These are termed co-minimal pairs and we show that any non-empty finite set in an arbitrary free abelian group belongs to some co-minimal pair. We also construct infinite sets forming co-minimal pairs. Finally, we remark that a result of Kwon on the existence of minimal self-complements in $\mathbb{Z}$, also holds in any abelian group.
Tue, 09 Jul 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25092019-07-09T00:00:00ZBiswas, ArindamSaha, Jyoti PrakashA pair of non-empty subsets $(W,W')$ in an abelian group $G$ is a complement pair if $W+W'=G$. $W'$ is said to be minimal to $W$ if $W+(W'\setminus \{w'\}) \neq G, \forall \,w'\in W'$. In general, given an arbitrary subset in a group, the existence of minimal complement(s) depends on its structure. The dual problem asks that given such a set, if it is a minimal complement to some subset. We study tightness property of complement pairs $(W,W')$ such that both $W$ and $W'$ are minimal to each other. These are termed co-minimal pairs and we show that any non-empty finite set in an arbitrary free abelian group belongs to some co-minimal pair. We also construct infinite sets forming co-minimal pairs. Finally, we remark that a result of Kwon on the existence of minimal self-complements in $\mathbb{Z}$, also holds in any abelian group.A Quantitative Analysis of the “Lion-Man” Game
http://publications.mfo.de/handle/mfo/2508
A Quantitative Analysis of the “Lion-Man” Game
Kohlenbach, Ulrich; López-Acedo, Genaro; Nicolae, Adriana
In this paper we analyze, based on an interplay between ideas and techniques from logic and geometric analysis, a pursuit-evasion game. More precisely, we focus on a discrete lion and man game with an $\varepsilon$-capture criterion. We prove that in uniformly convex bounded domains the lion always wins and, using ideas stemming from proof mining, we extract a uniform rate of
convergence for the successive distances between the lion and the man. As a byproduct of our analysis, we study the relation among different convexity properties in the setting of geodesic spaces.
Mon, 08 Jul 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25082019-07-08T00:00:00ZKohlenbach, UlrichLópez-Acedo, GenaroNicolae, AdrianaIn this paper we analyze, based on an interplay between ideas and techniques from logic and geometric analysis, a pursuit-evasion game. More precisely, we focus on a discrete lion and man game with an $\varepsilon$-capture criterion. We prove that in uniformly convex bounded domains the lion always wins and, using ideas stemming from proof mining, we extract a uniform rate of
convergence for the successive distances between the lion and the man. As a byproduct of our analysis, we study the relation among different convexity properties in the setting of geodesic spaces.On Residuals of Finite Groups
http://publications.mfo.de/handle/mfo/1423
On Residuals of Finite Groups
Aivazidis, Stefanos; Müller, Thomas
A theorem of Dolfi, Herzog, Kaplan, and Lev [DHKL07, Thm. C] asserts that in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order, and that the inequality is sharp. Inspired by this result and some of the arguments in [DHKL07], we establish the following generalisation: if ${\mathfrak{X}}$ is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and $\overline{\mathfrak{X}}$ is the extension-closure of $\mathfrak{X}$, then there exists an (optimal) constant $\gamma$ depending only on $\mathfrak{X}$ such that, for all non-trivial finite groups G with trivial $\mathfrak{X}$-radical, ${\vert}G{\vert}^{\overline{\mathfrak{X}}} > {\vert}G{\vert}^\gamma$, where $G^{\overline{\mathfrak{X}}}$ is the ${\overline{\mathfrak{X}}}$-residual of $G$. When ${\mathfrak{X}}={\mathfrak{N}}$, the class of finite nilpotent groups, it follows that $\overline{\mathfrak{X}} = \mathfrak{S}$, the class of finite soluble groups, thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J. G. Thompson's classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations X of full characteristic such that $\mathfrak{S}
\subset \overline{\mathfrak{X}} \subset \mathfrak{E}$, thus providing applications of our main result beyond the reach of [DHKL07, Thm. C]
Tue, 28 May 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14232019-05-28T00:00:00ZAivazidis, StefanosMüller, ThomasA theorem of Dolfi, Herzog, Kaplan, and Lev [DHKL07, Thm. C] asserts that in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order, and that the inequality is sharp. Inspired by this result and some of the arguments in [DHKL07], we establish the following generalisation: if ${\mathfrak{X}}$ is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and $\overline{\mathfrak{X}}$ is the extension-closure of $\mathfrak{X}$, then there exists an (optimal) constant $\gamma$ depending only on $\mathfrak{X}$ such that, for all non-trivial finite groups G with trivial $\mathfrak{X}$-radical, ${\vert}G{\vert}^{\overline{\mathfrak{X}}} > {\vert}G{\vert}^\gamma$, where $G^{\overline{\mathfrak{X}}}$ is the ${\overline{\mathfrak{X}}}$-residual of $G$. When ${\mathfrak{X}}={\mathfrak{N}}$, the class of finite nilpotent groups, it follows that $\overline{\mathfrak{X}} = \mathfrak{S}$, the class of finite soluble groups, thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J. G. Thompson's classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations X of full characteristic such that $\mathfrak{S}
\subset \overline{\mathfrak{X}} \subset \mathfrak{E}$, thus providing applications of our main result beyond the reach of [DHKL07, Thm. C]Congruences Associated with Families of Nilpotent Subgroups and a Theorem of Hirsch
http://publications.mfo.de/handle/mfo/1422
Congruences Associated with Families of Nilpotent Subgroups and a Theorem of Hirsch
Aivazidis, Stefanos; Müller, Thomas
Our main result associates a family of congruences with each suitable system of nilpotent subgroups of a finite group. Using this result, we complete and correct the proof of a theorem of Hirsch concerning the class number of a finite group of odd order.
Mon, 27 May 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14222019-05-27T00:00:00ZAivazidis, StefanosMüller, ThomasOur main result associates a family of congruences with each suitable system of nilpotent subgroups of a finite group. Using this result, we complete and correct the proof of a theorem of Hirsch concerning the class number of a finite group of odd order.Experimenting with Symplectic Hypergeometric Monodromy Groups
http://publications.mfo.de/handle/mfo/1420
Experimenting with Symplectic Hypergeometric Monodromy Groups
Detinko, Alla; Flannery, Dane; Hulpke, Alexander
We present new computational results for symplectic monodromy groups of hypergeometric differential equations. In particular, we compute the arithmetic closure of each group, sometimes justifying arithmeticity. The results are obtained by extending our previous algorithms for Zariski dense groups, based on the strong approximation and congruence subgroup properties.
Wed, 22 May 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14202019-05-22T00:00:00ZDetinko, AllaFlannery, DaneHulpke, AlexanderWe present new computational results for symplectic monodromy groups of hypergeometric differential equations. In particular, we compute the arithmetic closure of each group, sometimes justifying arithmeticity. The results are obtained by extending our previous algorithms for Zariski dense groups, based on the strong approximation and congruence subgroup properties.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.