Oberwolfach Publications
http://publications.mfo.de:8080
The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Wed, 21 Aug 2019 11:12:17 GMT2019-08-21T11:12:17ZOberwolfach Publicationshttp://publications.mfo.de/themes/Mirage2/images/apple-touch-icon.png
http://publications.mfo.de:8080
A 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 $\gamma = 2^{9}d^{6}(d+1)^{2}$.Random permutations
http://publications.mfo.de/handle/mfo/2510
Random permutations
Betz, Volker
100 people leave their hats at the door at a party and
pick up a completely random hat when they leave.
How likely is it that at least one of them will get
back their own hat? If the hats carry name tags,
how difficult is it to arrange for all hats to be returned
to their owner? These classical questions of
probability theory can be answered relatively easily.
But if a geometric component is added, answering
the same questions immediately becomes very hard,
and little is known about them. We present some
of the open questions and give an overview of what
current research can say about them.
Fri, 12 Jul 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25102019-07-12T00:00:00ZBetz, Volker100 people leave their hats at the door at a party and
pick up a completely random hat when they leave.
How likely is it that at least one of them will get
back their own hat? If the hats carry name tags,
how difficult is it to arrange for all hats to be returned
to their owner? These classical questions of
probability theory can be answered relatively easily.
But if a geometric component is added, answering
the same questions immediately becomes very hard,
and little is known about them. We present some
of the open questions and give an overview of what
current research can say about them.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.Mathematical Instruments between Material Artifacts and Ideal Machines: Their Scientific and Social Role before 1950
http://publications.mfo.de/handle/mfo/2507
Mathematical Instruments between Material Artifacts and Ideal Machines: Their Scientific and Social Role before 1950
Since 1950, mathematicians have become increasingly familiar with the digital computer in their professional practice. Previously, however, many other instruments, now mostly forgotten, were commonly used to compute numerical solutions, generate geometrical objects, investigate mathematical problems, derive new results, and apply mathematics in a variety of scientific contexts. The problem of characterizing the mathematical objects that can be constructed with a given set of instruments frequently prompted deep theoretical investigations, from the Euclidean geometry of constructions with straightedge and compass, to Shannon’s theorem which, in 1941, stated that the functions constructible with a differential analyzer are exactly the solutions of algebraic differential equations. Beyond these mathematical considerations, instruments should also be viewed as social objects of a given time period and cultural tradition that can amalgamate the perspectives of the inventor, the maker, the user, and the collector; in this sense, mathematical instruments are an important part of the mathematical cultural heritage and are thus widely used in many science museums to demonstrate the cultural value of mathematics to the public. This workshop brought together mathematicians, historians, philosophers, collection curators, and scholars of education to address the various approaches to the history of mathematical instruments and compare the definition and role of these instruments over time, with the following fundamental questions in mind – What is mathematical in a mathematical instrument? What kind of mathematics is involved? What does it mean to embody mathematics in a material artefact, and how do non-mathematicians engage with this kind of embodied mathematics?
Sun, 01 Jan 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25072017-01-01T00:00:00ZSince 1950, mathematicians have become increasingly familiar with the digital computer in their professional practice. Previously, however, many other instruments, now mostly forgotten, were commonly used to compute numerical solutions, generate geometrical objects, investigate mathematical problems, derive new results, and apply mathematics in a variety of scientific contexts. The problem of characterizing the mathematical objects that can be constructed with a given set of instruments frequently prompted deep theoretical investigations, from the Euclidean geometry of constructions with straightedge and compass, to Shannon’s theorem which, in 1941, stated that the functions constructible with a differential analyzer are exactly the solutions of algebraic differential equations. Beyond these mathematical considerations, instruments should also be viewed as social objects of a given time period and cultural tradition that can amalgamate the perspectives of the inventor, the maker, the user, and the collector; in this sense, mathematical instruments are an important part of the mathematical cultural heritage and are thus widely used in many science museums to demonstrate the cultural value of mathematics to the public. This workshop brought together mathematicians, historians, philosophers, collection curators, and scholars of education to address the various approaches to the history of mathematical instruments and compare the definition and role of these instruments over time, with the following fundamental questions in mind – What is mathematical in a mathematical instrument? What kind of mathematics is involved? What does it mean to embody mathematics in a material artefact, and how do non-mathematicians engage with this kind of embodied mathematics?Network Models: Structure and Function
http://publications.mfo.de/handle/mfo/2506
Network Models: Structure and Function
The focus of the meeting was on the mathematical analysis of complex networks, both on how networks emerge through microscopic interaction rules as well as on dynamic processes and optimization problems on networks, including random walks, interacting particle systems and search algorithms. Topics that were addressed included: percolation on graphs and critical regimes for the emergence of a giant component; graph limits and graphons; epidemics, propagation and competition; trees and forests; dynamic random graphs; local versus global algorithms; statistical learning on networks.
Sun, 01 Jan 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25062017-01-01T00:00:00ZThe focus of the meeting was on the mathematical analysis of complex networks, both on how networks emerge through microscopic interaction rules as well as on dynamic processes and optimization problems on networks, including random walks, interacting particle systems and search algorithms. Topics that were addressed included: percolation on graphs and critical regimes for the emergence of a giant component; graph limits and graphons; epidemics, propagation and competition; trees and forests; dynamic random graphs; local versus global algorithms; statistical learning on networks.Classical and Quantum Mechanical Models of Many-Particle Systems
http://publications.mfo.de/handle/mfo/2505
Classical and Quantum Mechanical Models of Many-Particle Systems
This workshop was dedicated to the presentation of recent results in the field of the mathematical study of kinetic theory and its naturalextensions (statistical physics and fluid mechanics). The main models are the Vlasov(-Poisson) equation and the Boltzmann equation, which are obtainedas limits of many-body equations (Newton’s equations in the classical case and Schrödinger’s equation in the quantum case) thanks to the mean-field and Boltzmann-Grad scalings. Numerical aspects and applications to mechanics, physics, engineering and biology were also discussed.
Sun, 01 Jan 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25052017-01-01T00:00:00ZThis workshop was dedicated to the presentation of recent results in the field of the mathematical study of kinetic theory and its naturalextensions (statistical physics and fluid mechanics). The main models are the Vlasov(-Poisson) equation and the Boltzmann equation, which are obtainedas limits of many-body equations (Newton’s equations in the classical case and Schrödinger’s equation in the quantum case) thanks to the mean-field and Boltzmann-Grad scalings. Numerical aspects and applications to mechanics, physics, engineering and biology were also discussed.Reflection Positivity
http://publications.mfo.de/handle/mfo/2504
Reflection Positivity
The main theme of the workshop was reflection positivity and its occurences in various areas of mathematics and physics, such as Representation Theory, Quantum Field Theory, Noncommutative Geometry, Dynamical Systems, Analysis and Statistical Mechanics. Accordingly, the program was intrinsically interdisciplinary and included talks covering different aspects of reflection positivity.
Sun, 01 Jan 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25042017-01-01T00:00:00ZThe main theme of the workshop was reflection positivity and its occurences in various areas of mathematics and physics, such as Representation Theory, Quantum Field Theory, Noncommutative Geometry, Dynamical Systems, Analysis and Statistical Mechanics. Accordingly, the program was intrinsically interdisciplinary and included talks covering different aspects of reflection positivity.Variational Methods for Evolution
http://publications.mfo.de/handle/mfo/2503
Variational Methods for Evolution
Many evolutionary systems, as for example gradient flows or Hamiltonian systems, can be formulated in terms of variational principles or can be approximated using time-incremental minimization. Hence they can be studied using the mathematical techniques of the field of calculus of variations. This viewpoint has led to many discoveries and rapid expansion of the field over the last two decades. Relevant applications arise in mechanics of fluids and solids, in reaction-diffusion systems, in biology, in many-particle models, as well as in emerging uses in data science.
This workshop brought together a broad spectrum of researchers from calculus of variations, partial differential equations, metric geometry, and stochastics, as well as applied and computational scientists to discuss and exchange ideas. It focused on variational tools such as minimizing movement schemes, Gamma convergence, optimal transport, gradient flows, and large-deviation principles for time-continuous Markov processes.
Sun, 01 Jan 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25032017-01-01T00:00:00ZMany evolutionary systems, as for example gradient flows or Hamiltonian systems, can be formulated in terms of variational principles or can be approximated using time-incremental minimization. Hence they can be studied using the mathematical techniques of the field of calculus of variations. This viewpoint has led to many discoveries and rapid expansion of the field over the last two decades. Relevant applications arise in mechanics of fluids and solids, in reaction-diffusion systems, in biology, in many-particle models, as well as in emerging uses in data science.
This workshop brought together a broad spectrum of researchers from calculus of variations, partial differential equations, metric geometry, and stochastics, as well as applied and computational scientists to discuss and exchange ideas. It focused on variational tools such as minimizing movement schemes, Gamma convergence, optimal transport, gradient flows, and large-deviation principles for time-continuous Markov processes.