Oberwolfach Publications
http://publications.mfo.de:80
The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Thu, 01 Jun 2023 05:03:11 GMT2023-06-01T05:03:11ZOberwolfach Publicationshttp://publications.mfo.de/themes/Mirage2/images/apple-touch-icon.png
http://publications.mfo.de:80
Computer Algebra with GAP
http://publications.mfo.de/handle/mfo/4023
Computer Algebra with GAP
Piterman, Kevin I.; Vendramin, Leandro
This monograph includes the following topics: a basic introduction to the language, basic arithmetic, permutations, matrices, polynomial rings, finite fields, finite and finitely presented groups, small groups, group representations and character theory, and simple groups. Advanced topics include testing several open conjectures and theorems. In addition, each chapter ends with an extensive list of problems. We hope the reader will find some problems challenging and exciting as they are based on outstanding research papers. Selected solutions can be found at the end of the book.
Thu, 13 Apr 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40232023-04-13T00:00:00ZPiterman, Kevin I.Vendramin, LeandroThis monograph includes the following topics: a basic introduction to the language, basic arithmetic, permutations, matrices, polynomial rings, finite fields, finite and finitely presented groups, small groups, group representations and character theory, and simple groups. Advanced topics include testing several open conjectures and theorems. In addition, each chapter ends with an extensive list of problems. We hope the reader will find some problems challenging and exciting as they are based on outstanding research papers. Selected solutions can be found at the end of the book.Wave Phenomena
http://publications.mfo.de/handle/mfo/4025
Wave Phenomena; Mathematical Analysis and Numerical Approximation
Dörfler, Willy; Hochbruck, Marlis; Köhler, Jonas; Rieder, Andreas; Schnaubelt, Roland; Wieners, Christian
This book presents the notes from the seminar on wave phenomena given in 2019 at the Mathematical Research Center in Oberwolfach.
The research on wave-type problems is a fascinating and emerging field in mathematical research with many challenging applications in sciences and engineering. Profound investigations on waves require a strong interaction of several mathematical disciplines including functional analysis, partial differential equations, mathematical modeling, mathematical physics, numerical analysis, and scientific computing.
The goal of this book is to present a comprehensive introduction to the research on wave phenomena. Starting with basic models for acoustic, elastic, and electro-magnetic waves, topics such as the existence of solutions for linear and some nonlinear material laws, efficient discretizations and solution methods in space and time, and the application to inverse parameter identification problems are covered. The aim of this book is to intertwine analysis and numerical mathematics for wave-type problems promoting thus cooperative research projects in this field.
Thu, 30 Mar 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40252023-03-30T00:00:00ZDörfler, WillyHochbruck, MarlisKöhler, JonasRieder, AndreasSchnaubelt, RolandWieners, ChristianThis book presents the notes from the seminar on wave phenomena given in 2019 at the Mathematical Research Center in Oberwolfach.
The research on wave-type problems is a fascinating and emerging field in mathematical research with many challenging applications in sciences and engineering. Profound investigations on waves require a strong interaction of several mathematical disciplines including functional analysis, partial differential equations, mathematical modeling, mathematical physics, numerical analysis, and scientific computing.
The goal of this book is to present a comprehensive introduction to the research on wave phenomena. Starting with basic models for acoustic, elastic, and electro-magnetic waves, topics such as the existence of solutions for linear and some nonlinear material laws, efficient discretizations and solution methods in space and time, and the application to inverse parameter identification problems are covered. The aim of this book is to intertwine analysis and numerical mathematics for wave-type problems promoting thus cooperative research projects in this field.Real Enumerative Invariants Relative to the Anti-Canonical Divisor and their Refinement
http://publications.mfo.de/handle/mfo/4022
Real Enumerative Invariants Relative to the Anti-Canonical Divisor and their Refinement
Itenberg, Ilia; Shustin, Eugenii
We introduce new invariants of the projective plane (and, more generally, of
certain toric surfaces) that arise from the appropriate enumeration of real
elliptic curves. These invariants admit a refinement (according to the quantum
index) similar to the one introduced by Grigory Mikhalkin in the rational case.
We also construct tropical counterparts of the refined elliptic invariants
under consideration and establish a tropical algorithm allowing one to compute,
$via$ a suitable version of the correspondence theorem, the above
invariants.
Fri, 24 Mar 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40222023-03-24T00:00:00ZItenberg, IliaShustin, EugeniiWe introduce new invariants of the projective plane (and, more generally, of
certain toric surfaces) that arise from the appropriate enumeration of real
elliptic curves. These invariants admit a refinement (according to the quantum
index) similar to the one introduced by Grigory Mikhalkin in the rational case.
We also construct tropical counterparts of the refined elliptic invariants
under consideration and establish a tropical algorithm allowing one to compute,
$via$ a suitable version of the correspondence theorem, the above
invariants.Flag-Accurate Arrangements
http://publications.mfo.de/handle/mfo/4012
Flag-Accurate Arrangements
Mücksch, Paul; Röhrle, Gerhard; Tran, Tan Nhat
In [MR21], the first two authors introduced the notion of an accurate arrangement, a particular notion of freeness. In this paper, we consider a special subclass, where the property of accuracy stems from a flag of flats in the intersection lattice of the underlying arrangement. Members of this family are called flag-accurate. One relevance of this new notion is that it entails divisional freeness. There are a number of important natural classes which are flag-accurate, the most prominent one among them is the one consisting of Coxeter arrangements. This warrants a systematic study which is put forward in the present paper. More specifically, let $\mathscr A$ be a free arrangement of rank $\ell$. Suppose that for every $1\leq d \leq \ell$, the first $d$ exponents of $\mathscr A$ - when listed in increasing order - are realized as the exponents of a free restriction of $\mathscr A$ to some intersection of reflecting hyperplanes of $\mathscr A$ of dimension $d$. Following [MR21], we call such an arrangement $\mathscr A$ with this natural property accurate. If in addition the flats involved can be chosen to form a flag, we call $\mathscr A$ flag-accurate. We investigate flag-accuracy among reflection arrangements, extended Shi and extended Catalan arrangements, and further for various families of graphic and digraphic arrangements. We pursue these both from theoretical and computational perspectives. Along the way we present examples of accurate arrangements that are not flag-accurate. The main result of [MR21] shows that MAT-free arrangements are accurate. We provide strong evidence for the conjecture that MAT-freeness actually entails flag-accuracy.
Mon, 13 Feb 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40122023-02-13T00:00:00ZMücksch, PaulRöhrle, GerhardTran, Tan NhatIn [MR21], the first two authors introduced the notion of an accurate arrangement, a particular notion of freeness. In this paper, we consider a special subclass, where the property of accuracy stems from a flag of flats in the intersection lattice of the underlying arrangement. Members of this family are called flag-accurate. One relevance of this new notion is that it entails divisional freeness. There are a number of important natural classes which are flag-accurate, the most prominent one among them is the one consisting of Coxeter arrangements. This warrants a systematic study which is put forward in the present paper. More specifically, let $\mathscr A$ be a free arrangement of rank $\ell$. Suppose that for every $1\leq d \leq \ell$, the first $d$ exponents of $\mathscr A$ - when listed in increasing order - are realized as the exponents of a free restriction of $\mathscr A$ to some intersection of reflecting hyperplanes of $\mathscr A$ of dimension $d$. Following [MR21], we call such an arrangement $\mathscr A$ with this natural property accurate. If in addition the flats involved can be chosen to form a flag, we call $\mathscr A$ flag-accurate. We investigate flag-accuracy among reflection arrangements, extended Shi and extended Catalan arrangements, and further for various families of graphic and digraphic arrangements. We pursue these both from theoretical and computational perspectives. Along the way we present examples of accurate arrangements that are not flag-accurate. The main result of [MR21] shows that MAT-free arrangements are accurate. We provide strong evidence for the conjecture that MAT-freeness actually entails flag-accuracy.Geometrie
http://publications.mfo.de/handle/mfo/4010
Geometrie
The workshop Geometrie, organized by Aaron Naber (Evanston),
André Neves (Chicago) and Burkhard Wilking (Münster) was well attended with over 42 participants (35 in person and 7 online) with broad geographic representation from all continents, and held in a very active atmosphere. During the meeting, various interesting topics in geometry were discussed, such as geometric flows, Einstein manifolds and spaces with sectional curvature bounds.
Sat, 01 Jan 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40102022-01-01T00:00:00ZThe workshop Geometrie, organized by Aaron Naber (Evanston),
André Neves (Chicago) and Burkhard Wilking (Münster) was well attended with over 42 participants (35 in person and 7 online) with broad geographic representation from all continents, and held in a very active atmosphere. During the meeting, various interesting topics in geometry were discussed, such as geometric flows, Einstein manifolds and spaces with sectional curvature bounds.Hilbert Complexes: Analysis, Applications, and Discretizations
http://publications.mfo.de/handle/mfo/4008
Hilbert Complexes: Analysis, Applications, and Discretizations
In this workshop 70 (43 at MFO, 27 online) leading mathematicians from Europe, United States, China, and Australia
met at the MFO to discuss and present new developments
in the mathematical and numerical analysis including discretizations
of Hilbert complexes related to systems of partial differential equations,
in particular the well known de Rham complex
and the complexes of elasticity and the biharmonic equations.
The report at hand offers the extended abstracts of their talks.
Sat, 01 Jan 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40082022-01-01T00:00:00ZIn this workshop 70 (43 at MFO, 27 online) leading mathematicians from Europe, United States, China, and Australia
met at the MFO to discuss and present new developments
in the mathematical and numerical analysis including discretizations
of Hilbert complexes related to systems of partial differential equations,
in particular the well known de Rham complex
and the complexes of elasticity and the biharmonic equations.
The report at hand offers the extended abstracts of their talks.Convolution in Dual Cesàro Sequence Spaces
http://publications.mfo.de/handle/mfo/4002
Convolution in Dual Cesàro Sequence Spaces
Curbera, Guillermo P.; Ricker, Werner J.
We investigate convolution operators in the sequence spaces $d_p$, for 1 $\leqslant p<\infty$. These spaces, for $p$ > 1, arise as dual spaces of the Cesàro sequence spaces $ces_p$ thoroughly investigated by G. Bennett. A detailed study is also made of the algebra of those sequences which convolve $d_p$ into $d_p$. It turns out that such multiplier spaces exhibit features which are very different to the classical multiplier spaces of $l^{p}$.
Fri, 16 Dec 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40022022-12-16T00:00:00ZCurbera, Guillermo P.Ricker, Werner J.We investigate convolution operators in the sequence spaces $d_p$, for 1 $\leqslant p<\infty$. These spaces, for $p$ > 1, arise as dual spaces of the Cesàro sequence spaces $ces_p$ thoroughly investigated by G. Bennett. A detailed study is also made of the algebra of those sequences which convolve $d_p$ into $d_p$. It turns out that such multiplier spaces exhibit features which are very different to the classical multiplier spaces of $l^{p}$.Hutchinson's Intervals and Entire Functions from the Laguerre-Pólya Class
http://publications.mfo.de/handle/mfo/4001
Hutchinson's Intervals and Entire Functions from the Laguerre-Pólya Class
Nguyen, Thu Hien; Vishnyakova, Anna
We find the intervals $[\alpha, \beta (\alpha)]$ such that if a univariate real polynomial or entire function $f(z) = a_0 + a_1 z + a_2 z^2 + \cdots $ with positive coefficients satisfy the conditions $ \frac{a_{k-1}^2}{a_{k-2}a_{k}} \in [\alpha, \beta(\alpha)]$ for all $k \geq 2,$ then $f$ belongs to the Laguerre-Pólya class. For instance, from J.I. Hutchinson's theorem, one can observe that $f$ belongs to the Laguerre-Pólya class (has only real zeros) when $q_k(f) \in [4, + \infty).$ We are interested in finding those intervals which are not subsets of $[4, + \infty).$
Mon, 12 Dec 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40012022-12-12T00:00:00ZNguyen, Thu HienVishnyakova, AnnaWe find the intervals $[\alpha, \beta (\alpha)]$ such that if a univariate real polynomial or entire function $f(z) = a_0 + a_1 z + a_2 z^2 + \cdots $ with positive coefficients satisfy the conditions $ \frac{a_{k-1}^2}{a_{k-2}a_{k}} \in [\alpha, \beta(\alpha)]$ for all $k \geq 2,$ then $f$ belongs to the Laguerre-Pólya class. For instance, from J.I. Hutchinson's theorem, one can observe that $f$ belongs to the Laguerre-Pólya class (has only real zeros) when $q_k(f) \in [4, + \infty).$ We are interested in finding those intervals which are not subsets of $[4, + \infty).$Universality: Random Matrices, Random Geometry and SPDEs
http://publications.mfo.de/handle/mfo/3999
Universality: Random Matrices, Random Geometry and SPDEs
The postulate that large random systems can be described by limiting
objects whose characteristic do not depend on the exact details of the
models one started from is central in probability theory, under the
name of universality. This workshop was aimed at uncovering the latest
developments of this concept in the various topics where it is
relevant, namely statistical physics, stochastic partial differential
equations, random geometries and random matrices. It was in particular
the occasion to feature some important recently introduced universal objects like the
stochastic quantization of the Yang-Mills measure in dimensions 2 and
3, the KPZ fixed point, Liouville quantum gravity metrics and other
objects connected to the Gaussian free field.
Sat, 01 Jan 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39992022-01-01T00:00:00ZThe postulate that large random systems can be described by limiting
objects whose characteristic do not depend on the exact details of the
models one started from is central in probability theory, under the
name of universality. This workshop was aimed at uncovering the latest
developments of this concept in the various topics where it is
relevant, namely statistical physics, stochastic partial differential
equations, random geometries and random matrices. It was in particular
the occasion to feature some important recently introduced universal objects like the
stochastic quantization of the Yang-Mills measure in dimensions 2 and
3, the KPZ fixed point, Liouville quantum gravity metrics and other
objects connected to the Gaussian free field.Closed geodesics on surfaces
http://publications.mfo.de/handle/mfo/3998
Closed geodesics on surfaces
Dozier, Benjamin
We consider surfaces of three types: the sphere, the torus, and many-holed tori. These surfaces naturally admit geometries of positive, zero, and negative curvature, respectively. It is interesting to study straight line paths, known as geodesics, in these geometries. We discuss the issue of counting closed geodesics; this is particularly rich for hyperbolic (negatively curved) surfaces.
Thu, 08 Dec 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39982022-12-08T00:00:00ZDozier, BenjaminWe consider surfaces of three types: the sphere, the torus, and many-holed tori. These surfaces naturally admit geometries of positive, zero, and negative curvature, respectively. It is interesting to study straight line paths, known as geodesics, in these geometries. We discuss the issue of counting closed geodesics; this is particularly rich for hyperbolic (negatively curved) surfaces.