Oberwolfach Publications
http://publications.mfo.de:80
The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Wed, 29 Mar 2023 19:58:38 GMT2023-03-29T19:58:38ZOberwolfach Publicationshttp://publications.mfo.de/themes/Mirage2/images/apple-touch-icon.png
http://publications.mfo.de:80
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.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 $[4, + \infty).$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.Route planning for bacteria
http://publications.mfo.de/handle/mfo/3997
Route planning for bacteria
Hellmuth, Kathrin; Klingenberg, Christian
Bacteria have been fascinating biologists since their discovery in the late 17th century. By analysing their movements, mathematical models have been developed as a tool to understand their behaviour. However, adapting these models to real situations can be challenging, because the model coefficients cannot be observed directly. In this snapshot, we study this question mathematically and explain how the idea of “route planning” can be used to determine these model coefficients.
Thu, 08 Dec 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39972022-12-08T00:00:00ZHellmuth, KathrinKlingenberg, ChristianBacteria have been fascinating biologists since their discovery in the late 17th century. By analysing their movements, mathematical models have been developed as a tool to understand their behaviour. However, adapting these models to real situations can be challenging, because the model coefficients cannot be observed directly. In this snapshot, we study this question mathematically and explain how the idea of “route planning” can be used to determine these model coefficients.Characterizations of intrinsic volumes on convex bodies and convex functions
http://publications.mfo.de/handle/mfo/3996
Characterizations of intrinsic volumes on convex bodies and convex functions; Charakterisierungen von inneren Volumina auf konvexen Körpern und konvexen Funktionen
Mussnig, Fabian
If we want to express the size of a two-dimensional shape with a number, then we usually think about its area or circumference. But what makes these quantities so special? We give an answer to this question in terms of classical mathematical results. We also take a look at applications and new generalizations to the setting of functions.; Wenn wir die Größe einer zweidimensionalen Form mittels einer Zahl ausdrücken wollen, dann denken wir gewöhnlich an ihren Flächeninhalt oder ihren Umfang. Aber was macht diese Kennzahlen so besonders? Wir beantworten diese Frage anhand klassischer mathematischer Resultate und werfen einen Blick auf Anwendungen und Verallgemeinerungen dieser Theorie.
Thu, 08 Dec 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39962022-12-08T00:00:00ZMussnig, FabianIf we want to express the size of a two-dimensional shape with a number, then we usually think about its area or circumference. But what makes these quantities so special? We give an answer to this question in terms of classical mathematical results. We also take a look at applications and new generalizations to the setting of functions.
Wenn wir die Größe einer zweidimensionalen Form mittels einer Zahl ausdrücken wollen, dann denken wir gewöhnlich an ihren Flächeninhalt oder ihren Umfang. Aber was macht diese Kennzahlen so besonders? Wir beantworten diese Frage anhand klassischer mathematischer Resultate und werfen einen Blick auf Anwendungen und Verallgemeinerungen dieser Theorie.Quasi-Equilibria and Click Times for a Variant of Muller's Ratchet
http://publications.mfo.de/handle/mfo/3993
Quasi-Equilibria and Click Times for a Variant of Muller's Ratchet
González Casanova, Adrian; Smadi, Charline; Wakolbinger, Anton
Consider a population of $N$ individuals, each of them carrying a type in $\mathbb N_0$. The population evolves according to a Moran dynamics with selection and mutation, where an individual of type $k$ has the same selective advantage over all individuals with type $k' > k$, and type $k$ mutates to type
$k+1$ at a constant rate. This model is thus a variation of the classical Muller's ratchet: there the selective advantage is proportional to $k' - k$. For a regime of selection strength and mutation rates which is between the regimes of weak and strong selection/mutation, we obtain the asymptotic rate of the click times of the ratchet (i.e. the times at which the hitherto minimal ('best') type in the population is lost), and reveal the quasi-stationary type frequency profile between clicks. The large population limit of this profile is characterized as the normalized attractor of a "dual" hierarchical multitype logistic system, and also via the distribution of the final minimal
displacement in a branching random walk with one-sided steps. An important role in the proofs is played by a graphical representation of the model, both forward and backward in time, and a central tool is the ancestral selection graph decorated by mutations.
Wed, 30 Nov 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39932022-11-30T00:00:00ZGonzález Casanova, AdrianSmadi, CharlineWakolbinger, AntonConsider a population of $N$ individuals, each of them carrying a type in $\mathbb N_0$. The population evolves according to a Moran dynamics with selection and mutation, where an individual of type $k$ has the same selective advantage over all individuals with type $k' > k$, and type $k$ mutates to type
$k+1$ at a constant rate. This model is thus a variation of the classical Muller's ratchet: there the selective advantage is proportional to $k' - k$. For a regime of selection strength and mutation rates which is between the regimes of weak and strong selection/mutation, we obtain the asymptotic rate of the click times of the ratchet (i.e. the times at which the hitherto minimal ('best') type in the population is lost), and reveal the quasi-stationary type frequency profile between clicks. The large population limit of this profile is characterized as the normalized attractor of a "dual" hierarchical multitype logistic system, and also via the distribution of the final minimal
displacement in a branching random walk with one-sided steps. An important role in the proofs is played by a graphical representation of the model, both forward and backward in time, and a central tool is the ancestral selection graph decorated by mutations.Jahresbericht | Annual Report - 2021
http://publications.mfo.de/handle/mfo/3991
Jahresbericht | Annual Report - 2021
Mathematisches Forschungsinstitut Oberwolfach
Sat, 01 Jan 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39912022-01-01T00:00:00ZMathematisches Forschungsinstitut OberwolfachA tale of three curves
http://publications.mfo.de/handle/mfo/3986
A tale of three curves
Balakrishnan, Jennifer S.
In this snapshot, we give a survey of some problems in the study of rational points on higher genus curves, discussing questions ranging from the era of the ancient Greeks to a few posed by mathematicians of the 20th century. To answer these questions, we describe a selection of techniques in modern number theory that can be used to determine the set of rational points on a curve.
Thu, 27 Oct 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39862022-10-27T00:00:00ZBalakrishnan, Jennifer S.In this snapshot, we give a survey of some problems in the study of rational points on higher genus curves, discussing questions ranging from the era of the ancient Greeks to a few posed by mathematicians of the 20th century. To answer these questions, we describe a selection of techniques in modern number theory that can be used to determine the set of rational points on a curve.