Oberwolfach Publications
http://publications.mfo.de:80
The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Mon, 24 Jan 2022 19:08:26 GMT2022-01-24T19:08:26ZOberwolfach Publicationshttp://publications.mfo.de/themes/Mirage2/images/apple-touch-icon.png
http://publications.mfo.de:80
Describing distance: from the plane to spectral triples
http://publications.mfo.de/handle/mfo/3912
Describing distance: from the plane to spectral triples
Arici, Francesca; Mesland, Bram
Geometry draws its power from the abstract structures that govern the shapes found in the real world. These abstractions often provide deeper insights into the underlying mathematical objects. In this snapshot, we give a glimpse into how certain “curved spaces” called manifolds can be better understood by looking at the (complex) differentiable functions they admit.
Fri, 31 Dec 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39122021-12-31T00:00:00ZArici, FrancescaMesland, BramGeometry draws its power from the abstract structures that govern the shapes found in the real world. These abstractions often provide deeper insights into the underlying mathematical objects. In this snapshot, we give a glimpse into how certain “curved spaces” called manifolds can be better understood by looking at the (complex) differentiable functions they admit.Reflection Positivity and Hankel Operators- the Multiplicity Free Case
http://publications.mfo.de/handle/mfo/3906
Reflection Positivity and Hankel Operators- the Multiplicity Free Case
Adamo, Maria Stella; Neeb, Karl-Hermann; Schober, Jonas
We analyze reflection positive representations in terms of positive Hankel operators. This is motivated by the fact that positive Hankel operators are described in terms of their Carleson measures, whereas the compatibility condition between representations and reflection positive Hilbert spaces is quite intricate. This leads us to the concept of a Hankel positive representation of triples $(G,S,\tau)$, where $G$ is a group, $\tau$ an involutive automorphism of $G$ and $S \subseteq G$ a subsemigroup with $\tau(S) = S^{-1}$. For the triples $(\mathbb Z,\mathbb N,-id_{\mathbb Z})$, corresponding to reflection positive operators, and $(\mathbb R,\mathbb R_+,-id_{\mathbb R})$, corresponding to reflection positive one-parameter groups, we show that every Hankel positive representation can be made reflection positive by a slight change of the scalar product. A key method
consists in using the measure $\mu_H$ on $\mathbb R_+$ defined by a positive Hankel operator $H$ on $H^2(\mathbb C_+)$ to define a Pick function whose imaginary part, restricted to the imaginary axis, provides an operator symbol for $H$.
Wed, 15 Dec 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39062021-12-15T00:00:00ZAdamo, Maria StellaNeeb, Karl-HermannSchober, JonasWe analyze reflection positive representations in terms of positive Hankel operators. This is motivated by the fact that positive Hankel operators are described in terms of their Carleson measures, whereas the compatibility condition between representations and reflection positive Hilbert spaces is quite intricate. This leads us to the concept of a Hankel positive representation of triples $(G,S,\tau)$, where $G$ is a group, $\tau$ an involutive automorphism of $G$ and $S \subseteq G$ a subsemigroup with $\tau(S) = S^{-1}$. For the triples $(\mathbb Z,\mathbb N,-id_{\mathbb Z})$, corresponding to reflection positive operators, and $(\mathbb R,\mathbb R_+,-id_{\mathbb R})$, corresponding to reflection positive one-parameter groups, we show that every Hankel positive representation can be made reflection positive by a slight change of the scalar product. A key method
consists in using the measure $\mu_H$ on $\mathbb R_+$ defined by a positive Hankel operator $H$ on $H^2(\mathbb C_+)$ to define a Pick function whose imaginary part, restricted to the imaginary axis, provides an operator symbol for $H$.Jahresbericht | Annual Report - 2020
http://publications.mfo.de/handle/mfo/3905
Jahresbericht | Annual Report - 2020
Mathematisches Forschungsinstitut Oberwolfach
Fri, 01 Jan 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39052021-01-01T00:00:00ZMathematisches Forschungsinstitut OberwolfachFundamental Theorem of Projective Geometry over Semirings
http://publications.mfo.de/handle/mfo/3892
Fundamental Theorem of Projective Geometry over Semirings
Tewari, Ayush Kumar
We state the fundamental theorem of projective geometry for semimodules over semirings, which is facilitated by recent work in the study of bases in semimodules defined over semirings. In the process we explore in detail the linear algebra setup over semirings. We also provide more explicit results to understand the implications of our main theorem on maps between tropical lines in the tropical plane. Along with this we also look at geometrical connections to the rich theory of tropical geometry.
Mon, 11 Oct 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38922021-10-11T00:00:00ZTewari, Ayush KumarWe state the fundamental theorem of projective geometry for semimodules over semirings, which is facilitated by recent work in the study of bases in semimodules defined over semirings. In the process we explore in detail the linear algebra setup over semirings. We also provide more explicit results to understand the implications of our main theorem on maps between tropical lines in the tropical plane. Along with this we also look at geometrical connections to the rich theory of tropical geometry.Finite geometries: pure mathematics close to applications
http://publications.mfo.de/handle/mfo/3889
Finite geometries: pure mathematics close to applications
Storme, Leo
The research field of finite geometries investigates structures with a finite number of objects. Classical examples include vector spaces, projective spaces, and affine spaces over finite fields. Although many of these structures are studied for their geometrical importance, they are also of great interest in other, more applied domains of mathematics. In this snapshot, finite vector spaces are introduced. We discuss the geometrical concept of partial t-spreads together with its implications for the “packing problem” and a recent application in the existence of “cooling codes”.
Wed, 22 Sep 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38892021-09-22T00:00:00ZStorme, LeoThe research field of finite geometries investigates structures with a finite number of objects. Classical examples include vector spaces, projective spaces, and affine spaces over finite fields. Although many of these structures are studied for their geometrical importance, they are also of great interest in other, more applied domains of mathematics. In this snapshot, finite vector spaces are introduced. We discuss the geometrical concept of partial t-spreads together with its implications for the “packing problem” and a recent application in the existence of “cooling codes”.Lagrangian mean curvature flow
http://publications.mfo.de/handle/mfo/3884
Lagrangian mean curvature flow
Lotay, Jason D.
Lagrangian mean curvature flow is a powerful tool in modern mathematics with connections to topics in analysis, geometry, topology and mathematical physics. I will describe some of the key aspects of Lagrangian mean curvature flow, some recent progress, and some major open problems.
Thu, 16 Sep 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38842021-09-16T00:00:00ZLotay, Jason D.Lagrangian mean curvature flow is a powerful tool in modern mathematics with connections to topics in analysis, geometry, topology and mathematical physics. I will describe some of the key aspects of Lagrangian mean curvature flow, some recent progress, and some major open problems.Reflections on hyperbolic space
http://publications.mfo.de/handle/mfo/3876
Reflections on hyperbolic space
Haensch, Anna
In school, we learn that the interior angles of any triangle sum up to pi. However, there exist spaces different from the usual Euclidean space in which this is not true. One of these spaces is the ''hyperbolic space'', which has another geometry than the classical Euclidean geometry. In this snapshot, we consider the geometry of hyperbolic polytopes, for example polygons, how they tile hyperbolic space, and how reflections along the faces of polytopes give rise to important mathematical structures. The classification of these structures is an open area of research.
Tue, 24 Aug 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38762021-08-24T00:00:00ZHaensch, AnnaIn school, we learn that the interior angles of any triangle sum up to pi. However, there exist spaces different from the usual Euclidean space in which this is not true. One of these spaces is the ''hyperbolic space'', which has another geometry than the classical Euclidean geometry. In this snapshot, we consider the geometry of hyperbolic polytopes, for example polygons, how they tile hyperbolic space, and how reflections along the faces of polytopes give rise to important mathematical structures. The classification of these structures is an open area of research.The Enigma behind the Good–Turing formula
http://publications.mfo.de/handle/mfo/3875
The Enigma behind the Good–Turing formula
Balabdaoui, Fadoua; Kulagina, Yulia
Finding the total number of species in a population
based on a finite sample is a difficult but practically
important problem. In this snapshot, we will attempt
to shed light on how during World War II, two
cryptanalysts, Irving J. Good and Alan M. Turing,
discovered one of the most widely applied formulas in
statistics. The formula estimates the probability of
missing some of the species in a sample drawn from
a heterogeneous population. We will provide some
intuition behind the formula, show its wide range of
applications, and give a few technical details.
Fri, 16 Jul 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38752021-07-16T00:00:00ZBalabdaoui, FadouaKulagina, YuliaFinding the total number of species in a population
based on a finite sample is a difficult but practically
important problem. In this snapshot, we will attempt
to shed light on how during World War II, two
cryptanalysts, Irving J. Good and Alan M. Turing,
discovered one of the most widely applied formulas in
statistics. The formula estimates the probability of
missing some of the species in a sample drawn from
a heterogeneous population. We will provide some
intuition behind the formula, show its wide range of
applications, and give a few technical details.Weak*-Continuity of Invariant Means on Spaces of Matrix Coefficients
http://publications.mfo.de/handle/mfo/3873
Weak*-Continuity of Invariant Means on Spaces of Matrix Coefficients
de Laat, Tim; Zadeh, Safoura
With every locally compact group $G$, one can associate several interesting bi-invariant subspaces $X(G)$ of the weakly almost periodic functions $\mathrm{WAP}(G)$ on $G$, each of which captures parts of the representation theory of $G$. Under certain natural assumptions, such a space $X(G)$ carries a unique invariant mean and has a natural predual, and we view the weak$^*$-continuity of this mean as a rigidity property of $G$. Important examples of such spaces $X(G)$, which we study explicitly, are the algebra $M_{\mathrm{cb}}A_p(G)$ of $p$-completely bounded multipliers of the Figà-Talamanca-Herz algebra $A_p(G)$ and the $p$-Fourier-Stieltjes algebra $B_p(G)$. In the setting of connected Lie groups $G$, we relate the weak$^*$-continuity of the mean on these spaces to structural properties of $G$. Our results generalise results of Bekka, Kaniuth, Lau and Schlichting.
Tue, 13 Jul 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38732021-07-13T00:00:00Zde Laat, TimZadeh, SafouraWith every locally compact group $G$, one can associate several interesting bi-invariant subspaces $X(G)$ of the weakly almost periodic functions $\mathrm{WAP}(G)$ on $G$, each of which captures parts of the representation theory of $G$. Under certain natural assumptions, such a space $X(G)$ carries a unique invariant mean and has a natural predual, and we view the weak$^*$-continuity of this mean as a rigidity property of $G$. Important examples of such spaces $X(G)$, which we study explicitly, are the algebra $M_{\mathrm{cb}}A_p(G)$ of $p$-completely bounded multipliers of the Figà-Talamanca-Herz algebra $A_p(G)$ and the $p$-Fourier-Stieltjes algebra $B_p(G)$. In the setting of connected Lie groups $G$, we relate the weak$^*$-continuity of the mean on these spaces to structural properties of $G$. Our results generalise results of Bekka, Kaniuth, Lau and Schlichting.Ultrafilter methods in combinatorics
http://publications.mfo.de/handle/mfo/3870
Ultrafilter methods in combinatorics
Goldbring, Isaac
Given a set X, ultrafilters determine which subsets
of X should be considered as large. We illustrate
the use of ultrafilter methods in combinatorics by
discussing two cornerstone results in Ramsey theory,
namely Ramsey’s theorem itself and Hindman’s theorem.
We then present a recent result in combinatorial
number theory that verifies a conjecture of Erdos
known as the “B + C conjecture”.
Fri, 25 Jun 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38702021-06-25T00:00:00ZGoldbring, IsaacGiven a set X, ultrafilters determine which subsets
of X should be considered as large. We illustrate
the use of ultrafilter methods in combinatorics by
discussing two cornerstone results in Ramsey theory,
namely Ramsey’s theorem itself and Hindman’s theorem.
We then present a recent result in combinatorial
number theory that verifies a conjecture of Erdos
known as the “B + C conjecture”.