Pub Date : 2024-05-16DOI: 10.1007/s40598-024-00249-6
Gabriel Katz
{"title":"Doodles and Blobs on a Ruled Page: Convex Quasi-envelops of Traversing Flows on Surfaces","authors":"Gabriel Katz","doi":"10.1007/s40598-024-00249-6","DOIUrl":"https://doi.org/10.1007/s40598-024-00249-6","url":null,"abstract":"","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140970959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-04DOI: 10.1007/s40598-024-00247-8
Martin T. Luu
As part of the development of the orbit method, Kirillov has counted the number of strictly upper triangular matrices with coefficients in a finite field of q elements and fixed Jordan type. One obtains polynomials with respect to q with many interesting properties and close relation to type A representation theory. In the present work, we develop the corresponding theory for the exceptional Lie algebra (mathfrak g_2). In particular, we show that the leading coefficient can be expressed in terms of the Springer correspondence.
作为轨道方法发展的一部分,基里洛夫计算了在具有 q 个元素和固定约旦类型的有限域中具有系数的严格上三角矩阵的数量。我们得到了关于 q 的多项式,它们具有许多有趣的性质,并与 A 型表示理论密切相关。在本研究中,我们发展了例外李代数(mathfrak g_2)的相应理论。我们特别指出,前导系数可以用 Springer 对应关系来表示。
{"title":"Kirillov Polynomials for the Exceptional Lie Algebra (mathfrak g_{2})","authors":"Martin T. Luu","doi":"10.1007/s40598-024-00247-8","DOIUrl":"10.1007/s40598-024-00247-8","url":null,"abstract":"<div><p>As part of the development of the orbit method, Kirillov has counted the number of strictly upper triangular matrices with coefficients in a finite field of <i>q</i> elements and fixed Jordan type. One obtains polynomials with respect to <i>q</i> with many interesting properties and close relation to type A representation theory. In the present work, we develop the corresponding theory for the exceptional Lie algebra <span>(mathfrak g_2)</span>. In particular, we show that the leading coefficient can be expressed in terms of the Springer correspondence.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-024-00247-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140744435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-15DOI: 10.1007/s40598-024-00245-w
Andrey Smirnov, Alexander Varchenko
In this note, we discuss an integral representation for the vertex function of the cotangent bundle over the Grassmannian, (X=T^{*}{text {Gr}}(k,n)). This integral representation can be used to compute the (hbar rightarrow infty ) limit of the vertex function, where (hbar ) denotes the equivariant parameter of a torus acting on X by dilating the cotangent fibers. We show that in this limit, the integral turns into the standard mirror integral representation of the A-series of the Grassmannian ({text {Gr}}(k,n)) with the Laurent polynomial Landau–Ginzburg superpotential of Eguchi, Hori and Xiong.
{"title":"Polynomial Superpotential for Grassmannian ({text {Gr}}(k,n)) from a Limit of Vertex Function","authors":"Andrey Smirnov, Alexander Varchenko","doi":"10.1007/s40598-024-00245-w","DOIUrl":"10.1007/s40598-024-00245-w","url":null,"abstract":"<div><p>In this note, we discuss an integral representation for the vertex function of the cotangent bundle over the Grassmannian, <span>(X=T^{*}{text {Gr}}(k,n))</span>. This integral representation can be used to compute the <span>(hbar rightarrow infty )</span> limit of the vertex function, where <span>(hbar )</span> denotes the equivariant parameter of a torus acting on <i>X</i> by dilating the cotangent fibers. We show that in this limit, the integral turns into the standard mirror integral representation of the <i>A</i>-series of the Grassmannian <span>({text {Gr}}(k,n))</span> with the Laurent polynomial Landau–Ginzburg superpotential of Eguchi, Hori and Xiong.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140237665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-01DOI: 10.1007/s40598-024-00244-x
Zhengyi Zhou
We use spinal open books to construct contact manifolds with infinitely many different Weinstein fillings in any odd dimension (> 1,) which were previously unknown for dimensions equal to (4n+1.) The argument does not involve understanding factorizations in the symplectic mapping class group.
{"title":"A Note on Contact Manifolds with Infinite Fillings","authors":"Zhengyi Zhou","doi":"10.1007/s40598-024-00244-x","DOIUrl":"10.1007/s40598-024-00244-x","url":null,"abstract":"<div><p>We use spinal open books to construct contact manifolds with infinitely many different Weinstein fillings in any odd dimension <span>(> 1,)</span> which were previously unknown for dimensions equal to <span>(4n+1.)</span> The argument does not involve understanding factorizations in the symplectic mapping class group.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142409338","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-22DOI: 10.1007/s40598-023-00243-4
Peter Gibson, Nicolau Saldanha, Carlos Tomei
We present higher dimensional versions of the classical results of Euler and Fuss, both of which are special cases of the celebrated Poncelet porism. Our results concern polytopes, specifically simplices, parallelotopes and cross polytopes, inscribed in a given ellipsoid and circumscribed to another. The statements and proofs use the language of linear algebra. Without loss, one of the ellipsoids is the unit sphere and the other one is also centered at the origin. Let A be the positive symmetric matrix taking the outer ellipsoid to the inner one. If ({text {tr}}, A = 1), there exists a bijection between the orthogonal group O(n) and the set of such labeled simplices. Similarly, if ({text {tr}}, A^2 = 1), there are families of parallelotopes and of cross polytopes, also indexed by O(n).
我们提出了欧拉和福斯经典结果的高维版本,这两个结果都是著名的庞斯莱孔主义的特例。我们的结果涉及多面体,特别是简面、平行多面体和交叉多面体,它们都刻在给定的椭球体上,并以另一个椭球体为圆心。陈述和证明使用线性代数语言。在不损失任何信息的情况下,其中一个椭圆体是单位球面,另一个椭圆体也以原点为中心。设 A 是将外椭球面取为内椭球面的正对称矩阵。如果 ({text {tr}}, A = 1), 那么在正交群 O(n) 和这样的标注简约集之间存在一个双射。类似地,如果 ({text {tr}}, A^2 = 1), 则存在平行多面体族和交叉多面体族,同样以 O(n) 为索引。
{"title":"Higher Dimensional Versions of Theorems of Euler and Fuss","authors":"Peter Gibson, Nicolau Saldanha, Carlos Tomei","doi":"10.1007/s40598-023-00243-4","DOIUrl":"10.1007/s40598-023-00243-4","url":null,"abstract":"<div><p>We present higher dimensional versions of the classical results of Euler and Fuss, both of which are special cases of the celebrated Poncelet porism. Our results concern polytopes, specifically simplices, parallelotopes and cross polytopes, inscribed in a given ellipsoid and circumscribed to another. The statements and proofs use the language of linear algebra. Without loss, one of the ellipsoids is the unit sphere and the other one is also centered at the origin. Let <i>A</i> be the positive symmetric matrix taking the outer ellipsoid to the inner one. If <span>({text {tr}}, A = 1)</span>, there exists a bijection between the orthogonal group <i>O</i>(<i>n</i>) and the set of such labeled simplices. Similarly, if <span>({text {tr}}, A^2 = 1)</span>, there are families of parallelotopes and of cross polytopes, also indexed by <i>O</i>(<i>n</i>).</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142413037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-03DOI: 10.1007/s40598-023-00242-5
Anton Ayzenberg, Vladimir Gorchakov
We characterize the actions of compact tori on smooth closed manifolds for which the orbit space is a topological manifold (either closed or with boundary). For closed manifolds, the result was originally proved by Styrt in 2009. We give a new proof for closed manifolds which is also applicable to manifolds with boundary. In our arguments, we use the result of Provan and Billera who characterized matroid complexes which are pseudomanifolds. We study the combinatorial structure of torus actions whose orbit spaces are manifolds. In two appendix sections, we give an overview of two theories related to our work. The first one is the combinatorial theory of Leontief substitution systems from mathematical economics. The second one is the topological Kaluza–Klein model of Dirac’s monopole studied by Atiyah. The aim of these sections is to draw some bridges between disciplines and motivate further studies in toric topology.
{"title":"Toric Orbit Spaces Which are Manifolds","authors":"Anton Ayzenberg, Vladimir Gorchakov","doi":"10.1007/s40598-023-00242-5","DOIUrl":"10.1007/s40598-023-00242-5","url":null,"abstract":"<div><p>We characterize the actions of compact tori on smooth closed manifolds for which the orbit space is a topological manifold (either closed or with boundary). For closed manifolds, the result was originally proved by Styrt in 2009. We give a new proof for closed manifolds which is also applicable to manifolds with boundary. In our arguments, we use the result of Provan and Billera who characterized matroid complexes which are pseudomanifolds. We study the combinatorial structure of torus actions whose orbit spaces are manifolds. In two appendix sections, we give an overview of two theories related to our work. The first one is the combinatorial theory of Leontief substitution systems from mathematical economics. The second one is the topological Kaluza–Klein model of Dirac’s monopole studied by Atiyah. The aim of these sections is to draw some bridges between disciplines and motivate further studies in toric topology.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142409577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-08DOI: 10.1007/s40598-023-00241-6
Jesús Palma-Márquez
We generalize the construction of a toric variety associated with an integer convex polyhedron to construct generalized analytic varieties associated with polyhedra with not necessarily rational vertices. For germs of generalized analytic functions with a given Newton polyhedron (Gamma ), the generalized analytic variety associated with (Gamma ) provides a stratified resolution of singularities of these functions; also ensuring a full resolution for almost all of them. Thus, this constructive and elementary approach replaces the non-effective previous proof of this result based on consecutive blow-ups.
{"title":"Newton Polyhedra and Stratified Resolution of Singularities in the Class of Generalized Power Series","authors":"Jesús Palma-Márquez","doi":"10.1007/s40598-023-00241-6","DOIUrl":"10.1007/s40598-023-00241-6","url":null,"abstract":"<div><p>We generalize the construction of a toric variety associated with an integer convex polyhedron to construct generalized analytic varieties associated with polyhedra with not necessarily rational vertices. For germs of generalized analytic functions with a given Newton polyhedron <span>(Gamma )</span>, the generalized analytic variety associated with <span>(Gamma )</span> provides a stratified resolution of singularities of these functions; also ensuring a full resolution for almost all of them. Thus, this constructive and elementary approach replaces the non-effective previous proof of this result based on consecutive blow-ups.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138587734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-28DOI: 10.1007/s40598-023-00240-7
Antonio Lerario, Michele Stecconi
We explain how to use the probabilistic method to prove the existence of real polynomial singularities with rich topology, i.e., with total Betti number of the maximal possible order. We show how similar ideas can be used to produce real algebraic projective hypersurfaces with a rich structure of umbilical points.
{"title":"The Probabilistic Method in Real Singularity Theory","authors":"Antonio Lerario, Michele Stecconi","doi":"10.1007/s40598-023-00240-7","DOIUrl":"10.1007/s40598-023-00240-7","url":null,"abstract":"<div><p>We explain how to use the probabilistic method to prove the existence of real polynomial singularities with rich topology, i.e., with total Betti number of the maximal possible order. We show how similar ideas can be used to produce real algebraic projective hypersurfaces with a rich structure of umbilical points.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-023-00240-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139353370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-24DOI: 10.1007/s40598-023-00239-0
V. A. Vassiliev
We develop a technique for calculating the cohomology groups of spaces of complex parametric knots in ({{mathbb {C}}}^k), (k ge 3), and obtain these groups of low dimensions.
我们开发了一种计算 ({{mathbb {C}}^k),(k ge 3) 中复杂参数结空间的同调群的技术,并得到了这些群的低维度。
{"title":"Cohomology of Spaces of Complex Knots","authors":"V. A. Vassiliev","doi":"10.1007/s40598-023-00239-0","DOIUrl":"10.1007/s40598-023-00239-0","url":null,"abstract":"<div><p>We develop a technique for calculating the cohomology groups of spaces of complex parametric knots in <span>({{mathbb {C}}}^k)</span>, <span>(k ge 3)</span>, and obtain these groups of low dimensions.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135273712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-13DOI: 10.1007/s40598-023-00238-1
L. A. Lichtenfelz, T. Tauchi, T. Yoneda
The existence of a conjugate point on the volume-preserving diffeomorphism group of a compact Riemannian manifold M is related to the Lagrangian stability of a solution of the incompressible Euler equation on M. The Misiołek curvature is a reasonable criterion for the existence of a conjugate point on the volume-preserving diffeomorphism group corresponding to a stationary solution of the incompressible Euler equation. In this article, we introduce a class of stationary solutions on an arbitrary Riemannian manifold whose behavior is nice with respect to the Misiołek curvature and give a positivity result of the Misiołek curvature for solutions belonging to this class. Moreover, we also show the existence of a conjugate point in the three-dimensional ellipsoid case as its corollary.
紧凑黎曼流形 M 的保体积衍射群上共轭点的存在与 M 上不可压缩欧拉方程解的拉格朗日稳定性有关。Misiołek 曲率是不可压缩欧拉方程静止解对应的保体积衍射群上共轭点存在的合理标准。在本文中,我们介绍了一类任意黎曼流形上的静止解,其行为与米西奥韦克曲率有关,并给出了属于该类解的米西奥韦克曲率的正定结果。此外,我们还证明了在三维椭球体情况下共轭点的存在,作为其推论。
{"title":"Existence of a Conjugate Point in the Incompressible Euler Flow on a Three-Dimensional Ellipsoid","authors":"L. A. Lichtenfelz, T. Tauchi, T. Yoneda","doi":"10.1007/s40598-023-00238-1","DOIUrl":"10.1007/s40598-023-00238-1","url":null,"abstract":"<div><p>The existence of a conjugate point on the volume-preserving diffeomorphism group of a compact Riemannian manifold <i>M</i> is related to the Lagrangian stability of a solution of the incompressible Euler equation on <i>M</i>. The Misiołek curvature is a reasonable criterion for the existence of a conjugate point on the volume-preserving diffeomorphism group corresponding to a stationary solution of the incompressible Euler equation. In this article, we introduce a class of stationary solutions on an arbitrary Riemannian manifold whose behavior is nice with respect to the Misiołek curvature and give a positivity result of the Misiołek curvature for solutions belonging to this class. Moreover, we also show the existence of a conjugate point in the three-dimensional ellipsoid case as its corollary.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135853005","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}