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Horn conditions for quiver subrepresentations and the moment map 颤振子表示的角条件和矩映射
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-11-20 DOI: 10.4310/pamq.2023.v19.n4.a1
Velleda Baldoni, Michèle Vergne, Michael Walter
{"title":"Horn conditions for quiver subrepresentations and the moment map","authors":"Velleda Baldoni, Michèle Vergne, Michael Walter","doi":"10.4310/pamq.2023.v19.n4.a1","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a1","url":null,"abstract":"We give inductive conditions that characterize the Schubert positions of subrepresentations of a general quiver representation. Our results generalize Belkale’s criterion for the intersection of Schubert varieties in Grassmannians and refine Schofield’s characterization of the dimension vectors of general subrepresentations. This implies Horn type inequalities for the moment cone associated to the linear representation of the group $G = prod_x mathrm{GL}(n_x)$ associated to a quiver and a dimension vector $n = (n_x)$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"193 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Quantum Witten localization 量子书写局域化
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-11-20 DOI: 10.4310/pamq.2023.v19.n4.a9
Eduardo González, Chris T. Woodward
{"title":"Quantum Witten localization","authors":"Eduardo González, Chris T. Woodward","doi":"10.4310/pamq.2023.v19.n4.a9","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a9","url":null,"abstract":"We prove a quantum version of the localization formula of Witten $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1185834}{[31]}$, see also $[href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1792291}{28}$, $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1722000}{22}$, $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=2198772}{35}$], that relates invariants of a GIT quotient with the equivariant invariants of the action.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"196 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Bohr–Sommerfeld quantization of $b$-symplectic toric manifolds b -辛环流形的Bohr-Sommerfeld量化
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-11-20 DOI: 10.4310/pamq.2023.v19.n4.a15
Pau Mir, Eva Miranda, Jonathan Weitsman
{"title":"Bohr–Sommerfeld quantization of $b$-symplectic toric manifolds","authors":"Pau Mir, Eva Miranda, Jonathan Weitsman","doi":"10.4310/pamq.2023.v19.n4.a15","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a15","url":null,"abstract":"We introduce a Bohr–Sommerfeld quantization for bsymplectic toric manifolds and show that it coincides with the formal geometric quantization of $href{ https://mathscinet.ams.org/mathscinet/relay-station?mr=3804693}{[textrm{GMW18b}]}$. In particular, we prove that its dimension is given by a signed count of the integral points in the moment polytope of the torus action on the manifold.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"197 3","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Singular Lie filtrations and weightings 奇异谎言过滤和加权
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-11-20 DOI: 10.4310/pamq.2023.v19.n4.a14
Yiannis Loizides, Eckhard Meinrenken
{"title":"Singular Lie filtrations and weightings","authors":"Yiannis Loizides, Eckhard Meinrenken","doi":"10.4310/pamq.2023.v19.n4.a14","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a14","url":null,"abstract":"We study weightings (a.k.a. quasi-homogeneous structures) arising from manifolds with singular Lie filtrations. This generalizes constructions of Choi–Ponge, Van Erp–Yuncken, and Haj–Higson for (regular) Lie filtrations.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"197 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Stability and bifurcations of symmetric tops 对称顶的稳定性和分岔
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-11-20 DOI: 10.4310/pamq.2023.v19.n4.a12
Eugene Lerman
{"title":"Stability and bifurcations of symmetric tops","authors":"Eugene Lerman","doi":"10.4310/pamq.2023.v19.n4.a12","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a12","url":null,"abstract":"We study the stability and bifurcation of relative equilibria of a particle on the Lie group $SO(3)$ whose motion is governed by an $SO(3) times SO(2)$ invariant metric and an $SO(2) times SO(2)$ invariant potential. Our method is to reduce the number of degrees of freedom at <i>singular</i> values of the $SO(2) times SO(2)$ momentum map and study the stability of the equilibria of the reduced systems as a function of spin. The result is an elementary analysis of the fast/slow transition in the Lagrange and Kirchhoff tops. More generally, since an $SO(2) times SO(2)$ invariant potential on $SO(3)$ can be thought of as $mathbb{Z}_2$ invariant function on a circle, we analyze the stability and bifurcation of relative equilibria of the system in terms of the second and fourth derivative of the function.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"196 3","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Lower bounds for Steklov eigenfunctions Steklov特征函数的下界
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-11-20 DOI: 10.4310/pamq.2023.v19.n4.a7
Jeffrey Galkowski, John A. Toth
{"title":"Lower bounds for Steklov eigenfunctions","authors":"Jeffrey Galkowski, John A. Toth","doi":"10.4310/pamq.2023.v19.n4.a7","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a7","url":null,"abstract":"Let $(Omega,g)$ be a compact, real analytic Riemannian manifold with real analytic boundary $partial Omega = M$. We give $L^2$-lower bounds for Steklov eigenfunctions and their restrictions to interior hypersurfaces $H subset Omega^circ$ in a geometrically defined neighborhood of $M$. Our results are optimal in the entire geometric neighborhood and complement the results on eigenfunction upper bounds in $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=3897008}{[textrm{GT19}]}$","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"195 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495292","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 28
Classical and Quantum mechanics on 3D contact manifolds 三维接触流形的经典力学和量子力学
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-11-20 DOI: 10.4310/pamq.2023.v19.n4.a5
Yves Colin de Verdière
{"title":"Classical and Quantum mechanics on 3D contact manifolds","authors":"Yves Colin de Verdière","doi":"10.4310/pamq.2023.v19.n4.a5","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a5","url":null,"abstract":"In this survey paper, I describe some aspects of the dynamics and the spectral theory of sub-Riemannian 3D contact manifolds. We use Toeplitz quantization of the characteristic cone as introduced by Louis Boutet de Monvel and Victor Guillemin. We also discuss trace formulae following our work as well as the Duistermaat–Guillemin trace formula.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"194 3","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Generalizing the Mukai Conjecture to the symplectic category and the Kostant game 将Mukai猜想推广到辛范畴和Kostant对策
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-11-20 DOI: 10.4310/pamq.2023.v19.n4.a4
Alexander Caviedes Castro, Milena Pabiniak, Silvia Sabatini
{"title":"Generalizing the Mukai Conjecture to the symplectic category and the Kostant game","authors":"Alexander Caviedes Castro, Milena Pabiniak, Silvia Sabatini","doi":"10.4310/pamq.2023.v19.n4.a4","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a4","url":null,"abstract":"In this paper we pose the question of whether the (generalized) Mukai inequalities hold for compact, positive monotone symplectic manifolds. We first provide a method that enables one to check whether the (generalized) Mukai inequalities hold true. This only makes use of the almost complex structure of the manifold and the analysis of the zeros of the so-called generalized Hilbert polynomial, which takes into account the Atiyah-Singer indices of all possible line bundles. We apply this method to generalized flag varieties. In order to find the zeros of the corresponding generalized Hilbert polynomial we introduce a modified version of the Kostant game and study its combinatorial properties.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"194 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Lie–Rinehart algebra in general relativity 广义相对论中的李-莱因哈特代数
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-11-20 DOI: 10.4310/pamq.2023.v19.n4.a2
Christian Blohmann, Michele Schiavina, Alan Weinstein
{"title":"A Lie–Rinehart algebra in general relativity","authors":"Christian Blohmann, Michele Schiavina, Alan Weinstein","doi":"10.4310/pamq.2023.v19.n4.a2","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a2","url":null,"abstract":"We construct a Lie–Rinehart algebra over an infinitesimal extension of the space of initial value fields for Einstein’s equations. The bracket relations in this algebra are precisely those of the constraints for the initial value problem. The Lie–Rinehart algebra comes from a slight generalization of a Lie algebroid in which the algebra consists of sections of a sheaf rather than a vector bundle. (An actual Lie algebroid had been previously constructed by Blohmann, Fernandes, and Weinstein over a much larger extension.) The construction uses the BV–BFV (Batalin–Fradkin–Vilkovisky) approach to boundary value problems, starting with the Einstein equations themselves, to construct an $L_infty$-algebroid over a graded manifold which extends the initial data. The Lie–Rinehart algebra is then constructed by a change of variables. One of the consequences of the BV–BFV approach is a proof that the coisotropic property of the constraint set follows from the invariance of the Einstein equations under space-time diffeomorphisms.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"193 3","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Symplectic geometric flows 辛几何流
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-11-20 DOI: 10.4310/pamq.2023.v19.n4.a6
Teng Fei, Duong H. Phong
{"title":"Symplectic geometric flows","authors":"Teng Fei, Duong H. Phong","doi":"10.4310/pamq.2023.v19.n4.a6","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a6","url":null,"abstract":"Several geometric flows on symplectic manifolds are introduced which are potentially of interest in symplectic geometry and topology. They are motivated by the Type IIA flow and T‑duality between flows in symplectic geometry and flows in complex geometry. Examples include the Hitchin gradient flow on symplectic manifolds, and a new flow which is called the dual Ricci flow.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"195 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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