Pub Date : 2023-11-20DOI: 10.4310/pamq.2023.v19.n4.a1
Velleda Baldoni, Michèle Vergne, Michael Walter
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 $n = (n_x)$.
{"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}
Pub Date : 2023-11-20DOI: 10.4310/pamq.2023.v19.n4.a9
Eduardo González, Chris T. Woodward
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.
{"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}
Pub Date : 2023-11-20DOI: 10.4310/pamq.2023.v19.n4.a15
Pau Mir, Eva Miranda, Jonathan Weitsman
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.
{"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}
Pub Date : 2023-11-20DOI: 10.4310/pamq.2023.v19.n4.a14
Yiannis Loizides, Eckhard Meinrenken
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.
{"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}
Pub Date : 2023-11-20DOI: 10.4310/pamq.2023.v19.n4.a12
Eugene Lerman
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 singular 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.
{"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}
Pub Date : 2023-11-20DOI: 10.4310/pamq.2023.v19.n4.a7
Jeffrey Galkowski, John A. Toth
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 $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=3897008}{[textrm{GT19}]}$
{"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}
Pub Date : 2023-11-20DOI: 10.4310/pamq.2023.v19.n4.a5
Yves Colin de Verdière
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.
在这篇综述论文中，我描述了亚黎曼三维接触流形的动力学和谱理论的一些方面。我们使用了由Louis Boutet de Monvel和Victor Guillemin引入的特征锥的Toeplitz量化。我们还讨论了我们工作之后的示踪公式以及Duistermaat-Guillemin示踪公式。
{"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}
Pub Date : 2023-11-20DOI: 10.4310/pamq.2023.v19.n4.a4
Alexander Caviedes Castro, Milena Pabiniak, Silvia Sabatini
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.
{"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}
Pub Date : 2023-11-20DOI: 10.4310/pamq.2023.v19.n4.a2
Christian Blohmann, Michele Schiavina, Alan Weinstein
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.
{"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}
Pub Date : 2023-11-20DOI: 10.4310/pamq.2023.v19.n4.a16
Dadi Ni, Jiahao Cheng, Zhuo Chen, Chen He
$defDerL{operatorname{Der}(L)}$A Lie pair is an inclusion $A$ to $L$ of Lie algebroids over the same base manifold. In an earlier work, the third author with Bandiera, Stiénon, and Xu introduced a canonical $L_{leqslant 3}$ algebra $Gamma (wedge^bullet A^vee otimes L/A)$ whose unary bracket is the Chevalley–Eilenberg differential arising from every Lie pair $(L,A)$. In this note, we prove that to such a Lie pair there is an associated Lie algebra action by $operatorname{Der}(L)$ on the $L_{leqslant 3}$ algebra $Gamma (wedge^bullet A^vee otimes L/A)$. Here $DerL$ is the space of derivations on the Lie algebroid $L$, or infinitesimal automorphisms of $L$. The said action gives rise to a larger scope of gauge equivalences of Maurer–Cartan elements in $Gamma (wedge^bullet A^vee otimes L/A)$, and for this reason we elect to call the $DerL$-action internal symmetry of $Gamma (wedge^bullet A^vee otimes L/A)$.
{"title":"Internal symmetry of the $L_{leqslant 3}$ algebra arising from a Lie pair","authors":"Dadi Ni, Jiahao Cheng, Zhuo Chen, Chen He","doi":"10.4310/pamq.2023.v19.n4.a16","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a16","url":null,"abstract":"$defDerL{operatorname{Der}(L)}$A Lie pair is an inclusion $A$ to $L$ of Lie algebroids over the same base manifold. In an earlier work, the third author with Bandiera, Stiénon, and Xu introduced a canonical $L_{leqslant 3}$ algebra $Gamma (wedge^bullet A^vee otimes L/A)$ whose unary bracket is the Chevalley–Eilenberg differential arising from every Lie pair $(L,A)$. In this note, we prove that to such a Lie pair there is an associated Lie algebra action by $operatorname{Der}(L)$ on the $L_{leqslant 3}$ algebra $Gamma (wedge^bullet A^vee otimes L/A)$. Here $DerL$ is the space of derivations on the Lie algebroid $L$, or infinitesimal automorphisms of $L$. The said action gives rise to a larger scope of gauge equivalences of Maurer–Cartan elements in $Gamma (wedge^bullet A^vee otimes L/A)$, and for this reason we elect to call the $DerL$-action internal symmetry of $Gamma (wedge^bullet A^vee otimes L/A)$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"263 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138517072","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}
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