Pub Date : 2024-01-30DOI: 10.4310/pamq.2023.v19.n5.a5
Roberto Longo, Edward Witten
Von Neumann entropy has a natural extension to the case of an arbitrary semifinite von Neumann algebra, as was considered by I. E. Segal. We relate this entropy to the relative entropy and show that the entropy increase for an inclusion of von Neumann factors is bounded by the logarithm of the Jones index. The bound is optimal if the factors are infinite dimensional.
冯-诺依曼熵可以自然扩展到任意半有限冯-诺依曼代数的情况，正如 I. E. Segal 所考虑的那样。我们将这一熵与相对熵联系起来，并证明包含冯-诺依曼因子的熵增加受琼斯指数对数的约束。如果因子是无限维的，那么这个界限就是最佳的。
{"title":"A note on continuous entropy","authors":"Roberto Longo, Edward Witten","doi":"10.4310/pamq.2023.v19.n5.a5","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n5.a5","url":null,"abstract":"Von Neumann entropy has a natural extension to the case of an arbitrary semifinite von Neumann algebra, as was considered by I. E. Segal. We relate this entropy to the relative entropy and show that the entropy increase for an inclusion of von Neumann factors is bounded by the logarithm of the Jones index. The bound is optimal if the factors are infinite dimensional.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"3 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139647268","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 : 2024-01-30DOI: 10.4310/pamq.2023.v19.n5.a7
Nicolai Reshetikhin
The Quantum Calogero–Moser spin system is a superintegrable system with the spectrum of commuting Hamiltonians that can be described entirely in terms of representation theory of the corresponding simple Lie group. Here we describe its natural generalization known as quantum Calogero–Moser spin chain or $N$-spin Calogero–Moser system. In the first part of this paper we show that quantum Calogero–Moser spin chain is a quantum superintegrable systems. Then we show that the Euclidean multi-time propagator for this model can be written as a partition function of a two-dimensional Yang–Mills theory on a cylinder. Then we argue that the two-dimensional Yang–Mills theory withWilson loops with “outer ends” should be regarded as the theory on space times with non-removable corners. Partition functions of such theory satisfy non-stationary Calogero–Moser equations. In this paper the underlying Lie group $G$ is a compact connected, simply connected simple Lie group.
{"title":"Spin Calogero-Moser periodic chains and two dimensional Yang-Mills theory with corners","authors":"Nicolai Reshetikhin","doi":"10.4310/pamq.2023.v19.n5.a7","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n5.a7","url":null,"abstract":"The Quantum Calogero–Moser spin system is a superintegrable system with the spectrum of commuting Hamiltonians that can be described entirely in terms of representation theory of the corresponding simple Lie group. Here we describe its natural generalization known as quantum Calogero–Moser spin chain or $N$-spin Calogero–Moser system. In the first part of this paper we show that quantum Calogero–Moser spin chain is a quantum superintegrable systems. Then we show that the Euclidean multi-time propagator for this model can be written as a partition function of a two-dimensional Yang–Mills theory on a cylinder. Then we argue that the two-dimensional Yang–Mills theory withWilson loops with “outer ends” should be regarded as the theory on space times with non-removable corners. Partition functions of such theory satisfy non-stationary Calogero–Moser equations. In this paper the underlying Lie group $G$ is a compact connected, simply connected simple Lie group.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"68 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139647269","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 : 2024-01-30DOI: 10.4310/pamq.2023.v19.n6.a16
Mark Green, Phillip Griffiths
The global behavior of period mappings defined on generally non-complete algebraic varieties $B$ as well as their local behavior around points in the boundary $Z = overline{B}setminus B$ of smooth completions of $B$ have been extensively investigated. In this paper we shall study the global behavior of period mappings in neighborhoods of the entire boundary $Z$ when $dim B = 2$. One method will be to decompose the dual graph of the boundary into basic building blocks of cycles and trees and analyze these separately. A main tool will be a global version of the classical nilpotent orbit theorem of Schmid.
{"title":"Global behavior at infinity of period mappings defined on algebraic surface","authors":"Mark Green, Phillip Griffiths","doi":"10.4310/pamq.2023.v19.n6.a16","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n6.a16","url":null,"abstract":"The global behavior of period mappings defined on generally non-complete algebraic varieties $B$ as well as their local behavior around points in the boundary $Z = overline{B}setminus B$ of smooth completions of $B$ have been extensively investigated. In this paper we shall study the <i>global</i> behavior of period mappings in neighborhoods of the entire boundary $Z$ when $dim B = 2$. One method will be to decompose the dual graph of the boundary into basic building blocks of cycles and trees and analyze these separately. A main tool will be a global version of the classical nilpotent orbit theorem of Schmid.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"24 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656127","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 : 2024-01-30DOI: 10.4310/pamq.2023.v19.n5.a4
Mikhail Khovanov, Robert Laugwitz
A pair of biadjoint functors between two categories produces a collection of elements in the centers of these categories, one for each isotopy class of nested circles in the plane. If the centers are equipped with a trace map into the ground field, then one assigns an element of that field to a diagram of nested circles. We focus on the self-adjoint functor case of this construction and study the reverse problem of recovering such a functor and a category given values associated to diagrams of nested circles.
{"title":"Planar diagrammatics of self-adjoint functors and recognizable tree series","authors":"Mikhail Khovanov, Robert Laugwitz","doi":"10.4310/pamq.2023.v19.n5.a4","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n5.a4","url":null,"abstract":"A pair of biadjoint functors between two categories produces a collection of elements in the centers of these categories, one for each isotopy class of nested circles in the plane. If the centers are equipped with a trace map into the ground field, then one assigns an element of that field to a diagram of nested circles. We focus on the self-adjoint functor case of this construction and study the reverse problem of recovering such a functor and a category given values associated to diagrams of nested circles.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"71 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139647318","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 : 2024-01-30DOI: 10.4310/pamq.2023.v19.n5.a8
Hans Wenzl
$defEnd{operatorname{End}}$$defRep{operatorname{Rep}}$$defsl{mathfrak{sl}}$Let $V = mathbb{C}^N$ with $N$ odd.We construct a $q$-deformation of $End_{Sp(N-1)}(V^{otimes n})$ which contains $End_{U_q sl_N} (V^{otimes n})$. It is a quotient of an abstract two-variable algebra which is defined by adding one more generator to the generators of the Hecke algebras $H_n$. These results suggest the existence of module categories of $Rep(U_q sl_N)$ which may not come from already known coideal subalgebras of $ U_q sl_N$. We moreover indicate how this can be used to construct module categories of the associated fusion tensor categories as well as subfactors, along the lines of previous work for inclusions $Sp(N) subset SL(N)$.
{"title":"On module categories related to $Sp(N-1) subset Sl(N)$","authors":"Hans Wenzl","doi":"10.4310/pamq.2023.v19.n5.a8","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n5.a8","url":null,"abstract":"$defEnd{operatorname{End}}$$defRep{operatorname{Rep}}$$defsl{mathfrak{sl}}$Let $V = mathbb{C}^N$ with $N$ odd.We construct a $q$-deformation of $End_{Sp(N-1)}(V^{otimes n})$ which contains $End_{U_q sl_N} (V^{otimes n})$. It is a quotient of an abstract two-variable algebra which is defined by adding one more generator to the generators of the Hecke algebras $H_n$. These results suggest the existence of module categories of $Rep(U_q sl_N)$ which may not come from already known coideal subalgebras of $ U_q sl_N$. We moreover indicate how this can be used to construct module categories of the associated fusion tensor categories as well as subfactors, along the lines of previous work for inclusions $Sp(N) subset SL(N)$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"14 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139647720","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 : 2024-01-30DOI: 10.4310/pamq.2023.v19.n5.a9
Feng Xu
This paper contains my previously unpublished work on three problems proposed by Vaughan Jones.
本文包含我之前未发表的关于沃恩-琼斯提出的三个问题的研究成果。
{"title":"On three homework problems from Vaughan Jones","authors":"Feng Xu","doi":"10.4310/pamq.2023.v19.n5.a9","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n5.a9","url":null,"abstract":"This paper contains my previously unpublished work on three problems proposed by Vaughan Jones.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"85 1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139647313","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 : 2024-01-30DOI: 10.4310/pamq.2023.v19.n6.a4
Barbara Drinovec Drnovšek, Franc Forstnerič
The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space $mathbb{R}^n$, $n geq 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashi’s pseudometric on complex manifolds, which is defined in terms of holomorphic discs. They showed that on the unit ball of $mathbb{R}^n$, this minimal metric coincides with the classical Beltrami–Cayley–Klein metric. In the present paper we investigate properties of the minimal pseudometric and give sufficient conditions for a domain to be (complete) hyperbolic, meaning that the minimal pseudometric is a (complete) metric. We show in particular that a convex domain is complete hyperbolic if and only if it does not contain any affine $2$-planes. One of our main results is that a domain with a negative minimal plurisubharmonic exhaustion function is hyperbolic, and a bounded strongly minimally convex domain is complete hyperbolic. We also prove a localization theorem for the minimal pseudometric.
{"title":"Hyperbolic domains in real Euclidean spaces","authors":"Barbara Drinovec Drnovšek, Franc Forstnerič","doi":"10.4310/pamq.2023.v19.n6.a4","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n6.a4","url":null,"abstract":"The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space $mathbb{R}^n$, $n geq 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashi’s pseudometric on complex manifolds, which is defined in terms of holomorphic discs. They showed that on the unit ball of $mathbb{R}^n$, this minimal metric coincides with the classical Beltrami–Cayley–Klein metric. In the present paper we investigate properties of the minimal pseudometric and give sufficient conditions for a domain to be (complete) hyperbolic, meaning that the minimal pseudometric is a (complete) metric. We show in particular that a convex domain is complete hyperbolic if and only if it does not contain any affine $2$-planes. One of our main results is that a domain with a negative minimal plurisubharmonic exhaustion function is hyperbolic, and a bounded strongly minimally convex domain is complete hyperbolic. We also prove a localization theorem for the minimal pseudometric.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"64 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656087","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.a13
Yi Lin, Yiannis Loizides, Reyer Sjamaar, Yanli Song
We extend the Marsden–Weinstein reduction theorem and the Darboux–Moser–Weinstein theorem to symplectic Lie algebroids. We also obtain a coisotropic embedding theorem for symplectic Lie algebroids.
{"title":"Symplectic reduction and a Darboux–Moser–Weinstein theorem for Lie algebroids","authors":"Yi Lin, Yiannis Loizides, Reyer Sjamaar, Yanli Song","doi":"10.4310/pamq.2023.v19.n4.a13","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a13","url":null,"abstract":"We extend the Marsden–Weinstein reduction theorem and the Darboux–Moser–Weinstein theorem to symplectic Lie algebroids. We also obtain a coisotropic embedding theorem for symplectic Lie algebroids.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"197 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495287","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.a11
Peer Christian Kunstmann, Eric Todd Quinto, Andreas Rieder
Generalized Radon transforms are Fourier integral operators which are used, for instance, as imaging models in geophysical exploration. They appear naturally when linearizing about a known background compression wave speed. In this work we first consider a linearly increasing background velocity in two spatial dimensions. We verify the Bolker condition for the zero-offset scanning geometry and provide meaningful arguments for it to hold even if the common offset is positive. Based on this result we suggest an imaging operator for which we calculate the top order symbol in the zero-offset case to study how it maps singularities. Second, to support the usage of background models obtained from linear regression we present a stability result for the Bolker condition under perturbations of the background velocity and of the offset.
{"title":"Seismic imaging with generalized Radon transforms: stability of the Bolker condition","authors":"Peer Christian Kunstmann, Eric Todd Quinto, Andreas Rieder","doi":"10.4310/pamq.2023.v19.n4.a11","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a11","url":null,"abstract":"Generalized Radon transforms are Fourier integral operators which are used, for instance, as imaging models in geophysical exploration. They appear naturally when linearizing about a known background compression wave speed. In this work we first consider a linearly increasing background velocity in two spatial dimensions. We verify the Bolker condition for the zero-offset scanning geometry and provide meaningful arguments for it to hold even if the common offset is positive. Based on this result we suggest an imaging operator for which we calculate the top order symbol in the zero-offset case to study how it maps singularities. Second, to support the usage of background models obtained from linear regression we present a stability result for the Bolker condition under perturbations of the background velocity and of the offset.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"196 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495289","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.a10
David Kazhdan, Alexander Polishchuk
We prove an efficient version of the Wagner’s theorem on almost invariant subspaces (see $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1608090}{[5]}$) and deduce some consequences in the context of Galois extensions.
{"title":"Almost invariant subspaces and operators","authors":"David Kazhdan, Alexander Polishchuk","doi":"10.4310/pamq.2023.v19.n4.a10","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a10","url":null,"abstract":"We prove an efficient version of the Wagner’s theorem on almost invariant subspaces (see $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1608090}{[5]}$) and deduce some consequences in the context of Galois extensions.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138517075","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：