Pub Date : 2024-01-30DOI: 10.4310/pamq.2023.v19.n6.a11
Dennis Sullivan
$defspinc{operatorname{spin}^mathrm{c}}$We give a 1965 era proof of the title assuming $M$ is $spinc$. The fact that any oriented four manifold is $spinc$ is a challenging result from 1995 whose interesting argument by Teichner–Vogt is analyzed and used in the appendix to show an analogous integral lift result about the top Wu class in $dim 4k$. This will be used in future work to study related complex structures on higher dimensional open manifolds.
{"title":"Any oriented non-closed connected $4$-manifold can be spread holomorphically over the complex projective plane minus a point","authors":"Dennis Sullivan","doi":"10.4310/pamq.2023.v19.n6.a11","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n6.a11","url":null,"abstract":"$defspinc{operatorname{spin}^mathrm{c}}$We give a 1965 era proof of the title assuming $M$ is $spinc$. The fact that any oriented four manifold is $spinc$ is a challenging result from 1995 whose interesting argument by Teichner–Vogt is analyzed and used in the appendix to show an analogous integral lift result about the top Wu class in $dim 4k$. This will be used in future work to study related complex structures on higher dimensional open manifolds.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139659454","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.a1
David Evans, Mathew Pugh
Spectral measures provide invariants for braided subfactors via fusion modules. In this paper we study joint spectral measures associated to the compact connected rank two Lie group $SO(5)$ and its double cover the compact connected, simply-connected rank two Lie group $Sp(2)$, including the McKay graphs for the irreducible representations of $Sp(2)$ and $SO(5)$ and their maximal tori, and fusion modules associated to the $Sp(2)$ modular invariants.
{"title":"Spectral measures for $Sp(2)$","authors":"David Evans, Mathew Pugh","doi":"10.4310/pamq.2023.v19.n5.a1","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n5.a1","url":null,"abstract":"Spectral measures provide invariants for braided subfactors via fusion modules. In this paper we study joint spectral measures associated to the compact connected rank two Lie group $SO(5)$ and its double cover the compact connected, simply-connected rank two Lie group $Sp(2)$, including the McKay graphs for the irreducible representations of $Sp(2)$ and $SO(5)$ and their maximal tori, and fusion modules associated to the $Sp(2)$ modular invariants.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139647312","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.a5
Claude LeBrun
The infimum of the Weyl functional is shown to be surprisingly small on many compact $4$-manifolds that admit positive-scalar-curvature metrics. Results are also proved that systematically compare the scalar and self-dual Weyl curvatures of certain almost-Kähler $4$-manifolds.
{"title":"Curvature in the balance: the Weyl functional and scalar curvature of $4$-manifolds","authors":"Claude LeBrun","doi":"10.4310/pamq.2023.v19.n6.a5","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n6.a5","url":null,"abstract":"The infimum of the Weyl functional is shown to be surprisingly small on many compact $4$-manifolds that admit positive-scalar-curvature metrics. Results are also proved that systematically compare the scalar and self-dual Weyl curvatures of certain almost-Kähler $4$-manifolds.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656351","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.a3
Pinhas Grossman, Masaki Izumi, Noah Snyder
$defZ{mathbb{Z}}$We classify certain $Z_2$-graded extensions of generalized Haagerup categories in terms of numerical invariants satisfying polynomial equations. In particular, we construct a number of new examples of fusion categories, including: $Z_2$-graded extensions of $Z_{2n}$ generalized Haagerup categories for all $n leq 5$; $Z_2 times Z_2$-graded extensions of the Asaeda-Haagerup categories; and extensions of the $Z_2 times Z_2$ generalized Haagerup category by its outer automorphism group $A_4$. The construction uses endomorphism categories of operator algebras, and in particular, free products of Cuntz algebras with free group $mathrm{C}^ast$-algebras.
{"title":"Graded extensions of generalized Haagerup categories","authors":"Pinhas Grossman, Masaki Izumi, Noah Snyder","doi":"10.4310/pamq.2023.v19.n5.a3","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n5.a3","url":null,"abstract":"$defZ{mathbb{Z}}$We classify certain $Z_2$-graded extensions of generalized Haagerup categories in terms of numerical invariants satisfying polynomial equations. In particular, we construct a number of new examples of fusion categories, including: $Z_2$-graded extensions of $Z_{2n}$ generalized Haagerup categories for all $n leq 5$; $Z_2 times Z_2$-graded extensions of the Asaeda-Haagerup categories; and extensions of the $Z_2 times Z_2$ generalized Haagerup category by its outer automorphism group $A_4$. The construction uses endomorphism categories of operator algebras, and in particular, free products of Cuntz algebras with free group $mathrm{C}^ast$-algebras.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139647537","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.a14
F. Reese Harvey, Kevin R. Payne
We discuss one of the many topics that illustrate the interaction of Blaine Lawson’s deep geometric and analytic insights. The first author is extremely grateful to have had the pleasure of collaborating with Blaine over many enjoyable years. The topic to be discussed concerns the fruitful interplay between nonlinear potential theory; that is, the study of subharmonics with respect to a general constraint set in the $2$-jet bundle and the study of subsolutions and supersolutions of a nonlinear (degenerate) elliptic PDE. The main results include (but are not limited to) the validity of the comparison principle and the existence and uniqueness to solutions to the relevant Dirichlet problems on domains which are suitably “pseudoconvex”. The methods employed are geometric and flexible as well as being very general on the potential theory side, which is interesting in its own right. Moreover, in many important geometric contexts no natural operator may be present. On the other hand, the potential theoretic approach can yield results on the PDE side in terms of non standard structural conditions on a given differential operator.
{"title":"Interplay between nonlinear potential theory and fully nonlinear elliptic PDEs","authors":"F. Reese Harvey, Kevin R. Payne","doi":"10.4310/pamq.2023.v19.n6.a14","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n6.a14","url":null,"abstract":"We discuss one of the many topics that illustrate the interaction of Blaine Lawson’s deep geometric and analytic insights. The first author is extremely grateful to have had the pleasure of collaborating with Blaine over many enjoyable years. The topic to be discussed concerns the fruitful interplay between <i>nonlinear potential theory</i>; that is, the study of subharmonics with respect to a general constraint set in the $2$-jet bundle and the study of subsolutions and supersolutions of a nonlinear (degenerate) elliptic PDE. The main results include (but are not limited to) the validity of the comparison principle and the existence and uniqueness to solutions to the relevant Dirichlet problems on domains which are suitably “pseudoconvex”. The methods employed are geometric and flexible as well as being very general on the potential theory side, which is interesting in its own right. Moreover, in many important geometric contexts no natural operator may be present. On the other hand, the potential theoretic approach can yield results on the PDE side in terms of non standard structural conditions on a given differential operator.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656344","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.a7
Pedro F. dos Santos, Paulo Lima-Filho
We investigate the restriction to fixed-points and change of coefficient functors in $RO(C_2)$-graded equivariant cohomology, with applications to the equivariant cohomology of spaces with a trivial $C_2$-action for $underline{mathbb{Z}}$ and $underline{mathbb{F}_2}$ coefficients. To this end, we study the nonequivariant spectra representing these theories and the corresponding functors. In particular, we show that the $RO(C2)$-graded homology class determined by a Real submanifold $Y$ (in the sense of Atiyah) of a Real compact manifold $X$ encodes the total Steenrod square of the dual to $Y^{C_2}$ in $X^{C_2}$.
{"title":"$RO(C_2)$-graded equivariant cohomology and classical Steenrod squares","authors":"Pedro F. dos Santos, Paulo Lima-Filho","doi":"10.4310/pamq.2023.v19.n6.a7","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n6.a7","url":null,"abstract":"We investigate the <i>restriction to fixed-points</i> and <i>change of coefficient functors</i> in $RO(C_2)$-graded equivariant cohomology, with applications to the equivariant cohomology of spaces with a trivial $C_2$-action for $underline{mathbb{Z}}$ and $underline{mathbb{F}_2}$ coefficients. To this end, we study the nonequivariant spectra representing these theories and the corresponding functors. In particular, we show that the $RO(C2)$-graded homology class determined by a Real submanifold $Y$ (in the sense of Atiyah) of a Real compact manifold $X$ encodes the total Steenrod square of the dual to $Y^{C_2}$ in $X^{C_2}$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656214","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.a12
Jonathan Rosenberg, Shmuel Weinberger
This paper is about positive scalar curvature on a compact manifold $X$ with non-empty boundary $partial X$. In some cases, we completely answer the question of when $X$ has a positive scalar curvature metric which is a product metric near $partial X$, or when $X$ has a positive scalar curvature metric with positive mean curvature on the boundary, and more generally, we study the relationship between boundary conditions on $partial X$ for positive scalar curvature metrics on $X$ and the positive scalar curvature problem for the double $M = operatorname{Dbl} (X, partial X)$.
{"title":"Positive scalar curvature on manifolds with boundary and their doubles","authors":"Jonathan Rosenberg, Shmuel Weinberger","doi":"10.4310/pamq.2023.v19.n6.a12","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n6.a12","url":null,"abstract":"This paper is about positive scalar curvature on a compact manifold $X$ with non-empty boundary $partial X$. In some cases, we completely answer the question of when $X$ has a positive scalar curvature metric which is a product metric near $partial X$, or when $X$ has a positive scalar curvature metric with positive mean curvature on the boundary, and more generally, we study the relationship between boundary conditions on $partial X$ for positive scalar curvature metrics on $X$ and the positive scalar curvature problem for the double $M = operatorname{Dbl} (X, partial X)$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656311","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.a2
Nigel Hitchin
We use the notion of the principal three-dimensional subgroup of a simple Lie group to identify certain special subspaces of the Lie algebra and address the question of whether these are calibrated for invariant forms on the group.
{"title":"A remark on calibrations and Lie groups","authors":"Nigel Hitchin","doi":"10.4310/pamq.2023.v19.n6.a2","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n6.a2","url":null,"abstract":"We use the notion of the principal three-dimensional subgroup of a simple Lie group to identify certain special subspaces of the Lie algebra and address the question of whether these are calibrated for invariant forms on the group.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656317","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.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":null,"pages":null},"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":null,"pages":null},"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}
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