Pub Date : 2024-01-30DOI: 10.4310/pamq.2023.v19.n5.a2
Stavros Garoufalidis, Rinat Kashaev
We discuss two realizations of the colored Jones polynomials of a knot, one appearing in an unnoticed work of the second author in 1994 on quantum R-matrices at roots of unity obtained from solutions of the pentagon identity, and another formulated in terms of a sequence of elements of the Habiro ring appearing in recent work of D. Zagier and the first author on the Refined Quantum Modularity Conjecture.
我们讨论了结的彩色琼斯多项式的两种实现方式，一种出现在第二作者 1994 年关于从五边形特性解中获得的统一根量子 R 矩阵的一项未被注意的工作中，另一种则出现在 D. Zagier 和第一作者关于精炼量子模块性猜想的最新工作中，以哈比罗环元素序列的形式制定。
{"title":"The descendant colored Jones polynomials","authors":"Stavros Garoufalidis, Rinat Kashaev","doi":"10.4310/pamq.2023.v19.n5.a2","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n5.a2","url":null,"abstract":"We discuss two realizations of the colored Jones polynomials of a knot, one appearing in an unnoticed work of the second author in 1994 on quantum R-matrices at roots of unity obtained from solutions of the pentagon identity, and another formulated in terms of a sequence of elements of the Habiro ring appearing in recent work of D. Zagier and the first author on the Refined Quantum Modularity Conjecture.","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":"139647281","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":"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":"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":"13 1","pages":""},"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":"7 1","pages":""},"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.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":"67 1","pages":""},"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":"44 1","pages":""},"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.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":"189 1","pages":""},"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":"15 1","pages":""},"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.a3
Chris Peters
The isometry class of the intersection form of a compact complex surface can be easily determined from complex-analytic invariants. For projective surfaces the primitive lattice is another naturally occurring lattice. The goal of this note is to show that it can be determined from the intersection lattice and the self-intersection of a primitive ample class, at least when the primitive lattice is indefinite. Examples include the Godeaux surfaces, the Kunev surface and a specific Horikawa surface. There are also some results concerning (negative) definite primitive lattices, especially for canonically polarized surfaces of general type.
{"title":"A note on the primitive cohomology lattice of a projective surface","authors":"Chris Peters","doi":"10.4310/pamq.2023.v19.n6.a3","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n6.a3","url":null,"abstract":"The isometry class of the intersection form of a compact complex surface can be easily determined from complex-analytic invariants. For projective surfaces the primitive lattice is another naturally occurring lattice. The goal of this note is to show that it can be determined from the intersection lattice and the self-intersection of a primitive ample class, at least when the primitive lattice is indefinite. Examples include the Godeaux surfaces, the Kunev surface and a specific Horikawa surface. There are also some results concerning (negative) definite primitive lattices, especially for canonically polarized surfaces of general type.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"11 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656318","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.a10
Xiuxiong Chen, Jingrui Cheng
In this paper, we consider a version of parabolic complex Monge–Ampère equations, and use a PDE approach similar to Phong et al. to establish $L^infty$ and Hölder estimates. We also generalize the $L^infty$ estimates to parabolic Hessian equations.
{"title":"The $L^infty$ estimates for parabolic complex Monge–Ampère and Hessian equations","authors":"Xiuxiong Chen, Jingrui Cheng","doi":"10.4310/pamq.2023.v19.n6.a10","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n6.a10","url":null,"abstract":"In this paper, we consider a version of parabolic complex Monge–Ampère equations, and use a PDE approach similar to Phong <i>et al.</i> to establish $L^infty$ and Hölder estimates. We also generalize the $L^infty$ estimates to parabolic Hessian equations.","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":"139656645","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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