Pub Date : 2021-10-31DOI: 10.2140/pjm.2022.318.109
Willi Kepplinger
We present an algorithm taking a Kirby diagram of a closed oriented $4$-manifold to a trisection diagram of the same manifold. This algorithm provides us with a large number of examples for trisection diagrams of closed oriented $4$-manifolds since many Kirby-diagrammatic descriptions of closed oriented $4$-manifolds are known. That being said, the algorithm does not necessarily provide particularly efficient trisection diagrams. We also extend this algorithm to work for the non-orientable case.
{"title":"An algorithm taking Kirby diagrams to trisection diagrams","authors":"Willi Kepplinger","doi":"10.2140/pjm.2022.318.109","DOIUrl":"https://doi.org/10.2140/pjm.2022.318.109","url":null,"abstract":"We present an algorithm taking a Kirby diagram of a closed oriented $4$-manifold to a trisection diagram of the same manifold. This algorithm provides us with a large number of examples for trisection diagrams of closed oriented $4$-manifolds since many Kirby-diagrammatic descriptions of closed oriented $4$-manifolds are known. That being said, the algorithm does not necessarily provide particularly efficient trisection diagrams. We also extend this algorithm to work for the non-orientable case.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41735235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the order of the canonical vector bundle over the configuration space is 2 for a general planar graph, and is 4 for a nonplanar graph.
证明了构型空间上正则向量束的阶数对于一般平面图是2阶，对于非平面图是4阶。
{"title":"Orders of the canonical vector bundles over configuration spaces of finite graphs","authors":"F. Cohen, R. Huang","doi":"10.2140/pjm.2022.316.53","DOIUrl":"https://doi.org/10.2140/pjm.2022.316.53","url":null,"abstract":"We prove that the order of the canonical vector bundle over the configuration space is 2 for a general planar graph, and is 4 for a nonplanar graph.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45340324","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the global weak solution problem of semilinear generalized Tricomi equations, II","authors":"Daoyin He, I. Witt, Huicheng Yin","doi":"10.2140/pjm.2021.314.29","DOIUrl":"https://doi.org/10.2140/pjm.2021.314.29","url":null,"abstract":"","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47740877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-12DOI: 10.2140/pjm.2021.313.343
Ady Cambraia Jr., Abigail Folha, C. Peñafiel
{"title":"On totally umbilical surfaces in the warped\u0000product 𝕄(κ)f× I","authors":"Ady Cambraia Jr., Abigail Folha, C. Peñafiel","doi":"10.2140/pjm.2021.313.343","DOIUrl":"https://doi.org/10.2140/pjm.2021.313.343","url":null,"abstract":"","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45485106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-05DOI: 10.2140/pjm.2022.317.297
F. Fang, C. Xia
We adopt the convention that each eigenvalue is repeated according to its multiplicity. An important issue in spectral geometry is to obtain good estimates for these and other eigenvalues in terms of the geometric data of the manifold M such as the volume, the diameter, the curvature, the isoperimetric constants, etc. See [1],[2],[10],[13],[31] for references. On the other hand, after the seminal works of Bleecker-Weiner [4] and Reilly [30], the following approach is developed: the manifold (M, g) is immersed isometrically into another Riemannian manifold. One then gets good estimates for λk(M), mostly for λ1(M), in termos of the extrinsic geometric quantities of M . See for example [4], [15], [16], [23], [24], [35], [37]. Especially relevant for us is the quoted work of Reilly [30], where he obtained the following remarkable isoperimetric inequality for the first positive eigenvalue λ1(M) in the case that M is embedded as a hypersurface bounding a domain Ω in R: λ1(M) ≤ n− 1 n2 · |M | 2 |Ω|2 . (1.1)
{"title":"Isoperimetric bounds for lower-order eigenvalues","authors":"F. Fang, C. Xia","doi":"10.2140/pjm.2022.317.297","DOIUrl":"https://doi.org/10.2140/pjm.2022.317.297","url":null,"abstract":"We adopt the convention that each eigenvalue is repeated according to its multiplicity. An important issue in spectral geometry is to obtain good estimates for these and other eigenvalues in terms of the geometric data of the manifold M such as the volume, the diameter, the curvature, the isoperimetric constants, etc. See [1],[2],[10],[13],[31] for references. On the other hand, after the seminal works of Bleecker-Weiner [4] and Reilly [30], the following approach is developed: the manifold (M, g) is immersed isometrically into another Riemannian manifold. One then gets good estimates for λk(M), mostly for λ1(M), in termos of the extrinsic geometric quantities of M . See for example [4], [15], [16], [23], [24], [35], [37]. Especially relevant for us is the quoted work of Reilly [30], where he obtained the following remarkable isoperimetric inequality for the first positive eigenvalue λ1(M) in the case that M is embedded as a hypersurface bounding a domain Ω in R: λ1(M) ≤ n− 1 n2 · |M | 2 |Ω|2 . (1.1)","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44643070","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-04DOI: 10.2140/pjm.2023.322.407
T. Przeździecki
We consider an"orientifold"generalization of Khovanov-Lauda-Rouquier algebras, depending on a quiver with an involution and a framing. Their representation theory is related, via a Schur-Weyl duality type functor, to Kac-Moody quantum symmetric pairs, and, via a categorification theorem, to highest weight modules over an algebra introduced by Enomoto and Kashiwara. Our first main result is a new shuffle realization of these highest weight modules and a combinatorial construction of their PBW and canonical bases in terms of Lyndon words. Our second main result is a classification of irreducible representations of orientifold KLR algebras and a computation of their global dimension in the case when the framing is trivial.
{"title":"Representations of orientifold\u0000Khovanov–Lauda–Rouquier algebras and the Enomoto–Kashiwara algebra","authors":"T. Przeździecki","doi":"10.2140/pjm.2023.322.407","DOIUrl":"https://doi.org/10.2140/pjm.2023.322.407","url":null,"abstract":"We consider an\"orientifold\"generalization of Khovanov-Lauda-Rouquier algebras, depending on a quiver with an involution and a framing. Their representation theory is related, via a Schur-Weyl duality type functor, to Kac-Moody quantum symmetric pairs, and, via a categorification theorem, to highest weight modules over an algebra introduced by Enomoto and Kashiwara. Our first main result is a new shuffle realization of these highest weight modules and a combinatorial construction of their PBW and canonical bases in terms of Lyndon words. Our second main result is a classification of irreducible representations of orientifold KLR algebras and a computation of their global dimension in the case when the framing is trivial.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43299820","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note, we use Warren-Yuan's super isoperimetric inequality on the level sets of subharmonic functions, which is available only in two dimensions, to derive a modified Hessian bound for solutions of the two dimensional Lagrangian mean curvature equation. We assume the Lagrangian phase to be supercritical with bounded second derivatives. Unlike the previous approach, the simplified approach in this proof does not require the Michael-Simon mean value and Sobolev inequalities on generalized submanifolds of $mathbb{R}^n$.
{"title":"A note on the two-dimensional Lagrangian mean curvature equation","authors":"A. Bhattacharya","doi":"10.2140/pjm.2022.318.43","DOIUrl":"https://doi.org/10.2140/pjm.2022.318.43","url":null,"abstract":"In this note, we use Warren-Yuan's super isoperimetric inequality on the level sets of subharmonic functions, which is available only in two dimensions, to derive a modified Hessian bound for solutions of the two dimensional Lagrangian mean curvature equation. We assume the Lagrangian phase to be supercritical with bounded second derivatives. Unlike the previous approach, the simplified approach in this proof does not require the Michael-Simon mean value and Sobolev inequalities on generalized submanifolds of $mathbb{R}^n$.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48851459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-24DOI: 10.2140/pjm.2022.317.339
Elia Fioravanti, Annette Karrer
Let a finitely generated group $G$ split as a graph of groups. If edge groups are undistorted and do not contribute to the Morse boundary $partial_MG$, we show that every connected component of $partial_MG$ with at least two points originates from the Morse boundary of a vertex group. Under stronger assumptions on the edge groups (such as wideness in the sense of Druc{t}u-Sapir), we show that Morse boundaries of vertex groups are topologically embedded in $partial_MG$.
{"title":"Connected components of Morse boundaries of graphs of groups","authors":"Elia Fioravanti, Annette Karrer","doi":"10.2140/pjm.2022.317.339","DOIUrl":"https://doi.org/10.2140/pjm.2022.317.339","url":null,"abstract":"Let a finitely generated group $G$ split as a graph of groups. If edge groups are undistorted and do not contribute to the Morse boundary $partial_MG$, we show that every connected component of $partial_MG$ with at least two points originates from the Morse boundary of a vertex group. Under stronger assumptions on the edge groups (such as wideness in the sense of Druc{t}u-Sapir), we show that Morse boundaries of vertex groups are topologically embedded in $partial_MG$.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48988941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-21DOI: 10.2140/pjm.2022.321.193
Hongjie Yu
Let F be a global function field with constant field $mathbb{F}_q$. Let G be a reductive group over $mathbb{F}_q$. We establish a variant of Arthur's truncated kernel for G and for its Lie algebra which generalizes Arthur's original construction. We establish a coarse geometric expansion for our variant truncation. As applications, we consider some existence and uniqueness problems of some cuspidal automorphic representations for the functions field of the projective line $mathbb{P}^1_{mathbb{F}_q}$ with two points of ramifications.
{"title":"A coarse geometric expansion of a variant of\u0000Arthur’s truncated traces and some applications","authors":"Hongjie Yu","doi":"10.2140/pjm.2022.321.193","DOIUrl":"https://doi.org/10.2140/pjm.2022.321.193","url":null,"abstract":"Let F be a global function field with constant field $mathbb{F}_q$. Let G be a reductive group over $mathbb{F}_q$. We establish a variant of Arthur's truncated kernel for G and for its Lie algebra which generalizes Arthur's original construction. We establish a coarse geometric expansion for our variant truncation. As applications, we consider some existence and uniqueness problems of some cuspidal automorphic representations for the functions field of the projective line $mathbb{P}^1_{mathbb{F}_q}$ with two points of ramifications.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68571168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-13DOI: 10.2140/pjm.2023.322.301
Yeongseong Jo
Let $F$ be a non-archimedean local field of odd characteristic $p>0$. In this paper, we consider local exterior square $L$-functions $L(s,pi,wedge^2)$, Bump-Friedberg $L$-functions $L(s,pi,BF)$, and Asai $L$-functions $L(s,pi,As)$ of an irreducible admissible representation $pi$ of $GL_m(F)$. In particular, we establish that those $L$-functions, via the theory of integral representations, are equal to their corresponding Artin $L$-functions $L(s,wedge^2(phi(pi)))$, $L(s+1/2,phi(pi))L(s,wedge^2(phi(pi)))$, and $L(s,As(phi(pi)))$ of the associated Langlands parameter $phi(pi)$ under the local Langlands correspondence. These are achieved by proving the identity for irreducible supercuspidal representations, exploiting the local to global argument due to Henniart and Lomeli.
{"title":"Local exterior square and Asai L-functions for\u0000GL(n) in odd characteristic","authors":"Yeongseong Jo","doi":"10.2140/pjm.2023.322.301","DOIUrl":"https://doi.org/10.2140/pjm.2023.322.301","url":null,"abstract":"Let $F$ be a non-archimedean local field of odd characteristic $p>0$. In this paper, we consider local exterior square $L$-functions $L(s,pi,wedge^2)$, Bump-Friedberg $L$-functions $L(s,pi,BF)$, and Asai $L$-functions $L(s,pi,As)$ of an irreducible admissible representation $pi$ of $GL_m(F)$. In particular, we establish that those $L$-functions, via the theory of integral representations, are equal to their corresponding Artin $L$-functions $L(s,wedge^2(phi(pi)))$, $L(s+1/2,phi(pi))L(s,wedge^2(phi(pi)))$, and $L(s,As(phi(pi)))$ of the associated Langlands parameter $phi(pi)$ under the local Langlands correspondence. These are achieved by proving the identity for irreducible supercuspidal representations, exploiting the local to global argument due to Henniart and Lomeli.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45283370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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