Pub Date : 2021-12-21DOI: 10.2140/pjm.2022.319.343
Jason Joseph, J. Meier, Maggie Miller, Alexander Zupan
We seek to connect ideas in the theory of bridge trisections with other well-studied facets of classical knotted surface theory. First, we show how the normal Euler number can be computed from a tri-plane diagram, and we use this to give a trisection-theoretic proof of the Whitney-Massey Theorem, which bounds the possible values of this number in terms of the Euler characteristic. Second, we describe in detail how to compute the fundamental group and related invariants from a tri-plane diagram, and we use this, together with an analysis of bridge trisections of ribbon surfaces, to produce an infinite family of knotted spheres that admit non-isotopic bridge trisections of minimal complexity.
{"title":"Bridge trisections and classical knotted surface theory","authors":"Jason Joseph, J. Meier, Maggie Miller, Alexander Zupan","doi":"10.2140/pjm.2022.319.343","DOIUrl":"https://doi.org/10.2140/pjm.2022.319.343","url":null,"abstract":"We seek to connect ideas in the theory of bridge trisections with other well-studied facets of classical knotted surface theory. First, we show how the normal Euler number can be computed from a tri-plane diagram, and we use this to give a trisection-theoretic proof of the Whitney-Massey Theorem, which bounds the possible values of this number in terms of the Euler characteristic. Second, we describe in detail how to compute the fundamental group and related invariants from a tri-plane diagram, and we use this, together with an analysis of bridge trisections of ribbon surfaces, to produce an infinite family of knotted spheres that admit non-isotopic bridge trisections of minimal complexity.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48501151","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}
Daniel Ballesteros-Chavez, W. Klingenberg, Ben Lambert
. We study the isometric spacelike embedding problem in scaled de Sitter space, and obtain Weyl-type estimates and the corresponding closedness in the space of embeddings.
{"title":"Weyl estimates for spacelike hypersurfaces in de Sitter space","authors":"Daniel Ballesteros-Chavez, W. Klingenberg, Ben Lambert","doi":"10.2140/pjm.2022.320.1","DOIUrl":"https://doi.org/10.2140/pjm.2022.320.1","url":null,"abstract":". We study the isometric spacelike embedding problem in scaled de Sitter space, and obtain Weyl-type estimates and the corresponding closedness in the space of embeddings.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43479448","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 study Hecke pairs using the coarse geometry of their coset space and their Schlichting completion. We prove new stability results for the Baum-Connes and the Novikov conjectures in the case where the pair is co-Haagerup. This allows to generalize previous results, while providing new examples of groups satisfying the Baum-Connes conjecture with coefficients. For instance, we show that for some S-arithmetic subgroups of Sp(5,1) and Sp(3,1) the conjecture with coefficients holds.
{"title":"Coarse geometry of Hecke pairs and the\u0000Baum–Connes conjecture","authors":"Cl'ement Dell'Aiera","doi":"10.2140/pjm.2023.322.21","DOIUrl":"https://doi.org/10.2140/pjm.2023.322.21","url":null,"abstract":"We study Hecke pairs using the coarse geometry of their coset space and their Schlichting completion. We prove new stability results for the Baum-Connes and the Novikov conjectures in the case where the pair is co-Haagerup. This allows to generalize previous results, while providing new examples of groups satisfying the Baum-Connes conjecture with coefficients. For instance, we show that for some S-arithmetic subgroups of Sp(5,1) and Sp(3,1) the conjecture with coefficients holds.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44379227","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}
The space of all weakly holomorphic modular forms and the space of all holomorphic period functions of a fixed weight for the modular group are realized as locally convex topological vector spaces that are topologically dual to each other. This framework is used to study the kernel and range of a linear differential operator that preserves modularity and to define and describe its adjoint. The main result is an index formula for such a differential operator that is holomorphic at infinity. The co-kernel of the operator is identified as a cohomology group of the modular group acting on the kernel.
{"title":"The index of a modular differential operator","authors":"W. Duke","doi":"10.2140/pjm.2021.315.45","DOIUrl":"https://doi.org/10.2140/pjm.2021.315.45","url":null,"abstract":"The space of all weakly holomorphic modular forms and the space of all holomorphic period functions of a fixed weight for the modular group are realized as locally convex topological vector spaces that are topologically dual to each other. This framework is used to study the kernel and range of a linear differential operator that preserves modularity and to define and describe its adjoint. The main result is an index formula for such a differential operator that is holomorphic at infinity. The co-kernel of the operator is identified as a cohomology group of the modular group acting on the kernel.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42694080","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":"Bessel quotients and Robin eigenvalues","authors":"P. Freitas","doi":"10.2140/pjm.2021.315.75","DOIUrl":"https://doi.org/10.2140/pjm.2021.315.75","url":null,"abstract":"","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44081923","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}
. The puzzle method was introduced by Choi and Park as an effective method for finding non-singular characteristic maps over wedged simplicial complexes K ( J ) obtained from a given simplicial complex K . We study further the mod 2 case of the puzzle method. We firstly describe it completely in terms of linear algebraic language which allows us to develop a constructive puzzle algorithm. We also analyze our algorithm and compare its performances with other known algorithms including the Garrison and Scott algorithm.
{"title":"An algorithmic strategy for finding characteristic maps over wedged simplicial complexes","authors":"Suyoung Choi, Mathieu Vall'ee","doi":"10.2140/pjm.2022.320.13","DOIUrl":"https://doi.org/10.2140/pjm.2022.320.13","url":null,"abstract":". The puzzle method was introduced by Choi and Park as an effective method for finding non-singular characteristic maps over wedged simplicial complexes K ( J ) obtained from a given simplicial complex K . We study further the mod 2 case of the puzzle method. We firstly describe it completely in terms of linear algebraic language which allows us to develop a constructive puzzle algorithm. We also analyze our algorithm and compare its performances with other known algorithms including the Garrison and Scott algorithm.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42558482","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 : 2021-11-10DOI: 10.2140/pjm.2021.314.311
Naveed Hussain, S. Yau, Huaiqing Zuo
. Let ( V, 0) = { ( z 1 , · · · , z n ) ∈ C n : f ( z 1 , · · · , z n ) = 0 } be an isolated hypersurface singularity with mult ( f ) = m . Let J k ( f ) be the ideal generated by all k -th order partial derivative of f . For 1 ≤ k ≤ m − 1, the new object L k ( V ) is defined to be the Lie algebra of derivations of the new k -th local algebra M k ( V ), where M k ( V ) := O n / ( f + J 1 ( f ) + · · · + J k ( f )). Its dimension is denoted as δ k ( V ). This number δ k ( V ) is a new numerical analytic invariant. In this article we compute L 3 ( V ) for fewnomial isolated singularities (binomial, trinomial) and obtain the formulas of δ 3 ( V ). We also formulate a sharp upper estimate conjecture for the δ k ( V ) of weighted homogeneous isolated hypersurface singularities and verify this conjecture for large class of singularities. Furthermore, we formulate another inequality conjecture: δ ( k +1) ( V ) < δ k ( V ) , k ≥ 1 and verify it for low-dimensional fewnomial singularities.
。设(V, 0) = {(z 1，···，z n)∈C n: f (z 1，···，z n) = 0}是一个孤立的超曲面奇点，其mult (f) = m。设J k (f)是由f的所有k阶偏导数生成的理想。对于1≤k≤m−1，定义新对象lk (V)为新k -局部代数m k (V)的派生李代数，其中m k (V):= O n / (f + J 1 (f) +···+ J k (f))。其维数记为δ k (V)。这个数字δ k (V)是一个新的数值解析不变量。本文计算了几种孤立奇异点(二项式、三项式)的δ 3 (V)，得到了δ 3 (V)的表达式。我们还对加权齐次孤立超曲面奇点的δ k (V)提出了一个尖锐的上估计猜想，并对大类奇点进行了验证。进一步，我们提出了另一个不等式猜想:δ (k +1) (V) < δ k (V)， k≥1，并对低维的少量奇异点进行了验证。
{"title":"Derivation Lie algebras of new k-th local\u0000algebras of isolated hypersurface singularities","authors":"Naveed Hussain, S. Yau, Huaiqing Zuo","doi":"10.2140/pjm.2021.314.311","DOIUrl":"https://doi.org/10.2140/pjm.2021.314.311","url":null,"abstract":". Let ( V, 0) = { ( z 1 , · · · , z n ) ∈ C n : f ( z 1 , · · · , z n ) = 0 } be an isolated hypersurface singularity with mult ( f ) = m . Let J k ( f ) be the ideal generated by all k -th order partial derivative of f . For 1 ≤ k ≤ m − 1, the new object L k ( V ) is defined to be the Lie algebra of derivations of the new k -th local algebra M k ( V ), where M k ( V ) := O n / ( f + J 1 ( f ) + · · · + J k ( f )). Its dimension is denoted as δ k ( V ). This number δ k ( V ) is a new numerical analytic invariant. In this article we compute L 3 ( V ) for fewnomial isolated singularities (binomial, trinomial) and obtain the formulas of δ 3 ( V ). We also formulate a sharp upper estimate conjecture for the δ k ( V ) of weighted homogeneous isolated hypersurface singularities and verify this conjecture for large class of singularities. Furthermore, we formulate another inequality conjecture: δ ( k +1) ( V ) < δ k ( V ) , k ≥ 1 and verify it for low-dimensional fewnomial singularities.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42555464","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 : 2021-11-10DOI: 10.2140/pjm.2021.314.259
D. Dummit, H. Kisilevsky
{"title":"The unit signature rank deficiency is unbounded over cyclotomic fields","authors":"D. Dummit, H. Kisilevsky","doi":"10.2140/pjm.2021.314.259","DOIUrl":"https://doi.org/10.2140/pjm.2021.314.259","url":null,"abstract":"","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43914084","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 : 2021-10-31DOI: 10.2140/pjm.2022.321.359
Eoin Mackall
We provide an alternative proof that the Chow group of $1$-cycles on a Severi--Brauer variety associated to a biquaternion division algebra is torsion-free. There are three proofs of this result in the literature, all of which are due to Karpenko and rely on a clever use of $K$-theory. The proof that we give here, by contrast, is geometric and uses degenerations of quartic elliptic normal curves.
{"title":"On the Chow groups of a biquaternion\u0000Severi–Brauer variety","authors":"Eoin Mackall","doi":"10.2140/pjm.2022.321.359","DOIUrl":"https://doi.org/10.2140/pjm.2022.321.359","url":null,"abstract":"We provide an alternative proof that the Chow group of $1$-cycles on a Severi--Brauer variety associated to a biquaternion division algebra is torsion-free. There are three proofs of this result in the literature, all of which are due to Karpenko and rely on a clever use of $K$-theory. The proof that we give here, by contrast, is geometric and uses degenerations of quartic elliptic normal curves.","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":"45818265","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 : 2021-10-31DOI: 10.2140/pjm.2022.321.415
Ehsan Shahoseini, A. Rajaei, A. Maarefparvar
For $L/K$ a finite Galois extension of number fields, the relative P'olya group $Po(L/K)$ coincides with the group of strongly ambiguous ideal classes in $L/K$. In this paper, using a well known exact sequence related to $Po(L/K)$, in the works of Brumer-Rosen and Zantema, we find short proofs for some classical results in the literatur. Then we define the ``Ostrowski quotient'' $Ost(L/K)$ as the cokernel of the capitulation map into $Po(L/K)$, and generalize some known results for $Po(L/mathbb{Q})$ to $Ost(L/K)$.
{"title":"Ostrowski quotients for finite extensions of number fields","authors":"Ehsan Shahoseini, A. Rajaei, A. Maarefparvar","doi":"10.2140/pjm.2022.321.415","DOIUrl":"https://doi.org/10.2140/pjm.2022.321.415","url":null,"abstract":"For $L/K$ a finite Galois extension of number fields, the relative P'olya group $Po(L/K)$ coincides with the group of strongly ambiguous ideal classes in $L/K$. In this paper, using a well known exact sequence related to $Po(L/K)$, in the works of Brumer-Rosen and Zantema, we find short proofs for some classical results in the literatur. Then we define the ``Ostrowski quotient'' $Ost(L/K)$ as the cokernel of the capitulation map into $Po(L/K)$, and generalize some known results for $Po(L/mathbb{Q})$ to $Ost(L/K)$.","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":"42229788","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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