We describe combinatorial Hopf (co‐)operadic models for the Swiss Cheese operads built from Feynman diagrams. This extends previous work of Kontsevich and Lambrechts‐Volić for the little disks operads.
{"title":"Models for the n ‐Swiss Cheese operads","authors":"T. Willwacher","doi":"10.1112/tlm3.12031","DOIUrl":"https://doi.org/10.1112/tlm3.12031","url":null,"abstract":"We describe combinatorial Hopf (co‐)operadic models for the Swiss Cheese operads built from Feynman diagrams. This extends previous work of Kontsevich and Lambrechts‐Volić for the little disks operads.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2015-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A bijection ψ is defined between the prime spectrum of quantum SL3 and the Poisson prime spectrum of SL3, and we verify that ψ and ψ−1 both preserve inclusions of primes, i.e. that ψ is in fact a homeomorphism between these two spaces. This is accomplished by developing a Poisson analogue of Brown and Goodearl's framework for describing the Zariski topology of spectra of quantum algebras, and then verifying directly that in the case of SL3 these give rise to identical pictures on both the quantum and Poisson sides. As part of this analysis, we study the Poisson primitive spectrum of O(SL3) and obtain explicit generating sets for all of the Poisson primitive ideals.
{"title":"The prime spectrum of quantum SL3 and the Poisson prime spectrum of its semiclassical limit","authors":"Zee Fryer","doi":"10.1112/tlm3.12004","DOIUrl":"https://doi.org/10.1112/tlm3.12004","url":null,"abstract":"A bijection ψ is defined between the prime spectrum of quantum SL3 and the Poisson prime spectrum of SL3, and we verify that ψ and ψ−1 both preserve inclusions of primes, i.e. that ψ is in fact a homeomorphism between these two spaces. This is accomplished by developing a Poisson analogue of Brown and Goodearl's framework for describing the Zariski topology of spectra of quantum algebras, and then verifying directly that in the case of SL3 these give rise to identical pictures on both the quantum and Poisson sides. As part of this analysis, we study the Poisson primitive spectrum of O(SL3) and obtain explicit generating sets for all of the Poisson primitive ideals.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A general one‐dimensional foliation in the complex projective space has finitely many singularities. For an appropriately good family of subschemes in ℙn , we study the loci in the space of foliations of degree d defined by the requirement that the singularities contain a member of the family. We give a formula for the dimensions of such loci. We show that their degrees are expressed by a polynomial in d . We compute it explicitly in a few examples. Next we provide a formula for the number of isolated singular points of a foliation containing a prescribed positive‐dimensional subscheme in its singular scheme under mild assumptions. We include an appendix by Steven L. Kleiman on a theorem of Bertini suitable for sections of vector bundles with rank equal to the dimension of the base.
复射影空间中的一般一维叶理具有有限多个奇异点。对于一个适当好的子方案族，我们研究了d次叶空间中的位置，该空间由奇异点包含该族的一个成员的要求来定义。我们给出了这类位点的尺寸公式。我们证明了它们的度数是用d的多项式表示的。我们在几个例子中明确地计算它。其次，在温和的假设条件下，我们给出了含有规定的正维子格式的叶理奇异格式的孤立奇点数的公式。我们包括了Steven L. Kleiman关于Bertini定理的附录，该定理适用于秩等于基维数的向量束的截面。
{"title":"Foliations singular along a curve","authors":"I. Vainsencher","doi":"10.1112/tlms/tlv004","DOIUrl":"https://doi.org/10.1112/tlms/tlv004","url":null,"abstract":"A general one‐dimensional foliation in the complex projective space has finitely many singularities. For an appropriately good family of subschemes in ℙn , we study the loci in the space of foliations of degree d defined by the requirement that the singularities contain a member of the family. We give a formula for the dimensions of such loci. We show that their degrees are expressed by a polynomial in d . We compute it explicitly in a few examples. Next we provide a formula for the number of isolated singular points of a foliation containing a prescribed positive‐dimensional subscheme in its singular scheme under mild assumptions. We include an appendix by Steven L. Kleiman on a theorem of Bertini suitable for sections of vector bundles with rank equal to the dimension of the base.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlms/tlv004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We use the framework of reverse mathematics to address the question of, given a mathematical problem, whether or not it is easier to find an infinite partial solution than it is to find a complete solution. Following Flood [‘Reverse mathematics and a Ramsey‐type König's lemma’, J. Symb. Log. 77 (2012) 1272–1280], we say that a Ramsey‐type variant of a problem is the problem with the same instances but whose solutions are the infinite partial solutions to the original problem. We study Ramsey‐type variants of problems related to König's lemma, such as restrictions of König's lemma, Boolean satisfiability problems and graph coloring problems. We find that sometimes the Ramsey‐type variant of a problem is strictly easier than the original problem (as Flood showed with weak König's lemma) and that sometimes the Ramsey‐type variant of a problem is equivalent to the original problem. We show that the Ramsey‐type variant of weak König's lemma is robust in the sense of Montalbán [‘Open questions in reverse mathematics’, Bull. Symb. Log. 17 (2011) 431–454]: it is equivalent to several perturbations. We also clarify the relationship between Ramsey‐type weak König's lemma and algorithmic randomness by showing that Ramsey‐type weak weak König's lemma is equivalent to the problem of finding diagonally non‐recursive functions and that these problems are strictly easier than Ramsey‐type weak König's lemma. This answers a question of Flood.
我们使用反向数学的框架来解决这个问题，给定一个数学问题，是否找到一个无限部分解比找到一个完整解更容易。“逆向数学和Ramsey - type König引理”，J. Symb。Log. 77(2012) 1272-1280]，我们说问题的Ramsey型变体是具有相同实例的问题，但其解是原始问题的无限部分解。我们研究了与König引理相关的问题的Ramsey‐型变体，如König引理的限制、布尔可满足性问题和图着色问题。我们发现有时问题的Ramsey型变体比原始问题严格容易(如Flood用弱König引理所示)，有时问题的Ramsey型变体与原始问题等效。我们证明了弱König引理的Ramsey‐型变体在Montalbán意义上是鲁棒的[' Open questions in reverse mathematics '， Bull]。Symb。Log. 17(2011) 431-454]:相当于几个摄动。我们还阐明了Ramsey - type weak König引理与算法随机性之间的关系，证明了Ramsey - type weak König引理等价于寻找对角非递归函数的问题，并且这些问题严格地比Ramsey - type weak König引理更容易。这回答了洪水的一个问题。
{"title":"On the logical strengths of partial solutions to mathematical problems","authors":"L. Bienvenu, Ludovic Patey, P. Shafer","doi":"10.1112/tlm3.12001","DOIUrl":"https://doi.org/10.1112/tlm3.12001","url":null,"abstract":"We use the framework of reverse mathematics to address the question of, given a mathematical problem, whether or not it is easier to find an infinite partial solution than it is to find a complete solution. Following Flood [‘Reverse mathematics and a Ramsey‐type König's lemma’, J. Symb. Log. 77 (2012) 1272–1280], we say that a Ramsey‐type variant of a problem is the problem with the same instances but whose solutions are the infinite partial solutions to the original problem. We study Ramsey‐type variants of problems related to König's lemma, such as restrictions of König's lemma, Boolean satisfiability problems and graph coloring problems. We find that sometimes the Ramsey‐type variant of a problem is strictly easier than the original problem (as Flood showed with weak König's lemma) and that sometimes the Ramsey‐type variant of a problem is equivalent to the original problem. We show that the Ramsey‐type variant of weak König's lemma is robust in the sense of Montalbán [‘Open questions in reverse mathematics’, Bull. Symb. Log. 17 (2011) 431–454]: it is equivalent to several perturbations. We also clarify the relationship between Ramsey‐type weak König's lemma and algorithmic randomness by showing that Ramsey‐type weak weak König's lemma is equivalent to the problem of finding diagonally non‐recursive functions and that these problems are strictly easier than Ramsey‐type weak König's lemma. This answers a question of Flood.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2014-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Triangles of groups have been introduced by Gersten and Stallings. They are, roughly speaking, a generalization of the amalgamated free product of two groups and occur in the framework of Corson diagrams. First, we prove an intersection theorem for Corson diagrams. Then, we focus on triangles of groups. It has been shown by Howie and Kopteva that the colimit of a hyperbolic triangle of groups contains a non‐abelian free subgroup. We give two natural conditions, each of which ensures that the colimit of a non‐spherical triangle of groups either contains a non‐abelian free subgroup or is virtually solvable.
{"title":"The Tits alternative for non‐spherical triangles of groups","authors":"J. Cuno, J. Lehnert","doi":"10.1112/tlms/tlv005","DOIUrl":"https://doi.org/10.1112/tlms/tlv005","url":null,"abstract":"Triangles of groups have been introduced by Gersten and Stallings. They are, roughly speaking, a generalization of the amalgamated free product of two groups and occur in the framework of Corson diagrams. First, we prove an intersection theorem for Corson diagrams. Then, we focus on triangles of groups. It has been shown by Howie and Kopteva that the colimit of a hyperbolic triangle of groups contains a non‐abelian free subgroup. We give two natural conditions, each of which ensures that the colimit of a non‐spherical triangle of groups either contains a non‐abelian free subgroup or is virtually solvable.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2014-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlms/tlv005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412890","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, universal concentration estimates are established for the local times of random walks on weighted graphs in terms of the resistance metric. As a particular application of these, a modulus of continuity for local times is provided in the case when the graphs in question satisfy a certain volume growth condition with respect to the resistance metric. Moreover, it is explained how these results can be applied to self‐similar fractals, for which they are shown to be useful for deriving scaling limits for local times and asymptotic bounds for the cover time distribution.
{"title":"Moduli of continuity of local times of random walks on graphs in terms of the resistance metric","authors":"D. Croydon","doi":"10.1112/tlms/tlv003","DOIUrl":"https://doi.org/10.1112/tlms/tlv003","url":null,"abstract":"In this article, universal concentration estimates are established for the local times of random walks on weighted graphs in terms of the resistance metric. As a particular application of these, a modulus of continuity for local times is provided in the case when the graphs in question satisfy a certain volume growth condition with respect to the resistance metric. Moreover, it is explained how these results can be applied to self‐similar fractals, for which they are shown to be useful for deriving scaling limits for local times and asymptotic bounds for the cover time distribution.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2014-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlms/tlv003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a new theory of Beurling regular variation (Part II). This includes the previously known theory of Beurling slow variation (Part I) to which we contribute by extending Bloom's theorem. Beurling slow variation arose in the classical theory of Karamata slow and regular variation. We show that the Beurling theory includes the Karamata theory.
{"title":"Beurling slow and regular variation","authors":"N. Bingham, A. Ostaszewski","doi":"10.1112/tlms/tlu002","DOIUrl":"https://doi.org/10.1112/tlms/tlu002","url":null,"abstract":"We give a new theory of Beurling regular variation (Part II). This includes the previously known theory of Beurling slow variation (Part I) to which we contribute by extending Bloom's theorem. Beurling slow variation arose in the classical theory of Karamata slow and regular variation. We show that the Beurling theory includes the Karamata theory.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlms/tlu002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We combine the method of Faltings (Arakelov, Paršin, Szpiro) with the Shimura–Taniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic curves. For several fundamental Diophantine problems, such as for example S ‐unit and Mordell equations, this gives an effective method which does not rely on Diophantine approximation or transcendence techniques.
{"title":"Integral points on moduli schemes of elliptic curves","authors":"Rafael von Känel","doi":"10.1112/tlms/tlu003","DOIUrl":"https://doi.org/10.1112/tlms/tlu003","url":null,"abstract":"We combine the method of Faltings (Arakelov, Paršin, Szpiro) with the Shimura–Taniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic curves. For several fundamental Diophantine problems, such as for example S ‐unit and Mordell equations, this gives an effective method which does not rely on Diophantine approximation or transcendence techniques.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlms/tlu003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study fundamental groups of clique complexes associated to random Erdős–Rényi graphs Γ . We establish thresholds for a number of properties of fundamental groups of these complexes XΓ . In particular, if p=nα , then we show that gdim(π1(XΓ))=cd(π1(XΓ))=1ifα<−12,gdim(π1(XΓ))=cd(π1(XΓ))=2if−12−13 . We prove that for −1130
{"title":"Fundamental groups of clique complexes of random graphs","authors":"A. Costa, M. Farber, Danijela Horak","doi":"10.1112/tlms/tlv001","DOIUrl":"https://doi.org/10.1112/tlms/tlv001","url":null,"abstract":"We study fundamental groups of clique complexes associated to random Erdős–Rényi graphs Γ . We establish thresholds for a number of properties of fundamental groups of these complexes XΓ . In particular, if p=nα , then we show that gdim(π1(XΓ))=cd(π1(XΓ))=1ifα<−12,gdim(π1(XΓ))=cd(π1(XΓ))=2if−12−13 . We prove that for −1130","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2013-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlms/tlv001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The algebraic unknotting number ua(K) of a knot K was introduced by Hitoshi Murakami. It equals the minimal number of crossing changes needed to turn K into an Alexander polynomial one knot. In a previous paper, the authors used the Blanchfield form of a knot K to define an invariant n(K) and proved that n(K)⩽ua(K) . They also showed that n(K) subsumes all previous classical lower bounds on the (algebraic) unknotting number. In this paper, we prove that n(K)=ua(K) .
{"title":"On the algebraic unknotting number","authors":"Maciej Borodzik, Stefan Friedl","doi":"10.1112/tlms/tlu004","DOIUrl":"https://doi.org/10.1112/tlms/tlu004","url":null,"abstract":"The algebraic unknotting number ua(K) of a knot K was introduced by Hitoshi Murakami. It equals the minimal number of crossing changes needed to turn K into an Alexander polynomial one knot. In a previous paper, the authors used the Blanchfield form of a knot K to define an invariant n(K) and proved that n(K)⩽ua(K) . They also showed that n(K) subsumes all previous classical lower bounds on the (algebraic) unknotting number. In this paper, we prove that n(K)=ua(K) .","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2013-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlms/tlu004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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