Canadian Journal of Mathematics最新文献

{"title":"Fiber functors and reconstruction of Hopf algebras","authors":"Simon Lentner, Martín Mombelli","doi":"10.4153/s0008414x24000531","DOIUrl":"https://doi.org/10.4153/s0008414x24000531","url":null,"abstract":"<p>The main objective of the present paper is to present a version of the Tannaka–Krein type reconstruction theorems: if <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910155856085-0229:S0008414X24000531:S0008414X24000531_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$F:{mathcal B}to {mathcal C}$</span></span></img></span></span> is an exact faithful monoidal functor of tensor categories, one would like to realize <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910155856085-0229:S0008414X24000531:S0008414X24000531_inline2.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal B}$</span></span></img></span></span> as category of representations of a braided Hopf algebra <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910155856085-0229:S0008414X24000531:S0008414X24000531_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$H(F)$</span></span></img></span></span> in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910155856085-0229:S0008414X24000531:S0008414X24000531_inline4.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal C}$</span></span></img></span></span>. We prove that this is the case iff <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910155856085-0229:S0008414X24000531:S0008414X24000531_inline5.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal B}$</span></span></img></span></span> has the additional structure of a monoidal <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910155856085-0229:S0008414X24000531:S0008414X24000531_inline6.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal C}$</span></span></img></span></span>-module category compatible with <span>F</span>, which equivalently means that <span>F</span> admits a monoidal section. For Hopf algebras, this reduces to a version of the Radford projection theorem. The Hopf algebra is constructed through the relative coend for module categories. We expect this basic result to have a wide range of applications, in particular in the absence of fiber functors, and we give some applications. One particular motivation was the logarithmic Kazhdan–Lusztig conjecture.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185043","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}

{"title":"Différentielles quadratiques à singularités prescrites","authors":"Quentin Gendron, Guillaume Tahar","doi":"10.4153/s0008414x24000506","DOIUrl":"https://doi.org/10.4153/s0008414x24000506","url":null,"abstract":"<p>The local invariants of a meromorphic quadratic differential on a compact Riemann surface are the orders of zeros and poles, and the residues at the poles of even orders. The main result of this paper is that with few exceptions, every pattern of local invariants can be obtained by a quadratic differential on some Riemann surface. The exceptions are completely classified and only occur in genera zero and one. Moreover, in the case of a nonconnected stratum, we show that, with three exceptions in genus one, each configuration of invariants can be realized in each non-hyperelliptic connected component of the stratum. In the hyperelliptic components with two poles the residues at both poles coincide. These results are obtained using the flat metric induced by the differentials. We give an application by bounding the number of disjoint cylinders on a primitive quadratic differential.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185044","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}

{"title":"The Galvin property under the ultrapower axiom","authors":"Tom Benhamou, Gabriel Goldberg","doi":"10.4153/s0008414x2400052x","DOIUrl":"https://doi.org/10.4153/s0008414x2400052x","url":null,"abstract":"<p>We continue the study of the Galvin property from Benhamou, Garti, and Shelah (2023, <span>Proceedings of the American Mathematical Society</span> 151, 1301–1309) and Benhamou (2023, <span>Saturation properties in canonical inner models</span>, submitted). In particular, we deepen the connection between certain diamond-like principles and non-Galvin ultrafilters. We also show that any Dodd sound non <span>p</span>-point ultrafilter is non-Galvin. We use these ideas to formulate what appears to be the optimal large cardinal hypothesis implying the existence of a non-Galvin ultrafilter, improving on a result from Benhamou and Dobrinen (2023, <span>Journal of Symbolic Logic</span>, 1–34). Finally, we use a strengthening of the Ultrapower Axiom to prove that in all the known canonical inner models, a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910160454271-0394:S0008414X2400052X:S0008414X2400052X_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$kappa $</span></span></img></span></span>-complete ultrafilter has the Galvin property if and only if it is an iterated sum of <span>p</span>-points.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224136","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}

{"title":"Random walks and the “Euclidean” association scheme in finite vector spaces","authors":"Charles Brittenham, Jonathan Pakianathan","doi":"10.4153/s0008414x24000518","DOIUrl":"https://doi.org/10.4153/s0008414x24000518","url":null,"abstract":"<p>In this paper, we provide an application to the random distance-<span>t</span> walk in finite planes and derive asymptotic formulas (as <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240824023603048-0335:S0008414X24000518:S0008414X24000518_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$q to infty $</span></span></img></span></span>) for the probability of return to start point after <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240824023603048-0335:S0008414X24000518:S0008414X24000518_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$ell $</span></span></img></span></span> steps based on the “vertical” equidistribution of Kloosterman sums established by N. Katz. This work relies on a “Euclidean” association scheme studied in prior work of W. M. Kwok, E. Bannai, O. Shimabukuro, and H. Tanaka. We also provide a self-contained computation of the P-matrix and intersection numbers of this scheme for convenience in our application as well as a more explicit form for the intersection numbers in the planar case.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185045","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}

{"title":"Hilbert direct integrals of monotone operators","authors":"Minh N. Bùi, Patrick L. Combettes","doi":"10.4153/s0008414x2400049x","DOIUrl":"https://doi.org/10.4153/s0008414x2400049x","url":null,"abstract":"<p>Finite Cartesian products of operators play a central role in monotone operator theory and its applications. Extending such products to arbitrary families of operators acting on different Hilbert spaces is an open problem, which we address by introducing the Hilbert direct integral of a family of monotone operators. The properties of this construct are studied, and conditions under which the direct integral inherits the properties of the factor operators are provided. The question of determining whether the Hilbert direct integral of a family of subdifferentials of convex functions is itself a subdifferential leads us to introducing the Hilbert direct integral of a family of functions. We establish explicit expressions for evaluating the Legendre conjugate, subdifferential, recession function, Moreau envelope, and proximity operator of such integrals. Next, we propose a duality framework for monotone inclusion problems involving integrals of linearly composed monotone operators and show its pertinence toward the development of numerical solution methods. Applications to inclusion and variational problems are discussed.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"72 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185046","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}

{"title":"A reproducing kernel approach to Lebesgue decomposition","authors":"Jashan Bal, Robert T.W. Martin, Fouad Naderi","doi":"10.4153/s0008414x24000488","DOIUrl":"https://doi.org/10.4153/s0008414x24000488","url":null,"abstract":"<p>We show that properties of pairs of finite, positive, and regular Borel measures on the complex unit circle such as domination, absolute continuity, and singularity can be completely described in terms of containment and intersection of their reproducing kernel Hilbert spaces of “Cauchy transforms” in the complex unit disk. This leads to a new construction of the classical Lebesgue decomposition and proof of the Radon–Nikodym theorem using reproducing kernel theory and functional analysis.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141171543","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}

{"title":"Invariant scalar-flat Kähler metrics on line bundles over generalized flag varieties","authors":"Qi Yao","doi":"10.4153/s0008414x24000464","DOIUrl":"https://doi.org/10.4153/s0008414x24000464","url":null,"abstract":"<p>Let <span>G</span> be a simply connected semisimple compact Lie group, let <span>X</span> be a simply connected compact Kähler manifold homogeneous under <span>G</span>, and let <span>L</span> be a negative holomorphic line bundle over <span>X</span>. We prove that all <span>G</span>-invariant Kähler metrics on the total space of <span>L</span> arise from the Calabi ansatz. Using this, we show that there exists a unique <span>G</span>-invariant scalar-flat Kähler metric in each <span>G</span>-invariant Kähler class of <span>L</span>. The <span>G</span>-invariant scalar-flat Kähler metrics are automatically asymptotically conical.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141195010","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}

{"title":"The Heisenberg covering of the Fermat curve","authors":"Debargha Banerjee, Loïc Merel","doi":"10.4153/s0008414x24000476","DOIUrl":"https://doi.org/10.4153/s0008414x24000476","url":null,"abstract":"<p>For <span>N</span> integer <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$ge 1$</span></span></img></span></span>, K. Murty and D. Ramakrishnan defined the <span>N</span>th Heisenberg curve, as the compactified quotient <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$X^{prime }_N$</span></span></img></span></span> of the upper half-plane by a certain non-congruence subgroup of the modular group. They ask whether the Manin–Drinfeld principle holds, namely, if the divisors supported on the cusps of those curves are torsion in the Jacobian. We give a model over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$mathbf {Z}[mu _N,1/N]$</span></span></img></span></span> of the <span>N</span>th Heisenberg curve as covering of the <span>N</span>th Fermat curve. We show that the Manin–Drinfeld principle holds for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$N=3$</span></span></img></span></span>, but not for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$N=5$</span></span></img></span></span>. We show that the description by generator and relations due to Rohrlich of the cuspidal subgroup of the Fermat curve is explained by the Heisenberg covering, together with a higher covering of a similar nature. The curves <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$X_N$</span></span></img></span></span> and the classical modular curves <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$X(n)$</span></span></img></span></span>, for <span>n</span> even integer, both dominate <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binar","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"69 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141257462","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}

{"title":"Mirror symmetry and Hitchin system on Deligne–Mumford curves: Strominger–Yau–Zaslow duality","authors":"Yonghong Huang","doi":"10.4153/s0008414x24000439","DOIUrl":"https://doi.org/10.4153/s0008414x24000439","url":null,"abstract":"<p>We systematically study the moduli stacks of Higgs bundles, spectral curves, and Norm maps on Deligne–Mumford curves. As an application, under some mild conditions, we prove the Strominger–Yau–Zaslow duality for the moduli spaces of Higgs bundles over a hyperbolic stacky curve.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"124 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141059516","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}

{"title":"Hyperbolic one-relator groups","authors":"Marco Linton","doi":"10.4153/s0008414x24000427","DOIUrl":"https://doi.org/10.4153/s0008414x24000427","url":null,"abstract":"<p>We introduce two families of two-generator one-relator groups called primitive extension groups and show that a one-relator group is hyperbolic if its primitive extension subgroups are hyperbolic. This reduces the problem of characterizing hyperbolic one-relator groups to characterizing hyperbolic primitive extension groups. These new groups, moreover, admit explicit decompositions as graphs of free groups with adjoined roots. In order to obtain this result, we characterize <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240914045819619-0491:S0008414X24000427:S0008414X24000427_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$2$</span></span></img></span></span>-free one-relator groups with exceptional intersection in terms of Christoffel words, show that hyperbolic one-relator groups have quasi-convex Magnus subgroup, and build upon the one-relator tower machinery developed in previous work of the author.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"103 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249654","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}