Journal of the London Mathematical Society-Second Series最新文献

{"title":"On Dedekind's problem, a sparse version of Sperner's theorem, and antichains of a given size in the Boolean lattice","authors":"Matthew Jenssen, Alexandru Malekshahian, Jinyoung Park","doi":"10.1112/jlms.70624","DOIUrl":"https://doi.org/10.1112/jlms.70624","url":null,"abstract":"<p>Dedekind's problem, dating back to 1897, asks for the total number <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ψ</mi>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$psi (n)$</annotation>\u0000 </semantics></math> of antichains contained in the Boolean lattice <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>B</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <annotation>$B_n$</annotation>\u0000 </semantics></math> on <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math> elements. We study Dedekind's problem using a recently developed method based on the cluster expansion from statistical physics, and as a result, obtain several new results on the number and typical structure of antichains in <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>B</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <annotation>$B_n$</annotation>\u0000 </semantics></math>. We obtain detailed estimates for both <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ψ</mi>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$psi (n)$</annotation>\u0000 </semantics></math> and the number of antichains of size <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>β</mi>\u0000 <mfenced>\u0000 <mfrac>\u0000 <mi>n</mi>\u0000 <mrow>\u0000 <mo>⌊</mo>\u0000 <mi>n</mi>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 <mo>⌋</mo>\u0000 </mrow>\u0000 </mfrac>\u0000 </mfenced>\u0000 </mrow>\u0000 <annotation>$beta binom{n}{lfloor n/2 rfloor }$</annotation>\u0000 </semantics></math> for any fixed <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>β</mi>\u0000 <mo>></mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$beta >0$</annotation>\u0000 </semantics></math>. We also establish a sparse version of Sperner's theorem: we determine the sharp threshold and scaling window for the property that almost every antichain of size <span></span><math>\u0000 <semantics>\u0000 <mi>m</mi>\u0000 <annotation>$m$</annotation>\u0000 </semantics><","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70624","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148462892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Wall–chamber decompositions for generalised Monge–Ampère equations","authors":"Sohaib Khalid,&nbsp;Zakarias Sjöström Dyrefelt","doi":"10.1112/jlms.70616","DOIUrl":"https://doi.org/10.1112/jlms.70616","url":null,"abstract":"<p>Generalised Monge–Ampère (gMA) equations form a large class of PDE including Donaldson's J-equation, inverse Hessian equations, some supercritical deformed Hermitian–Yang–Mills (dHYM) equations and some Z-critical equations. Solvability of these equations is characterised by numerical criteria involving intersection numbers over all subvarieties, and in this paper, we aim to characterise algebraically what happens when these non-linear Nakai–Moishezon type criteria fail. As a main result, we show that under mild positivity assumptions, there is a finite number of subvarieties violating the Nakai-Moishezon type criterion, and such subvarieties are rigid in a suitable sense. This gives the first effective solvability criteria for these families of PDE, thus improving on work of Chen, Datar–Pingali, Song and Fang–Ma, and provides the first existence results in higher dimensions of a natural PDE analogue of Bridgeland's locally finite wall–chamber decomposition.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70616","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148386704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An elegant model of the geodesic flow on the modular surface","authors":"Pierre Arnoux,&nbsp;Thomas A. Schmidt","doi":"10.1112/jlms.70588","DOIUrl":"https://doi.org/10.1112/jlms.70588","url":null,"abstract":"<p>Caroline Series' [<i>The modular surface and continued fractions</i>, J. Lond. Math. Soc. (2), <b>31</b>, no. 1, (1985), 69–80] gives a clear framework linking, in a deceptively simple way, the dynamics of the geodesic flow on the modular surface with the dynamics of the regular continued fraction, through a well-chosen symbolic coding. It has been called <i>required reading</i> for those interested in the symbolic dynamics of geodesic flows, and has had consequences in symbolic dynamics, ergodic theory, hyperbolic geometry, and continued fraction theory. In this overview, we give an indication of why this is so, sketch some of the history related to the paper, and also point to some later works.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70588","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148386703","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Collapse in noncommutative geometry and spectral continuity","authors":"Carla Farsi,&nbsp;Frédéric Latrémolière","doi":"10.1112/jlms.70628","DOIUrl":"https://doi.org/10.1112/jlms.70628","url":null,"abstract":"&lt;p&gt;In the classical realm of Riemannian geometry, informally, if two manifolds are close in the Gromov–Hausdorff distance, and belong to a class of closed manifolds with bounded curvature and diameter, then the spectra of their Laplacian or Dirac operators are also close under many scenarii. Of particular interest is the case where a sequence of manifolds converges for the Gromov–Hausdorff distance to a manifold of lower dimension, that is, &lt;i&gt;collapses&lt;/i&gt; to the limit. The question then arises of the continuity, in some sense, of the geometrically relevant operators and their spectra. This has been investigated at length in many classical papers. In the more general context of noncommutative geometry, the notion of convergence with respect to the variation of certain metric structures plays a very important role, as it allows the approximation of spaces and algebras over Hilbert spaces by simpler ones (e.g., finite-dimensional and/or matricial). So, if two quantum compact metric spaces are close in the metric sense, then how similar are they, as noncommutative spaces? In this paper, we initiate the study of the continuity of spectra and other properties of metric spectral triples on noncommutative &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;annotation&gt;$G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-bundles, for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;annotation&gt;$G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; compact Lie, under collapse in the “vertical” direction. As a first step in this study, we work with the &lt;i&gt;spectral propinquity&lt;/i&gt;, an analog of the Gromov–Hausdorff distance for metric spectral triples introduced by the second author. The spectral propinquity is a form of metric for differential structures. Inspired by results from collapse in Riemannian geometry, we study spectral triples that decompose, in some sense, in a vertical and a horizontal direction; they can be shown to be special Kasparov products. We perturb the (metric in the) vertical component by a parameter &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;annotation&gt;$varepsilon$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, and then we take the limit for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;annotation&gt;$varepsilon$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; approaching zero, thus obtaining a metric spectral triples convergence results. As a consequence, by the work of the second author, we also derive continuity results for the spectra of the Dirac operators of these spectral triples. Examples of applications of our work include collapse of products of spectral triples with one Abelian factor, spectral triples associated to principal &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;U&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148386657","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Simultaneous nonvanishing of Dirichlet \u0000 \u0000 L\u0000 $L$\u0000 -functions in Galois orbits","authors":"Hung M. Bui,&nbsp;Alexandra Florea,&nbsp;Hieu T. Ngo","doi":"10.1112/jlms.70611","DOIUrl":"https://doi.org/10.1112/jlms.70611","url":null,"abstract":"&lt;p&gt;Under the Generalized Riemann Hypothesis, we prove that given any two distinct imprimitive Dirichlet characters &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;η&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;η&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$eta _1, eta _2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; modulo &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$q=p^k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, a positive proportion of characters &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;χ&lt;/mi&gt;\u0000 &lt;annotation&gt;$chi$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; modulo &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;annotation&gt;$q$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; in a fixed Galois orbit of primitive characters satisfy the nonvanishing property that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;χ&lt;/mi&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;η&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;χ&lt;/mi&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;η&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;≠&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$L(1/2,chi eta _1) L(1/2,chi eta _2) ne 0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, as &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$k rightarrow infty$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; (with &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; fixed). Previously, only a positive proportion of nonvanishing result was available in Galois orbits (as opposed to simultane","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148386761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On elusive elliptic integrals","authors":"D. Masser,&nbsp;U. Zannier","doi":"10.1112/jlms.70625","DOIUrl":"https://doi.org/10.1112/jlms.70625","url":null,"abstract":"<p>In a recent paper, we found two counterexamples to a specialization conjecture of James Davenport involving integrals on algebraic curves. We showed that every such elliptic counterexample must involve complex multiplication; ours corresponded to the fields <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Q</mi>\u0000 <mo>(</mo>\u0000 <msqrt>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msqrt>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${bf Q}(sqrt {-1})$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Q</mi>\u0000 <mo>(</mo>\u0000 <msqrt>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 </msqrt>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${bf Q}(sqrt {-3})$</annotation>\u0000 </semantics></math>. Here, we prove that every imaginary <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Q</mi>\u0000 <mo>(</mo>\u0000 <msqrt>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 </msqrt>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${bf Q}(sqrt {-d})$</annotation>\u0000 </semantics></math> also turns up. This therefore becomes analogous to the situation with relative Manin-Mumford discovered by Daniel Bertrand, even though our underlying algebraic groups are rather different from his. We also give a complete description of all elliptic counterexamples (our paper could handle only one <span></span><math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$d$</annotation>\u0000 </semantics></math> at a time). A consequence is that there is a counterexample on an algebraic curve if and only if its Jacobian contains an elliptic curve with complex multiplication.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148386658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Generalised spin Calogero–Moser systems from Cherednik algebras","authors":"Misha Feigin,&nbsp;Mikhail Vasilev,&nbsp;Martin Vrabec","doi":"10.1112/jlms.70589","DOIUrl":"https://doi.org/10.1112/jlms.70589","url":null,"abstract":"<p>Integrable spin Calogero–Moser type systems with non-symmetric configurations of the singularities of the potential appeared in the work of Chalykh, Goncharenko and Veselov in 1999. We obtain various generalisations of these examples by making use of the representation theory of Cherednik algebras.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70589","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148386695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quantum \u0000 \u0000 K\u0000 $K$\u0000 -invariants via Quot schemes II","authors":"Shubham Sinha,&nbsp;Ming Zhang","doi":"10.1112/jlms.70623","DOIUrl":"https://doi.org/10.1112/jlms.70623","url":null,"abstract":"<p>We derive a <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>-theoretic analog of the Vafa–Intriligator formula, computing the (virtual) Euler characteristics of vector bundles over the Quot scheme that compactifies the space of degree <span></span><math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$d$</annotation>\u0000 </semantics></math> morphisms from a fixed projective curve to the Grassmannian <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Gr</mi>\u0000 <mo>(</mo>\u0000 <mi>r</mi>\u0000 <mo>,</mo>\u0000 <mi>N</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathrm{Gr}(r,N)$</annotation>\u0000 </semantics></math>. As an application, we deduce interesting vanishing results, used in our previous work to study the quantum <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>-ring of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Gr</mi>\u0000 <mo>(</mo>\u0000 <mi>r</mi>\u0000 <mo>,</mo>\u0000 <mi>N</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathrm{Gr}(r,N)$</annotation>\u0000 </semantics></math>. In the genus-zero case, we prove a simplified formula involving Schur functions, consistent with the Borel–Weil–Bott theorem in the degree-zero setting. These new formulas offer a novel approach for computing the structure constants of quantum <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>-products.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148386305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"String topology via the coHochschild complex and local intersections","authors":"Manuel Rivera,&nbsp;Alex Takeda","doi":"10.1112/jlms.70620","DOIUrl":"https://doi.org/10.1112/jlms.70620","url":null,"abstract":"<p>We construct an algebraic model for the Chas–Sullivan product and the Goresky–Hingston coproduct in string topology. The construction takes as its initial input a simplicial complex equipped with a local pairing on its simplicial chains, for instance, a homology manifold with its local intersection pairing. We define the two string topology operations on the coHochschild complex of a suitable coalgebra of chains, making use of local higher homotopies that control the compatibility of the pairing with the diagonal approximation coproduct. In the case of a closed oriented smooth manifold, we prove that our algebraic operations coincide, up to chain homotopy, with their geometric counterparts. The local nature of our constructions allows for arguments based on the method of acyclic models.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70620","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148386306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Inverse problems of polynomial and rational matrices involving the eigenstructure and data of the column and row spaces","authors":"Itziar Baragaña,&nbsp;Froilán M. Dopico,&nbsp;Silvia Marcaida,&nbsp;Alicia Roca","doi":"10.1112/jlms.70614","DOIUrl":"https://doi.org/10.1112/jlms.70614","url":null,"abstract":"<p>The eigenstructure of a rational matrix is comprised by its invariant rational functions, both finite and at infinity, and by the minimal indices of its left and right null spaces. These quantities arise in many applications and have been thoroughly studied in numerous references. There are two other fundamental subspaces of rational matrices that, in contrast, have received much less attention: the column and row spaces. They also have their associated minimal indices, which play a key role in the problem of factorizing polynomial matrices into low-degree factors. This work solves the problems of finding necessary and sufficient conditions for the existence of rational matrices in two scenarios: when the invariant rational functions and the minimal bases or minimal indices of the column and row spaces are prescribed, and when the eigenstructure together with the minimal indices of the column and row spaces are prescribed. The particular, but extremely important, cases of these problems for polynomial matrices are solved first and are the main tool for solving the general problems. The results in this work complete and nontrivially extend the necessary and sufficient conditions recently obtained for the existence of polynomial and rational matrices when only the eigenstructure is prescribed.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70614","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148386872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}