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

{"title":"Local spectral estimates and quantitative weak mixing for substitution \u0000 \u0000 Z\u0000 ${mathbb {Z}}$\u0000 -actions","authors":"Alexander I. Bufetov,&nbsp;Juan Marshall-Maldonado,&nbsp;Boris Solomyak","doi":"10.1112/jlms.70136","DOIUrl":"https://doi.org/10.1112/jlms.70136","url":null,"abstract":"<p>The paper investigates Hölder and log-Hölder regularity of spectral measures for weakly mixing substitutions and the related question of quantitative weak mixing. It is assumed that the substitution is primitive, aperiodic, and its substitution matrix is irreducible over the rationals. In the case when there are no eigenvalues of the substitution matrix on the unit circle, Theorem 2.2 says that a weakly mixing substitution <span></span><math>\u0000 <semantics>\u0000 <mi>Z</mi>\u0000 <annotation>${mathbb {Z}}$</annotation>\u0000 </semantics></math>-action has uniformly log-Hölder regular spectral measures, and hence admits power-logarithmic bounds for the rate of weak mixing. In the more delicate Salem substitution case, Theorem 2.5 says that Hölder regularity holds for spectral parameters from the respective number field, but the Hölder exponent cannot be chosen uniformly.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70136","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143770171","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":"A central limit theorem for the matching number of a sparse random graph","authors":"Margalit Glasgow,&nbsp;Matthew Kwan,&nbsp;Ashwin Sah,&nbsp;Mehtaab Sawhney","doi":"10.1112/jlms.70101","DOIUrl":"https://doi.org/10.1112/jlms.70101","url":null,"abstract":"<p>In 1981, Karp and Sipser proved a law of large numbers for the matching number of a sparse Erdős–Rényi random graph, in an influential paper pioneering the so-called <i>differential equation method</i> for analysis of random graph processes. Strengthening this classical result, and answering a question of Aronson, Frieze and Pittel, we prove a central limit theorem in the same setting: the fluctuations in the matching number of a sparse random graph are asymptotically Gaussian. Our new contribution is to prove this central limit theorem in the <i>subcritical</i> and <i>critical</i> regimes, according to a celebrated algorithmic phase transition first observed by Karp and Sipser. Indeed, in the <i>supercritical</i> regime, a central limit theorem has recently been proved in the PhD thesis of Kreačić, using a stochastic generalisation of the differential equation method (comparing the so-called <i>Karp–Sipser process</i> to a system of stochastic differential equations). Our proof builds on these methods, and introduces new techniques to handle certain degeneracies present in the subcritical and critical cases. Curiously, our new techniques lead to a <i>non-constructive</i> result: we are able to characterise the fluctuations of the matching number around its mean, despite these fluctuations being much smaller than the error terms in our best estimates of the mean. We also prove a central limit theorem for the rank of the adjacency matrix of a sparse random graph.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70101","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143762136","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":"Relative cubulation of relative strict hyperbolization","authors":"Jean-François Lafont,&nbsp;Lorenzo Ruffoni","doi":"10.1112/jlms.70093","DOIUrl":"https://doi.org/10.1112/jlms.70093","url":null,"abstract":"<p>We prove that many relatively hyperbolic groups obtained by relative strict hyperbolization admit a cocompact action on a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>CAT</mo>\u0000 <mo>(</mo>\u0000 <mn>0</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$operatorname{CAT}(0)$</annotation>\u0000 </semantics></math> cubical complex. Under suitable assumptions on the peripheral subgroups, these groups are residually finite and even virtually special. We include some applications to the theory of manifolds, such as the construction of new non-positively curved Riemannian manifolds with residually finite fundamental group, and the existence of non-triangulable aspherical manifolds with virtually special fundamental group.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70093","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143741249","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":"Hardy's uncertainty principle for Schrödinger equations with quadratic Hamiltonians","authors":"Elena Cordero,&nbsp;Gianluca Giacchi,&nbsp;Eugenia Malinnikova","doi":"10.1112/jlms.70134","DOIUrl":"https://doi.org/10.1112/jlms.70134","url":null,"abstract":"<p>Hardy's uncertainty principle is a classical result in harmonic analysis, stating that a function in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>d</mi>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$L^2(mathbb {R}^d)$</annotation>\u0000 </semantics></math> and its Fourier transform cannot both decay arbitrarily fast at infinity. In this paper, we extend this principle to the propagators of Schrödinger equations with quadratic Hamiltonians, known in the literature as metaplectic operators. These operators generalize the Fourier transform and have captured significant attention in recent years due to their wide-ranging applications in time-frequency analysis, quantum harmonic analysis, signal processing, and various other fields. However, the involved structure of these operators requires careful analysis, and most results obtained so far concern special propagators that can basically be reduced to rescaled Fourier transforms. The main contributions of this work are threefold: (1) we extend Hardy's uncertainty principle, covering all propagators of Schrödinger equations with quadratic Hamiltonians, (2) we provide concrete examples, such as fractional Fourier transforms, which arise when considering anisotropic harmonic oscillators, (3) we suggest Gaussian decay conditions in certain directions only, which are related to the geometry of the corresponding Hamiltonian flow.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70134","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143741250","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":"A Hörmander–Mikhlin theorem for higher rank simple Lie groups","authors":"José M. Conde-Alonso,&nbsp;Adrián M. González-Pérez,&nbsp;Javier Parcet,&nbsp;Eduardo Tablate","doi":"10.1112/jlms.70137","DOIUrl":"https://doi.org/10.1112/jlms.70137","url":null,"abstract":"<p>We establish regularity conditions for <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>L</mi>\u0000 <mi>p</mi>\u0000 </msub>\u0000 <annotation>$L_p$</annotation>\u0000 </semantics></math>-boundedness of Fourier multipliers on the group von Neumann algebras of higher rank simple Lie groups. This provides a natural Hörmander–Mikhlin (HM) criterion in terms of Lie derivatives of the symbol and a metric given by the adjoint representation. In line with Lafforgue/de la Salle's rigidity theorem, our condition imposes certain decay of the symbol at infinity. It refines and vastly generalizes a recent result by Parcet, Ricard, and de la Salle for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 <msub>\u0000 <mi>L</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>R</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$S L_n(mathbf {R})$</annotation>\u0000 </semantics></math>. Our approach is partly based on a sharp local HM theorem for arbitrary Lie groups, which follows in turn from recent estimates by the authors on singular non-Toeplitz Schur multipliers. We generalize the latter to arbitrary locally compact groups and refine the cocycle-based approach to Fourier multipliers in group algebras by Junge, Mei, and Parcet. A few related open problems are also discussed.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143741467","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 the Iitaka volumes of log canonical surfaces and threefolds","authors":"Guodu Chen,&nbsp;Jingjun Han,&nbsp;Wenfei Liu","doi":"10.1112/jlms.70132","DOIUrl":"https://doi.org/10.1112/jlms.70132","url":null,"abstract":"&lt;p&gt;Given positive integers &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;mo&gt;⩾&lt;/mo&gt;\u0000 &lt;mi&gt;κ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$dgeqslant kappa$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and a subset &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Γ&lt;/mi&gt;\u0000 &lt;mo&gt;⊂&lt;/mo&gt;\u0000 &lt;mo&gt;[&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&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;/mrow&gt;\u0000 &lt;annotation&gt;$Gamma subset [0,1]$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mo&gt;Ivol&lt;/mo&gt;\u0000 &lt;mi&gt;lc&lt;/mi&gt;\u0000 &lt;mi&gt;Γ&lt;/mi&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;κ&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$operatorname{Ivol}_mathrm{lc}^Gamma (d,kappa)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; denote the set of Iitaka volumes of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;annotation&gt;$d$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-dimensional projective log canonical pairs &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;B&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(X, B)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; such that the Iitaka–Kodaira dimension &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;κ&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mi&gt;B&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mi&gt;κ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$kappa (K_X+B)=kappa$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and the coefficients of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;B&lt;/mi&gt;\u0000 &lt;annotation&gt;$B$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; come from &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;Γ&lt;/mi&gt;\u0000 &lt;annotation&gt;$Gamma$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. In this paper, we show that, if &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;Γ&lt;/mi&gt;\u0000 &lt;annotation&gt;$Gamma$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; satisfies the descending chain condition, then so does &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143707271","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 Shehtman's two problems","authors":"Guram Bezhanishvili,&nbsp;Nick Bezhanishvili,&nbsp;Joel Lucero-Bryan,&nbsp;Jan van Mill","doi":"10.1112/jlms.70090","DOIUrl":"https://doi.org/10.1112/jlms.70090","url":null,"abstract":"<p>We provide partial solutions to two problems posed by Shehtman concerning the modal logic of the Čech–Stone compactification of an ordinal space. We use the Continuum Hypothesis to give a finite axiomatization of the modal logic of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>β</mi>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>ω</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$beta (omega ^2)$</annotation>\u0000 </semantics></math>, thus resolving Shehtman's first problem for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$n=2$</annotation>\u0000 </semantics></math>. We also characterize modal logics arising from the Čech–Stone compactification of an ordinal <span></span><math>\u0000 <semantics>\u0000 <mi>γ</mi>\u0000 <annotation>$gamma$</annotation>\u0000 </semantics></math> provided the Cantor normal form of <span></span><math>\u0000 <semantics>\u0000 <mi>γ</mi>\u0000 <annotation>$gamma$</annotation>\u0000 </semantics></math> satisfies an additional condition. This gives a partial solution of Shehtman's second problem.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70090","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143689741","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":"Twisted rational zeros of linear recurrence sequences","authors":"Yuri Bilu,&nbsp;Florian Luca,&nbsp;Joris Nieuwveld,&nbsp;Joël Ouaknine,&nbsp;James Worrell","doi":"10.1112/jlms.70126","DOIUrl":"https://doi.org/10.1112/jlms.70126","url":null,"abstract":"<p>We introduce the notion of a twisted rational zero of a nondegenerate linear recurrence sequence (LRS). We show that any nondegenerate LRS has only finitely many such twisted rational zeros. In the particular case of the Tribonacci sequence, we show that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$1/3$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>5</mn>\u0000 <mo>/</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$-5/3$</annotation>\u0000 </semantics></math> are the only twisted rational zeros that are not integral zeros.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70126","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143689354","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":"Lipschitz decompositions of domains with bilaterally flat boundaries","authors":"Jared Krandel","doi":"10.1112/jlms.70128","DOIUrl":"https://doi.org/10.1112/jlms.70128","url":null,"abstract":"&lt;p&gt;We study classes of domains in &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;mo&gt;⩾&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathbb {R}^{d+1}, d geqslant 2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with sufficiently flat boundaries that admit a decomposition or covering of bounded overlap by Lipschitz graph domains with controlled total surface area. This study is motivated by the following result proved by Peter Jones as a piece of his proof of the Analyst's Traveling Salesman Theorem in the complex plane: Any simply connected domain &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Ω&lt;/mi&gt;\u0000 &lt;mo&gt;⊆&lt;/mo&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Omega subseteq mathbb {C}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with finite boundary length &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;mi&gt;Ω&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;&lt;&lt;/mo&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal {H}^1(partial Omega) &lt; infty$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; can be decomposed into Lipschitz graph domains with total boundary length at most &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;mi&gt;Ω&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Mmathcal {H}^1(partial Omega)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for some &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$M &gt; 0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; independent of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;Ω&lt;/mi&gt;\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70128","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143689332","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":"A presentation of the torus-equivariant quantum \u0000 \u0000 K\u0000 $K$\u0000 -theory ring of flag manifolds of type \u0000 \u0000 A\u0000 $A$\u0000 , Part I: The defining ideal","authors":"Toshiaki Maeno,&nbsp;Satoshi Naito,&nbsp;Daisuke Sagaki","doi":"10.1112/jlms.70095","DOIUrl":"https://doi.org/10.1112/jlms.70095","url":null,"abstract":"<p>We give a presentation of the torus-equivariant (small) quantum <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>-theory ring of flag manifolds of type <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$A$</annotation>\u0000 </semantics></math>, as the quotient of a polynomial ring by an explicit ideal. This result is the torus-equivariant version of our previous one, which gives a presentation of the nonequivariant quantum <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>-theory ring of flag manifolds of type <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$A$</annotation>\u0000 </semantics></math>. However, the method of proof for the torus-equivariant one is entirely different from that for the nonequivariant one; our proof is based on the result in the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Q</mi>\u0000 <mo>=</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$Q = 0$</annotation>\u0000 </semantics></math> limit, and uses Nakayama-type arguments to upgrade it to the quantum situation. Also, in contrast to the nonequivariant case in which we used the Chevalley formula, we make use of the inverse Chevalley formula for the torus-equivariant <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>-group of semi-infinite flag manifolds to obtain relations that yield our presentation.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143689171","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}