Advances in Mathematics最新文献

{"title":"Simultaneous approximation in nilsystems and the multiplicative thickness of return-time sets","authors":"Daniel Glasscock","doi":"10.1016/j.aim.2024.109936","DOIUrl":"10.1016/j.aim.2024.109936","url":null,"abstract":"<div><p>In the topological dynamical system <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>T</mi><mo>)</mo></math></span>, a point <em>x</em> simultaneously approximates a point <em>y</em> if there exists a sequence <span><math><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, …of natural numbers for which <span><math><msup><mrow><mi>T</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msup><mi>x</mi></math></span>, <span><math><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msup><mi>x</mi></math></span>, …, <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>k</mi><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msup><mi>x</mi></math></span> all tend to <em>y</em>. In 1978, Furstenberg and Weiss showed that every system possesses a point which simultaneously approximates itself (a multiply recurrent point) and deduced refinements of van der Waerden's theorem on arithmetic progressions. In this paper, we study the denseness of the set of points that are simultaneously approximated by a given point. We show that in a minimal nilsystem, all points simultaneously approximate a <em>δ</em>-dense set of points under a necessarily restricted set of powers of <em>T</em>. We tie this theorem to the multiplicative combinatorial properties of return-time sets, showing that all nil-Bohr sets and typical return-time sets in a minimal system are multiplicatively thick in a coset of a multiplicative subsemigroup of the natural numbers. This yields an inhomogeneous multiple recurrence result that generalizes Furstenberg and Weiss' theorem and leads to new enhancements of van der Waerden's theorem. This work relies crucially on continuity in the prolongation relation (the closure of the orbit-closure relation) developed by Auslander, Akin, and Glasner; the theory of rational points and polynomials on nilmanifolds developed by Leibman, Green, and Tao; and the machinery of topological characteristic factors developed recently by Glasner, Huang, Shao, Weiss, and Ye.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"457 ","pages":"Article 109936"},"PeriodicalIF":1.5,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142162102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the minimal power of q in a Kazhdan–Lusztig polynomial","authors":"Christian Gaetz ,&nbsp;Yibo Gao","doi":"10.1016/j.aim.2024.109941","DOIUrl":"10.1016/j.aim.2024.109941","url":null,"abstract":"<div><p>For <em>w</em> in the symmetric group, we provide an exact formula for the smallest positive power <span><math><msup><mrow><mi>q</mi></mrow><mrow><mi>h</mi><mo>(</mo><mi>w</mi><mo>)</mo></mrow></msup></math></span> appearing in the Kazhdan–Lusztig polynomial <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>e</mi><mo>,</mo><mi>w</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span>. We also provide a tight upper bound on <span><math><mi>h</mi><mo>(</mo><mi>w</mi><mo>)</mo></math></span> in simply-laced types, resolving a conjecture of Billey–Postnikov from 2002.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"457 ","pages":"Article 109941"},"PeriodicalIF":1.5,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142162115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Skeleta and categories of algebras","authors":"Jonathan Beardsley,&nbsp;Tyler Lawson","doi":"10.1016/j.aim.2024.109944","DOIUrl":"10.1016/j.aim.2024.109944","url":null,"abstract":"<div><p>We define a notion of a connectivity structure on an ∞-category, analogous to a <em>t</em>-structure but applicable in unstable contexts—such as spaces, or algebras over an operad. This allows us to generalize notions of n-skeleta, minimal skeleta, and cellular approximation from the category of spaces. For modules over an Eilenberg–Mac Lane spectrum, these are closely related to the notion of projective amplitude.</p><p>We apply these to ring spectra, where they can be detected via the cotangent complex and higher Hochschild homology with coefficients. We show that the spectra <span><math><mi>Y</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> of chromatic homotopy theory are minimal skeleta for <span><math><mi>H</mi><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> in the category of associative ring spectra. Similarly, Ravenel's spectra <span><math><mi>T</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> are shown to be minimal skeleta for <em>BP</em> in the same way, which proves that these admit canonical associative algebra structures.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"457 ","pages":"Article 109944"},"PeriodicalIF":1.5,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142162103","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Models of Abelian varieties over valued fields, using model theory","authors":"Yatir Halevi","doi":"10.1016/j.aim.2024.109938","DOIUrl":"10.1016/j.aim.2024.109938","url":null,"abstract":"<div><p>Given an elliptic curve <em>E</em> over a perfect defectless henselian valued field <span><math><mo>(</mo><mi>F</mi><mo>,</mo><mrow><mi>val</mi></mrow><mo>)</mo></math></span> with perfect residue field <span><math><msub><mrow><mtext>k</mtext></mrow><mrow><mi>F</mi></mrow></msub></math></span> and valuation ring <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>F</mi></mrow></msub></math></span>, there exists an integral separated smooth group scheme <span><math><mi>E</mi></math></span> over <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>F</mi></mrow></msub></math></span> with <span><math><mi>E</mi><msub><mrow><mo>×</mo></mrow><mrow><mtext>Spec </mtext><msub><mrow><mi>O</mi></mrow><mrow><mi>F</mi></mrow></msub></mrow></msub><mtext>Spec </mtext><mi>F</mi><mo>≅</mo><mi>E</mi></math></span>. If <span><math><mrow><mi>char</mi></mrow><mo>(</mo><msub><mrow><mtext>k</mtext></mrow><mrow><mi>F</mi></mrow></msub><mo>)</mo><mo>≠</mo><mn>2</mn><mo>,</mo><mn>3</mn></math></span> then one can be found over <span><math><msub><mrow><mi>O</mi></mrow><mrow><msup><mrow><mi>F</mi></mrow><mrow><mi>a</mi><mi>l</mi><mi>g</mi></mrow></msup></mrow></msub></math></span> such that the definable group <span><math><mi>E</mi><mo>(</mo><mi>O</mi><mo>)</mo></math></span> is the maximal generically stable subgroup of <em>E</em>. We also give some partial results on general Abelian varieties over <em>F</em>.</p><p>The construction of <span><math><mi>E</mi></math></span> is by means of generating a birational group law over <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>F</mi></mrow></msub></math></span> by the aid of a generically stable generic type of a definable subgroup of <em>E</em>.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"457 ","pages":"Article 109938"},"PeriodicalIF":1.5,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142162116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the non-degenerate and degenerate generic singularities formed by mean curvature flow","authors":"Zhou Gang","doi":"10.1016/j.aim.2024.109937","DOIUrl":"10.1016/j.aim.2024.109937","url":null,"abstract":"<div><p>We study a neighborhood of generic singularities formed by mean curvature flow (MCF). For various possibilities when the singularities are modeled on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>×</mo><mi>R</mi></math></span>, we provide a detailed description for a small, but fixed, neighborhood of singularity, including proving that a small neighborhood is mean convex, and the singularity is isolated. For the remaining possibilities, we conjecture that an entire neighborhood of the singularity becomes singular at the time of blowup, and present evidence to support it. A key technique is that, when looking for a dominating direction for the rescaled MCF, we need a normal form transformation, as a result, the rescaled MCF is parametrized over some chosen curved cylinder, instead of a standard straight one.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"457 ","pages":"Article 109937"},"PeriodicalIF":1.5,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142162104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Incompressible tensor categories","authors":"Kevin Coulembier,&nbsp;Pavel Etingof,&nbsp;Victor Ostrik","doi":"10.1016/j.aim.2024.109935","DOIUrl":"10.1016/j.aim.2024.109935","url":null,"abstract":"&lt;div&gt;&lt;p&gt;A symmetric tensor category &lt;span&gt;&lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; over an algebraically closed field &lt;strong&gt;k&lt;/strong&gt; is called &lt;strong&gt;incompressible&lt;/strong&gt; if its objects have finite length (&lt;span&gt;&lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is pretannakian) and every tensor functor out of &lt;span&gt;&lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is an embedding of a tensor subcategory. E.g., the categories &lt;span&gt;&lt;math&gt;&lt;mi&gt;Vec&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;sVec&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of vector and supervector spaces are incompressible. Moreover, by Deligne's theorem &lt;span&gt;&lt;span&gt;[15]&lt;/span&gt;&lt;/span&gt;, if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;char&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; then any tensor category of moderate growth uniquely fibres over &lt;span&gt;&lt;math&gt;&lt;mi&gt;sVec&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. This implies that &lt;span&gt;&lt;math&gt;&lt;mi&gt;Vec&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;sVec&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; are the only incompressible categories over &lt;strong&gt;k&lt;/strong&gt; in this class, and perhaps altogether, as we expect that all incompressible categories have moderate growth.&lt;/p&gt;&lt;p&gt;Similarly, in characteristic &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, we also have incompressible Verlinde categories &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Ver&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;Ver&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt;, and by &lt;span&gt;&lt;span&gt;[10]&lt;/span&gt;&lt;/span&gt; any Frobenius exact category of moderate growth uniquely fibres over &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Ver&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, meaning that, in this class, the above categories are the only incompressible ones. More generally, the Verlinde categories &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Ver&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;Ver&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; introduced in &lt;span&gt;&lt;span&gt;[3]&lt;/span&gt;&lt;/span&gt;, &lt;span&gt;&lt;span&gt;[7]&lt;/span&gt;&lt;/span&gt; are incompressible, and a key conjecture is that every tensor category of moderate growth uniquely fibres over &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Ver&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. This would make the above the only incompressible categories in this class (and perhaps altogether).&lt;/p&gt;&lt;p&gt;We prove a part of this conjecture, showing that every tensor category of moderate growth fibres over an incompressible one. So it remains to understand incompressible categories, and we prove several results in this direction. Namely, let &lt;span&gt;&lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-Tann be the category of tensor categories that fibre over &lt;span&gt;&lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Then we say that &lt;span&gt;&lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is &lt;strong&gt;subterminal&lt;/strong&gt; if it is a terminal object o","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"457 ","pages":"Article 109935"},"PeriodicalIF":1.5,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S000187082400450X/pdfft?md5=7430145ca98feac2b6b24320a9ad17ce&pid=1-s2.0-S000187082400450X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142147908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Genus zero transverse foliations for weakly convex Reeb flows on the tight 3-sphere","authors":"Naiara V. de Paulo ,&nbsp;Umberto Hryniewicz ,&nbsp;Seongchan Kim ,&nbsp;Pedro A.S. Salomão","doi":"10.1016/j.aim.2024.109909","DOIUrl":"10.1016/j.aim.2024.109909","url":null,"abstract":"<div><p>A contact form on the tight 3-sphere <span><math><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> is called weakly convex if the Conley-Zehnder index of every Reeb orbit is at least 2. In this article, we study Reeb flows of weakly convex contact forms on <span><math><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> admitting a prescribed finite set of index-2 Reeb orbits, which are all hyperbolic and mutually unlinked. We present conditions so that these index-2 orbits are binding orbits of a genus zero transverse foliation whose additional binding orbits have index 3. In addition, we show in the real-analytic case that the topological entropy of the Reeb flow is positive if the branches of the stable/unstable manifolds of the index-2 orbits are mutually non-coincident.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"457 ","pages":"Article 109909"},"PeriodicalIF":1.5,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0001870824004249/pdfft?md5=f4345f01030ab23c2fdfe8102aa8fbb7&pid=1-s2.0-S0001870824004249-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142147907","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Almost splitting maps, transformation theorems and smooth fibration theorems","authors":"Hongzhi Huang ,&nbsp;Xian-Tao Huang","doi":"10.1016/j.aim.2024.109914","DOIUrl":"10.1016/j.aim.2024.109914","url":null,"abstract":"<div><p>In this paper, we introduce a notion, called generalized Reifenberg condition, under which we prove a smooth fibration theorem for collapsed manifolds with Ricci curvature bounded below, which gives a unified proof of smooth fibration theorems in many previous works (including the ones proved by Fukaya and Yamaguchi respectively). A key tool in the proof of this fibration theorem is the transformation technique for almost splitting maps, which originates from Cheeger-Naber (<span><span>[16]</span></span>) and Cheeger-Jiang-Naber (<span><span>[14]</span></span>). More precisely, we show that a transformation theorem of Cheeger-Jiang-Naber (see Proposition 7.7 in <span><span>[14]</span></span>) holds for possibly collapsed manifolds. Some other applications of the transformation theorems are given in this paper.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"457 ","pages":"Article 109914"},"PeriodicalIF":1.5,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142137458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Vertex algebras with big centre and a Kazhdan-Lusztig correspondence","authors":"Boris L. Feigin ,&nbsp;Simon D. Lentner","doi":"10.1016/j.aim.2024.109904","DOIUrl":"10.1016/j.aim.2024.109904","url":null,"abstract":"<div><p>We study the semiclassical limit <span><math><mi>κ</mi><mo>→</mo><mo>∞</mo></math></span> of the generalized quantum Langlands kernel associated to a Lie algebra <span><math><mi>g</mi></math></span> and an integer level <em>p</em>. This vertex algebra acquires a big centre, containing the ring of functions over the space of <span><math><mi>g</mi></math></span>-connections. We conjecture that the fibre over the zero connection is the Feigin-Tipunin vertex algebra, whose category of representations should be equivalent to the small quantum group, and that the other fibres are precisely its twisted modules, and that the entire category of representations is related to the quantum group with a big centre. In this sense we present a generalized Kazhdan-Lusztig conjecture, involving deformations by any <span><math><mi>g</mi></math></span>-connection. We prove our conjectures in small cases <span><math><mo>(</mo><mi>g</mi><mo>,</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mn>2</mn><mo>)</mo></math></span> by explicitly computing all vertex algebras and categories involved.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"457 ","pages":"Article 109904"},"PeriodicalIF":1.5,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142130176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Ergodic theory on coded shift spaces","authors":"Tamara Kucherenko ,&nbsp;Martin Schmoll ,&nbsp;Christian Wolf","doi":"10.1016/j.aim.2024.109913","DOIUrl":"10.1016/j.aim.2024.109913","url":null,"abstract":"<div><p>We study ergodic-theoretic properties of coded shift spaces. A coded shift space is defined as a closure of all bi-infinite concatenations of words from a fixed countable generating set. We derive sufficient conditions for the uniqueness of measures of maximal entropy and equilibrium states of Hölder continuous potentials based on the partition of the coded shift into its concatenation set (sequences that are concatenations of generating words) and its residual set (sequences added under the closure). In this case we provide a simple explicit description of the measure of maximal entropy. We also obtain flexibility results for the entropy on the concatenation and residual sets. Finally, we prove a local structure theorem for intrinsically ergodic coded shift spaces which shows that our results apply to a larger class of coded shift spaces compared to previous works by Climenhaga <span><span>[9]</span></span>, Climenhaga and Thompson <span><span>[10]</span></span>, <span><span>[11]</span></span>, and Pavlov <span><span>[25]</span></span>.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"457 ","pages":"Article 109913"},"PeriodicalIF":1.5,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142130097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}