Advances in Mathematics最新文献

{"title":"Lax functorialities of the comma construction for ω-categories","authors":"Dimitri Ara ,&nbsp;Léonard Guetta","doi":"10.1016/j.aim.2025.110762","DOIUrl":"10.1016/j.aim.2025.110762","url":null,"abstract":"<div><div>Motivated by the Grothendieck construction, we study the functorialities of the comma construction for strict <em>ω</em>-categories. To state the most general functorialities, we use the language of Gray <em>ω</em>-categories, that is, categories enriched in the category of strict <em>ω</em>-categories endowed with the oplax Gray tensor product. Our main result is that the comma construction of strict <em>ω</em>-categories defines a Gray <em>ω</em>-functor, that is, a morphism of Gray <em>ω</em>-categories. To makes sense of this statement, we prove that slices of Gray <em>ω</em>-categories exist. Coming back to the Grothendieck construction, we propose a definition in terms of the comma construction and, as a consequence, we get that the Grothendieck construction of strict <em>ω</em>-categories defines a Gray <em>ω</em>-functor. Finally, as a by-product, we get a notion of Grothendieck construction for Gray <em>ω</em>-functors, which we plan to investigate in future work.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110762"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981068","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 refined analyticity radius of 3-D generalized Navier-Stokes equations","authors":"Dong Li ,&nbsp;Ping Zhang","doi":"10.1016/j.aim.2025.110769","DOIUrl":"10.1016/j.aim.2025.110769","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We analyze the instantaneous growth of analyticity radius for three dimensional generalized Navier-Stokes equations. For the subcritical &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; case with &lt;span&gt;&lt;math&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;, we prove that there exists a positive time &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; so that for any &lt;span&gt;&lt;math&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, the radius of analyticity of the solution &lt;em&gt;u&lt;/em&gt; satisfies the pointwise-in-time lower bound&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;rad&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; as &lt;span&gt;&lt;math&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;. This in particular gives a nontrivial improvement of the previous result by Herbst and Skibsted in &lt;span&gt;&lt;span&gt;[17]&lt;/span&gt;&lt;/span&gt; for the case &lt;span&gt;&lt;math&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and also settles the decade-long open question in &lt;span&gt;&lt;span&gt;[17]&lt;/span&gt;&lt;/span&gt;, namely, whether or not&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;munder&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi&gt;lim&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mi&gt;inf&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi&gt;rad&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; for all &lt;span&gt;&lt;math&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;. In the critical case &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; we prove that there exists &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; so that for any &lt;span&gt;&lt;math&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"487 ","pages":"Article 110769"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928074","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":"Stability of the favorable Falkner-Skan profiles for the stationary Prandtl equations","authors":"Sameer Iyer","doi":"10.1016/j.aim.2025.110749","DOIUrl":"10.1016/j.aim.2025.110749","url":null,"abstract":"<div><div>The (favorable) Falkner-Skan boundary layer profiles are a one parameter (<span><math><mi>β</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo></math></span>) family of self-similar solutions to the stationary Prandtl system which describes the flow over a wedge with angle <span><math><mi>β</mi><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. The most famous member of this family is the endpoint Blasius profile, <span><math><mi>β</mi><mo>=</mo><mn>0</mn></math></span>, which exhibits pressureless flow over a flat plate. In contrast, the <span><math><mi>β</mi><mo>&gt;</mo><mn>0</mn></math></span> profiles are physically expected to exhibit a <em>favorable pressure gradient</em>, a common adage in the physics literature. In this work, we prove quantitative scattering estimates as <span><math><mi>x</mi><mo>→</mo><mo>∞</mo></math></span> which precisely captures the effect of this favorable gradient through the presence of new “CK” (Cauchy-Kovalevskaya) terms that appear in a quasilinear energy cascade.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"487 ","pages":"Article 110749"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886094","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":"The defect of the F-pure threshold","authors":"Alessandro De Stefani ,&nbsp;Luis Núñez-Betancourt ,&nbsp;Ilya Smirnov","doi":"10.1016/j.aim.2026.110792","DOIUrl":"10.1016/j.aim.2026.110792","url":null,"abstract":"<div><div>Introduced by Takagi and Watanabe, F-pure thresholds are invariants defined in terms of the Frobenius homomorphism. While they find applications in various settings, they are primarily used as a <em>local</em> invariant. The purpose of this note is to start filling this gap by opening the study of its behavior on a scheme. To this end, we define the <em>defect of the F-pure threshold</em> of a local ring <span><math><mo>(</mo><mi>R</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> by setting <span><math><mi>dfpt</mi><mo>(</mo><mi>R</mi><mo>)</mo><mo>=</mo><mi>dim</mi><mo>⁡</mo><mo>(</mo><mi>R</mi><mo>)</mo><mo>−</mo><mi>fpt</mi><mo>(</mo><mi>m</mi><mo>)</mo></math></span>. It turns out that this invariant defines an upper semi-continuous function on a scheme and satisfies Bertini-type theorems. We also study the behavior of the defect of the F-pure threshold under flat extensions and after blowing up the maximal ideal of a local ring.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110792"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039203","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":"A relative Nash-Tognoli theorem over Q and application to the Q-algebraicity problem","authors":"Enrico Savi","doi":"10.1016/j.aim.2025.110757","DOIUrl":"10.1016/j.aim.2025.110757","url":null,"abstract":"<div><div>We prove a relative version over <span><math><mi>Q</mi></math></span> of Nash-Tognoli theorem, that is: Let <em>M</em> be a compact smooth manifold with closed smooth submanifolds <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> in general position, then there exists a nonsingular real algebraic set <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with nonsingular algebraic subsets <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>ℓ</mi></mrow><mrow><mo>′</mo></mrow></msubsup></math></span> and a diffeomorphism <span><math><mi>h</mi><mo>:</mo><mi>M</mi><mo>→</mo><msup><mrow><mi>M</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> such that <span><math><mi>h</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>=</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup></math></span> for all <span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>ℓ</mi></math></span> such that <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>ℓ</mi></mrow><mrow><mo>′</mo></mrow></msubsup></math></span> are described, both globally and locally, by polynomial equations with rational coefficients. In addition, if <span><math><mi>M</mi><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> are nonsingular algebraic sets, then we prove the diffeomorphism <span><math><mi>h</mi><mo>:</mo><mi>M</mi><mo>→</mo><msup><mrow><mi>M</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> can be chosen semialgebraic and the result can be extended to the noncompact case. In the proof we describe also the <span><math><mi>Z</mi><mo>/</mo><mn>2</mn><mi>Z</mi></math></span>-homological cycles of real embedded Grassmannian manifolds by nonsingular algebraic representatives over <span><math><mi>Q</mi></math></span> via the Bott-Samelson resolution of Schubert varieties.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"487 ","pages":"Article 110757"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886096","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 related to regular representations of SL2","authors":"Dražen Adamović ,&nbsp;Antun Milas","doi":"10.1016/j.aim.2025.110763","DOIUrl":"10.1016/j.aim.2025.110763","url":null,"abstract":"<div><div>We construct a family of potentially quasi-lisse (non-rational) vertex algebras, denoted by <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, <span><math><mi>p</mi><mo>≥</mo><mn>2</mn></math></span>, which are closely related to the vertex algebra of chiral differential operators on <span><math><mi>S</mi><mi>L</mi><mo>(</mo><mn>2</mn><mo>)</mo></math></span> at level <span><math><mo>−</mo><mn>2</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></math></span>. We prove that for <span><math><mi>p</mi><mo>=</mo><mn>3</mn></math></span>, there is an isomorphism between <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> and the affine vertex algebra <span><math><msub><mrow><mi>L</mi></mrow><mrow><mo>−</mo><mn>5</mn><mo>/</mo><mn>3</mn></mrow></msub><mo>(</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> from the Cvitanović-Deligne series. Moreover, we also establish isomorphisms between <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> and certain affine <em>W</em>-algebras of types <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>8</mn></mrow></msub></math></span>, respectively. In this way, we resolve the problem of decomposing certain conformal embeddings of affine vertex algebras into affine <em>W</em>-algebras. An important feature is that <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> is <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mi>Z</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow></msub></math></span>-graded with finite-dimensional graded subspaces and convergent characters. Therefore, for all <span><math><mi>p</mi><mo>≥</mo><mn>2</mn></math></span>, we show that the characters of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> exhibit modularity, supporting the conjectural quasi-lisse property.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"487 ","pages":"Article 110763"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927995","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":"Topological Hochschild homology of the image of J","authors":"David Jongwon Lee ,&nbsp;Ishan Levy","doi":"10.1016/j.aim.2025.110759","DOIUrl":"10.1016/j.aim.2025.110759","url":null,"abstract":"<div><div>We compute the mod <span><math><mo>(</mo><mi>p</mi><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> and mod <span><math><mo>(</mo><mn>2</mn><mo>,</mo><mi>η</mi><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> THH of many variants of the image-of-<em>J</em> spectrum. In particular, we do this for <span><math><msub><mrow><mi>j</mi></mrow><mrow><mi>ζ</mi></mrow></msub></math></span>, whose TC is closely related to the <em>K</em>-theory of the <span><math><mi>K</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>-local sphere. We find in particular that the failure for THH to satisfy <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-Galois descent for the extension <span><math><msub><mrow><mi>j</mi></mrow><mrow><mi>ζ</mi></mrow></msub><mo>→</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> corresponds to the failure of the <em>p</em>-adic circle to be its own free loop space. For <span><math><mi>p</mi><mo>&gt;</mo><mn>2</mn></math></span>, we also prove the Segal conjecture for <span><math><msub><mrow><mi>j</mi></mrow><mrow><mi>ζ</mi></mrow></msub></math></span>, and we compute the <em>K</em>-theory of the <span><math><mi>K</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>-local sphere in degrees <span><math><mo>≤</mo><mn>4</mn><mi>p</mi><mo>−</mo><mn>6</mn></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"487 ","pages":"Article 110759"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886098","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":"Corrigendum to “Notes on Plücker's relations in geometric algebra” [Advances in Mathematics 363 (2020) 106959]","authors":"Garret Sobczyk","doi":"10.1016/j.aim.2026.110801","DOIUrl":"10.1016/j.aim.2026.110801","url":null,"abstract":"","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110801"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039195","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":"Counting rationals and diophantine approximation in missing-digit Cantor sets","authors":"Sam Chow ,&nbsp;Péter P. Varjú ,&nbsp;Han Yu","doi":"10.1016/j.aim.2026.110807","DOIUrl":"10.1016/j.aim.2026.110807","url":null,"abstract":"<div><div>We establish a new upper bound for the number of rationals up to a given height in a missing-digit set, making progress towards a conjecture of Broderick, Fishman, and Reich. This enables us to make novel progress towards another conjecture of those authors about the corresponding intrinsic diophantine approximation problem. Moreover, we make further progress towards conjectures of Bugeaud–Durand and Levesley–Salp–Velani on the distribution of diophantine exponents in missing-digit sets.</div><div>A key tool in our study is Fourier <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> dimension introduced by the last named author in Yu (2021) <span><span>[12]</span></span>. An important technical contribution of the paper is a method to compute this quantity.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110807"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039196","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":"Bounded common fundamental domains for two lattices","authors":"Sigrid Grepstad ,&nbsp;Mihail N. Kolountzakis","doi":"10.1016/j.aim.2025.110776","DOIUrl":"10.1016/j.aim.2025.110776","url":null,"abstract":"<div><div>We prove that for any two lattices <span><math><mi>L</mi><mo>,</mo><mi>M</mi><mo>⊆</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> of the same volume there exists a measurable, bounded, common fundamental domain of them. In other words, there exists a bounded measurable set <span><math><mi>E</mi><mo>⊆</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> such that <em>E</em> tiles <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> when translated by <em>L</em> or by <em>M</em>. A consequence of this is that the indicator function of <em>E</em> forms a Weyl–Heisenberg (Gabor) orthogonal basis of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> when translated by <em>L</em> and modulated by <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, the dual lattice of <em>M</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"487 ","pages":"Article 110776"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928073","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}