Advances in Mathematics最新文献

{"title":"Shifted symplectic structures on derived Quot-stacks II – derived Quot-schemes as dg manifolds","authors":"Dennis Borisov ,&nbsp;Ludmil Katzarkov ,&nbsp;Artan Sheshmani","doi":"10.1016/j.aim.2024.110092","DOIUrl":"10.1016/j.aim.2024.110092","url":null,"abstract":"<div><div>It is proved that derived <span><math><mi>Q</mi><mi>u</mi><mi>o</mi><mi>t</mi></math></span>-schemes, as defined by Ciocan-Fontanine and Kapranov, are represented by dg manifolds of finite type. This is the second part of a work aimed to analyze shifted symplectic structures on moduli spaces of coherent sheaves on Calabi–Yau manifolds. The first part related dg manifolds to derived schemes as defined by Toën and Vezzosi.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110092"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150249","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":"Dual boundary complexes of Betti moduli spaces over the two-sphere with one irregular singularity","authors":"Tao Su","doi":"10.1016/j.aim.2024.110101","DOIUrl":"10.1016/j.aim.2024.110101","url":null,"abstract":"<div><div>The weak geometric P=W conjecture of L. Katzarkov, A. Noll, P. Pandit, and C. Simpson states that, a smooth Betti moduli space of complex dimension <em>d</em> over a punctured Riemann surface has the dual boundary complex homotopy equivalent to a sphere of dimension <span><math><mi>d</mi><mo>−</mo><mn>1</mn></math></span>. Via a microlocal geometric perspective, we verify this conjecture for a class of rank <em>n</em> wild character varieties over the two-sphere with one puncture, associated with any Stokes Legendrian link defined by an <em>n</em>-strand positive braid.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110101"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150250","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 partition algebra and the plethysm coefficients II: Ramified plethysm","authors":"Chris Bowman ,&nbsp;Rowena Paget ,&nbsp;Mark Wildon","doi":"10.1016/j.aim.2024.110090","DOIUrl":"10.1016/j.aim.2024.110090","url":null,"abstract":"<div><div>The plethysm coefficient <span><math><mi>p</mi><mo>(</mo><mi>ν</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> is the multiplicity of the Schur function <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span> in the plethysm product <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>ν</mi></mrow></msub><mo>∘</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>μ</mi></mrow></msub></math></span>. In this paper we use Schur–Weyl duality between wreath products of symmetric groups and the ramified partition algebra to interpret an arbitrary plethysm coefficient as the multiplicity of an appropriate composition factor in the restriction of a module for the ramified partition algebra to the partition algebra. This result implies new stability phenomenon for plethysm coefficients when the first parts of <em>ν</em>, <em>μ</em> and <em>λ</em> are all large. In particular, it gives the first positive formula in the case when <em>ν</em> and <em>λ</em> are arbitrary and <em>μ</em> has one part. Corollaries include new explicit positive formulae and combinatorial interpretations for the plethysm coefficients <span><math><mi>p</mi><mo>(</mo><mo>(</mo><mi>n</mi><mo>−</mo><mi>b</mi><mo>,</mo><mi>b</mi><mo>)</mo><mo>,</mo><mo>(</mo><mi>m</mi><mo>)</mo><mo>,</mo><mo>(</mo><mi>m</mi><mi>n</mi><mo>−</mo><mi>r</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>)</mo></math></span>, and <span><math><mi>p</mi><mo>(</mo><mo>(</mo><mi>n</mi><mo>−</mo><mi>b</mi><mo>,</mo><msup><mrow><mn>1</mn></mrow><mrow><mi>b</mi></mrow></msup><mo>)</mo><mo>,</mo><mo>(</mo><mi>m</mi><mo>)</mo><mo>,</mo><mo>(</mo><mi>m</mi><mi>n</mi><mo>−</mo><mi>r</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>)</mo></math></span> when <em>m</em> and <em>n</em> are large.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110090"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143149611","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":"Inflations for representations of shifted quantum affine algebras","authors":"Théo Pinet","doi":"10.1016/j.aim.2024.110093","DOIUrl":"10.1016/j.aim.2024.110093","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Fix a finite-dimensional simple Lie algebra &lt;span&gt;&lt;math&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be a Lie subalgebra coming from a Dynkin diagram inclusion. Then, the corresponding restriction functor is not essentially surjective on finite-dimensional simple &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;–modules.&lt;/div&gt;&lt;div&gt;In this article, we study Finkelberg–Tsymbaliuk's shifted quantum affine algebras &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and the associated categories &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; (defined by Hernandez). In particular, we introduce natural subalgebras &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and obtain a functor &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; from &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;⨁&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; to &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;⨁&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mtext&gt;-Mod&lt;/mtext&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; using the canonical restriction functors. We then establish that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is essentially surjective on finite-dimensional simple objects by constructing notable preimages that we call &lt;em&gt;inflations&lt;/em&gt;.&lt;/div&gt;&lt;div&gt;We conjecture that all simple objects in &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; (which is the analog of &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; for the subalgebras &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;) admit some inflation and prove this for &lt;span&gt;&lt;math&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of type A–B or &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; a direct sum of copies of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;sl&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;sl&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110093"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143149614","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":"BKP-affine coordinates and emergent geometry of generalized Brézin-Gross-Witten tau-functions","authors":"Zhiyuan Wang ,&nbsp;Chenglang Yang ,&nbsp;Qingsheng Zhang","doi":"10.1016/j.aim.2024.110100","DOIUrl":"10.1016/j.aim.2024.110100","url":null,"abstract":"<div><div>Following Zhou's framework, we consider the emergent geometry of the generalized Brézin-Gross-Witten models whose partition functions are known to be a family of tau-functions of the BKP hierarchy. More precisely, we construct a spectral curve together with its special deformation, and show that the Eynard-Orantin topological recursion on this spectral curve emerges naturally from the Virasoro constraints for the generalized BGW tau-functions. Moreover, we give the explicit expressions for the BKP-affine coordinates of these tau-functions and their generating series. The BKP-affine coordinates and the topological recursion provide two different approaches towards the concrete computations of the connected <em>n</em>-point functions. Finally, we show that the quantum spectral curve of type <em>B</em> in the sense of Gukov-Sułkowski emerges from the BKP-affine coordinates and Eynard-Orantin topological recursion.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110100"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150244","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":"Equivariant enumerative geometry","authors":"Thomas Brazelton","doi":"10.1016/j.aim.2024.110082","DOIUrl":"10.1016/j.aim.2024.110082","url":null,"abstract":"<div><div>We formulate an <em>equivariant conservation of number</em>, which proves that a generalized Euler number of a complex equivariant vector bundle can be computed as a sum of local indices of an arbitrary section. This involves an expansion of the Pontryagin–Thom transfer in the equivariant setting. We leverage this result to commence a study of enumerative geometry in the presence of a group action. As an illustration of the power of this machinery, we prove that any smooth complex cubic surface defined by a symmetric polynomial has 27 lines whose orbit types under the <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-action on <span><math><mi>C</mi><msup><mrow><mtext>P</mtext></mrow><mrow><mn>3</mn></mrow></msup></math></span> are given by <span><math><mo>[</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>/</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>]</mo><mo>+</mo><mo>[</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>/</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>]</mo><mo>+</mo><mo>[</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>/</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>8</mn></mrow></msub><mo>]</mo></math></span>, where <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math></span> denote two non-conjugate cyclic subgroups of order two. As a consequence we demonstrate that a real symmetric cubic surface can only contain 3 or 27 real lines.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110082"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164284","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":"Homology cobordism and the geometry of hyperbolic three-manifolds","authors":"Francesco Lin","doi":"10.1016/j.aim.2024.110087","DOIUrl":"10.1016/j.aim.2024.110087","url":null,"abstract":"<div><div>A major challenge in the study of the structure of the three-dimensional homology cobordism group is to understand the interaction between hyperbolic geometry and homology cobordism. In this paper, for a hyperbolic homology sphere <em>Y</em> we derive explicit bounds on the relative grading between irreducible solutions to the Seiberg-Witten equations and the reducible one in terms of the spectral and Riemannian geometry of <em>Y</em>. Using this, we provide explicit bounds on some numerical invariants arising in monopole Floer homology (and its <span><math><mrow><mi>Pin</mi></mrow><mo>(</mo><mn>2</mn><mo>)</mo></math></span>-equivariant refinement). We apply this to study the subgroups of the homology cobordism group generated by hyperbolic homology spheres satisfying certain natural geometric constraints.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110087"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143149266","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":"Anisotropic symmetrization, convex bodies, and isoperimetric inequalities","authors":"Gabriele Bianchi,&nbsp;Andrea Cianchi,&nbsp;Paolo Gronchi","doi":"10.1016/j.aim.2024.110085","DOIUrl":"10.1016/j.aim.2024.110085","url":null,"abstract":"<div><div>This work is concerned with a Pólya-Szegö type inequality for anisotropic functionals of Sobolev functions. The relevant inequality entails a double-symmetrization involving both trial functions and functionals. A new approach uncovering geometric aspects of the inequality is proposed. It relies upon anisotropic isoperimetric inequalities, fine properties of Sobolev functions, and results from the Brunn-Minkowski theory of convex bodies. Importantly, unlike previously available proofs, the one offered in this paper does not require approximation arguments and hence allows for a characterization of extremal functions.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110085"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150242","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":"Tetrahedron instantons in Donaldson-Thomas theory","authors":"Nadir Fasola ,&nbsp;Sergej Monavari","doi":"10.1016/j.aim.2024.110099","DOIUrl":"10.1016/j.aim.2024.110099","url":null,"abstract":"<div><div>Inspired by the work of Pomoni-Yan-Zhang in String Theory, we introduce the moduli space of tetrahedron instantons as a Quot scheme on a singular threefold and describe it as a moduli space of quiver representations. We construct a virtual fundamental class and virtual structure sheaf à la Oh-Thomas, by which we define <em>K</em>-theoretic invariants. We show that the partition function of such invariants reproduces the one studied by Pomoni-Yan-Zhang, and explicitly determine it, as a product of shifted partition functions of rank one Donaldson-Thomas invariants of the three-dimensional affine space. Our geometric construction answers a series of questions of Pomoni-Yan-Zhang on the geometry of the moduli space of tetrahedron instantons and the behaviour of its partition function, and provides a new application of the recent work of Oh-Thomas.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110099"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150245","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":"The dispersion of dilated lacunary sequences, with applications in multiplicative Diophantine approximation","authors":"Eduard Stefanescu","doi":"10.1016/j.aim.2024.110062","DOIUrl":"10.1016/j.aim.2024.110062","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> be a lacunary sequence satisfying the Hadamard gap condition. We give upper bounds for the maximal gap of the set of dilates <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>α</mi><mo>}</mo></mrow><mrow><mi>n</mi><mo>≤</mo><mi>N</mi></mrow></msub></math></span> modulo 1, in terms of <em>N</em>. For any lacunary sequence <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> we prove the existence of a dilation factor <em>α</em> such that the maximal gap is of order at most <span><math><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>N</mi><mo>)</mo><mo>/</mo><mi>N</mi></math></span>, and we prove that for Lebesgue almost all <em>α</em> the maximal gap is of order at most <span><math><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>N</mi><mo>)</mo></mrow><mrow><mn>2</mn><mo>+</mo><mi>ε</mi></mrow></msup><mo>/</mo><mi>N</mi></math></span>. The metric result is generalized to other measures satisfying a certain Fourier decay assumption. Both upper bounds are optimal up to a factor of logarithmic order, and the latter result improves a recent result of Chow and Technau. Finally, we show that our result implies an improved upper bound in the inhomogeneous version of Littlewood's problem in multiplicative Diophantine approximation.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110062"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164283","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}