Advances in Mathematics最新文献

{"title":"Existence and uniqueness of the Levi-Civita connection on noncommutative differential forms","authors":"Bram Mesland ,&nbsp;Adam Rennie","doi":"10.1016/j.aim.2025.110207","DOIUrl":"10.1016/j.aim.2025.110207","url":null,"abstract":"<div><div>We combine Hilbert module and algebraic techniques to give necessary and sufficient conditions for the existence of an Hermitian torsion-free connection on the bimodule of differential one-forms of a first order differential calculus. In the presence of the extra structure of a bimodule connection, we give sufficient conditions for uniqueness.</div><div>We prove that any <em>θ</em>-deformation of a compact Riemannian manifold admits a unique Hermitian torsion-free bimodule connection and provide an explicit construction of it. Specialising to classical Riemannian manifolds yields a novel construction of the Levi-Civita connection on the cotangent bundle.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"468 ","pages":"Article 110207"},"PeriodicalIF":1.5,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143637760","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":"A Tukia-type theorem for nilpotent Lie groups and quasi-isometric rigidity of solvable groups","authors":"Tullia Dymarz ,&nbsp;David Fisher ,&nbsp;Xiangdong Xie","doi":"10.1016/j.aim.2025.110202","DOIUrl":"10.1016/j.aim.2025.110202","url":null,"abstract":"<div><div>In this paper we study uniform quasiconformal groups of Carnot-by-Carnot groups. We show that they can be conjugated into conformal groups provided the induced action on the space of distinct pairs is cocompact. Following the approach of Eskin-Fisher-Whyte <span><span>[17]</span></span> these results have applications to quasi-isometric rigidity of certain families of solvable groups.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"468 ","pages":"Article 110202"},"PeriodicalIF":1.5,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143636832","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":"Birman-Hilden theory for 3-manifolds","authors":"Trent Lucas","doi":"10.1016/j.aim.2025.110204","DOIUrl":"10.1016/j.aim.2025.110204","url":null,"abstract":"<div><div>Given a branched cover of manifolds, one can lift homeomorphisms along the cover to obtain a (virtual) homomorphism between mapping class groups. Following a question of Margalit-Winarski, we study the injectivity of this lifting map in the case of 3-manifolds. We show that in contrast to the case of surfaces, the lifting map is generally not injective for most regular branched covers of 3-manifolds. This includes the double cover of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> branched over the unlink, which generalizes the hyperelliptic branched cover of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. In this case, we find a finite normal generating set for the kernel of the lifting map.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"468 ","pages":"Article 110204"},"PeriodicalIF":1.5,"publicationDate":"2025-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143601058","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 decompositions of curvature measures via Hausdorff measures on singular sets of convex bodies","authors":"Xudong Wang,&nbsp;Baocheng Zhu","doi":"10.1016/j.aim.2025.110205","DOIUrl":"10.1016/j.aim.2025.110205","url":null,"abstract":"<div><div>This paper proves decomposition formulas for the curvature measures of convex bodies. The decomposition of a curvature measure of a convex body is with respect to Hausdorff measures of different dimensions restricted to the singular sets of the boundary of the convex body. The density functions and singular measures in the decomposition are explicitly given in terms of integrals of functions of the generalized curvatures of the convex body. A similar decomposition for curvature measures defined on the unit sphere is also proved.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"468 ","pages":"Article 110205"},"PeriodicalIF":1.5,"publicationDate":"2025-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143601059","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":"Iterated satellite operators on the knot concordance group","authors":"Jae Choon Cha ,&nbsp;Taehee Kim","doi":"10.1016/j.aim.2025.110203","DOIUrl":"10.1016/j.aim.2025.110203","url":null,"abstract":"<div><div>We show that for a winding number zero satellite operator <em>P</em> on the knot concordance group, if the axis of <em>P</em> has nontrivial self-pairing under the Blanchfield form of the pattern, then the image of the iteration <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> generates an infinite rank subgroup for each <em>n</em>. Furthermore, the graded quotients of the filtration of the knot concordance group associated with <em>P</em> have infinite rank at all levels. This gives an affirmative answer to a question of Hedden and Pinzón-Caicedo in many cases. We also show that under the same hypotheses, <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is not a homomorphism on the knot concordance group for each <em>n</em>. We use amenable <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-signatures to prove these results.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"468 ","pages":"Article 110203"},"PeriodicalIF":1.5,"publicationDate":"2025-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143601056","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":"Spinors and horospheres","authors":"Daniel V. Mathews","doi":"10.1016/j.aim.2025.110200","DOIUrl":"10.1016/j.aim.2025.110200","url":null,"abstract":"<div><div>We give an explicit bijective correspondence between nonzero pairs of complex numbers, which we regard as spinors or spin vectors, and horospheres in 3-dimensional hyperbolic space decorated with certain spinorial directions. This correspondence builds upon work of Penrose–Rindler and Penner. We show that the natural bilinear form on spin vectors describes a certain complex-valued distance between spin-decorated horospheres, generalising Penner's lambda lengths to 3 dimensions.</div><div>From this, we derive several applications. We show that the complex lambda lengths in a hyperbolic ideal tetrahedron satisfy a Ptolemy equation. We also obtain correspondences between certain spaces of hyperbolic ideal polygons and certain Grassmannian spaces, under which lambda lengths correspond to Plücker coordinates, illuminating the connection between Grassmannians, hyperbolic polygons, and type A cluster algebras.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"468 ","pages":"Article 110200"},"PeriodicalIF":1.5,"publicationDate":"2025-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143601060","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":"Some homological properties of category O, VII","authors":"Volodymyr Mazorchuk","doi":"10.1016/j.aim.2025.110201","DOIUrl":"10.1016/j.aim.2025.110201","url":null,"abstract":"<div><div>We describe Calabi-Yau objects in the regular block of the (parabolic) BGG category <span><math><mi>O</mi></math></span> associated to a semi-simple finite dimensional complex Lie algebra. Each such object comes with a natural transformation from the Serre functor to a shifted identity whose evaluation at that object is an isomorphism.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"468 ","pages":"Article 110201"},"PeriodicalIF":1.5,"publicationDate":"2025-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143601057","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":"On primes in arithmetic progressions and bounded gaps between many primes","authors":"Julia Stadlmann","doi":"10.1016/j.aim.2025.110190","DOIUrl":"10.1016/j.aim.2025.110190","url":null,"abstract":"<div><div>We prove that the primes below <em>x</em> are, on average, equidistributed in arithmetic progressions to smooth moduli of size up to <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>/</mo><mn>40</mn><mo>−</mo><mi>ϵ</mi></mrow></msup></math></span>. The exponent of distribution <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>40</mn></mrow></mfrac></math></span> improves on a result of Polymath <span><span>[13]</span></span>, who had previously obtained the exponent <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>7</mn></mrow><mrow><mn>300</mn></mrow></mfrac></math></span>. As a consequence, we improve results on intervals of bounded length which contain many primes, showing that<span><span><span><math><mrow><munder><mrow><mrow><mi>lim</mi></mrow><mspace></mspace><mrow><mi>inf</mi></mrow></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mo>(</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow></msub><mo>−</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><mi>exp</mi><mo>⁡</mo><mo>(</mo><mn>3.8075</mn><mi>m</mi><mo>)</mo><mo>)</mo><mo>.</mo></mrow></math></span></span></span> The main new ingredient of our proof is a modification of the <em>q</em>-van der Corput process. It allows us to exploit additional averaging for the exponential sums which appear in the Type I estimates of <span><span>[13]</span></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"468 ","pages":"Article 110190"},"PeriodicalIF":1.5,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143562272","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":"Kneser graphs are Hamiltonian","authors":"Arturo Merino ,&nbsp;Torsten Mütze ,&nbsp;Namrata","doi":"10.1016/j.aim.2025.110189","DOIUrl":"10.1016/j.aim.2025.110189","url":null,"abstract":"<div><div>For integers <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></math></span>, the Kneser graph <span><math><mi>K</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> has as vertices all <em>k</em>-element subsets of an <em>n</em>-element ground set, and an edge between any two disjoint sets. It has been conjectured since the 1970s that all Kneser graphs admit a Hamilton cycle, with one notable exception, namely the Petersen graph <span><math><mi>K</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. This problem received considerable attention in the literature, including a recent solution for the sparsest case <span><math><mi>n</mi><mo>=</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></math></span>. The main contribution of this paper is to prove the conjecture in full generality. We also extend this Hamiltonicity result to all connected generalized Johnson graphs (except the Petersen graph). The generalized Johnson graph <span><math><mi>J</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> has as vertices all <em>k</em>-element subsets of an <em>n</em>-element ground set, and an edge between any two sets whose intersection has size exactly <em>s</em>. Clearly, we have <span><math><mi>K</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>=</mo><mi>J</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></span>, i.e., generalized Johnson graphs include Kneser graphs as a special case. Our results imply that all known natural families of vertex-transitive graphs defined by intersecting set systems have a Hamilton cycle, which settles an interesting special case of Lovász' conjecture on Hamilton cycles in vertex-transitive graphs from 1970. Our main technical innovation is to study cycles in Kneser graphs by a kinetic system of multiple gliders that move at different speeds and that interact over time, reminiscent of the gliders in Conway's Game of Life, and to analyze this system combinatorially and via linear algebra.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"468 ","pages":"Article 110189"},"PeriodicalIF":1.5,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143562273","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":"Constructing smoothings of stable maps","authors":"Fatemeh Rezaee ,&nbsp;Mohan Swaminathan","doi":"10.1016/j.aim.2025.110188","DOIUrl":"10.1016/j.aim.2025.110188","url":null,"abstract":"<div><div>Let <em>X</em> be a smooth projective variety. Define a stable map <span><math><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>X</mi></math></span> to be <em>eventually smoothable</em> if there is an embedding <span><math><mi>X</mi><mo>↪</mo><msup><mrow><mi>P</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> such that <span><math><mo>(</mo><mi>C</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> occurs as the limit of a 1-parameter family of stable maps to <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> with smooth domain curves. Via an explicit deformation-theoretic construction, we produce a large class of stable maps (called <em>stable maps with model ghosts</em>), and show that they are eventually smoothable.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"467 ","pages":"Article 110188"},"PeriodicalIF":1.5,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552550","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}