Advances in Mathematics最新文献

{"title":"Order-2 Delaunay triangulations optimize angles","authors":"Herbert Edelsbrunner ,&nbsp;Alexey Garber ,&nbsp;Morteza Saghafian","doi":"10.1016/j.aim.2024.110055","DOIUrl":"10.1016/j.aim.2024.110055","url":null,"abstract":"<div><div>The <em>local angle property</em> of the (order-1) Delaunay triangulations of a generic set in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> asserts that the sum of two angles opposite a common edge is less than <em>π</em>. This paper extends this property to higher order and uses it to generalize two classic properties from order-1 to order-2: (1) among the complete level-2 hypertriangulations of a generic point set in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, the order-2 Delaunay triangulation lexicographically maximizes the sorted angle vector; (2) among the maximal level-2 hypertriangulations of a generic point set in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, the order-2 Delaunay triangulation is the only one that has the local angle property. We also use our method of establishing (2) to give a new short proof of the angle vector optimality for the (order-1) Delaunay triangulation. For order-1, both properties have been instrumental in numerous applications of Delaunay triangulations, and we expect that their generalization will make order-2 Delaunay triangulations more attractive to applications as well.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110055"},"PeriodicalIF":1.5,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142748634","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":"Quasilinear tropical compactifications","authors":"Nolan Schock","doi":"10.1016/j.aim.2024.110037","DOIUrl":"10.1016/j.aim.2024.110037","url":null,"abstract":"<div><div>The prototypical examples of tropical compactifications are compactifications of complements of hyperplane arrangements, which posses a number of remarkable properties not satisfied by more general tropical compactifications of closed subvarieties of tori. We introduce a broader class of tropical compactifications, which we call <em>quasilinear (tropical) compactifications</em>, and which continue to satisfy the desirable properties of compactifications of complements of hyperplane arrangements. In particular, we show any quasilinear compactification is schön, and its intersection theory is described entirely by the intersection theory of the corresponding tropical fan. As applications, we prove the quasilinearity of the moduli spaces of 6 lines in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and marked cubic surfaces, obtaining results on the geometry of the stable pair compactifications of these spaces.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110037"},"PeriodicalIF":1.5,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142748631","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 centre of the modular affine vertex algebra","authors":"Tomoyuki Arakawa ,&nbsp;Lewis Topley ,&nbsp;Juan J. Villarreal","doi":"10.1016/j.aim.2024.110052","DOIUrl":"10.1016/j.aim.2024.110052","url":null,"abstract":"<div><div>The Feigin–Frenkel theorem states that, over the complex numbers, the centre of the universal affine vertex algebra at the critical level is an infinite rank polynomial algebra. The first author and W. Wang observed that in positive characteristics, the universal affine vertex algebra contains a large central subalgebra known as the <em>p</em>-centre. They conjectured that at the critical level the centre should be generated by the Feigin–Frenkel centre and the <em>p</em>-centre. In this paper we prove the conjecture for classical simple Lie algebras for <em>p</em> larger than the Coxeter number, and for exceptional Lie algebras in large characteristics. Finally, we give an example which shows that at non-critical level the center is larger than the <em>p</em>-centre.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110052"},"PeriodicalIF":1.5,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142748632","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":"Bordism invariance of orientations and real APS index theory","authors":"Markus Upmeier","doi":"10.1016/j.aim.2024.110048","DOIUrl":"10.1016/j.aim.2024.110048","url":null,"abstract":"<div><div>We show that orientations and Floer gradings for elliptic differential operators can be propagated through bordisms. This is based on a new perspective on APS indices for elliptic boundary value problems over the real numbers. Several applications to moduli spaces of this new bordism-theoretic point of view will be given in the sequel.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110048"},"PeriodicalIF":1.5,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142748686","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":"Uniqueness up to inner automorphism of regular exact Borel subalgebras","authors":"Anna Rodriguez Rasmussen","doi":"10.1016/j.aim.2024.110049","DOIUrl":"10.1016/j.aim.2024.110049","url":null,"abstract":"<div><div>In <span><span>[18]</span></span>, Külshammer, König and Ovsienko proved that for any quasi-hereditary algebra <span><math><mo>(</mo><mi>A</mi><mo>,</mo><msub><mrow><mo>≤</mo></mrow><mrow><mi>A</mi></mrow></msub><mo>)</mo></math></span> there exists a Morita equivalent quasi-hereditary algebra <span><math><mo>(</mo><mi>R</mi><mo>,</mo><msub><mrow><mo>≤</mo></mrow><mrow><mi>R</mi></mrow></msub><mo>)</mo></math></span> containing a basic exact Borel subalgebra <em>B</em>. The Borel subalgebra <em>B</em> constructed in <span><span>[18]</span></span> is in fact a regular exact Borel subalgebra as defined in <span><span>[7]</span></span>. Later, Conde <span><span>[9]</span></span> showed that given a quasi-hereditary algebra <span><math><mo>(</mo><mi>R</mi><mo>,</mo><msub><mrow><mo>≤</mo></mrow><mrow><mi>R</mi></mrow></msub><mo>)</mo></math></span> with a basic regular exact Borel subalgebra <em>B</em> and a Morita equivalent quasi-hereditary algebra <span><math><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><msub><mrow><mo>≤</mo></mrow><mrow><msup><mrow><mi>R</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub><mo>)</mo></math></span> with a basic regular exact Borel subalgebra <span><math><msup><mrow><mi>B</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>, the algebras <em>R</em> and <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> are isomorphic, and Külshammer and Miemietz <span><span>[20]</span></span> showed that there is even an isomorphism <span><math><mi>φ</mi><mo>:</mo><mi>R</mi><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> such that <span><math><mi>φ</mi><mo>(</mo><mi>B</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>B</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>.</div><div>In this article, we show that if <span><math><mi>R</mi><mo>=</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>, then <em>φ</em> can be chosen to be an inner automorphism. Moreover, instead of just proving this for regular exact Borel subalgebras of quasi-hereditary algebras, we generalize this to an appropriate class of subalgebras of arbitrary finite-dimensional algebras. As an application, we show that if <span><math><mo>(</mo><mi>A</mi><mo>,</mo><msub><mrow><mo>≤</mo></mrow><mrow><mi>A</mi></mrow></msub><mo>)</mo></math></span> is a finite-dimensional algebra and <em>G</em> is a finite group acting on <em>A</em> via automorphisms, then under some natural compatibility conditions, there is a Morita equivalent quasi-hereditary algebra <span><math><mo>(</mo><mi>R</mi><mo>,</mo><msub><mrow><mo>≤</mo></mrow><mrow><mi>R</mi></mrow></msub><mo>)</mo></math></span> with a basic regular exact Borel subalgebra <em>B</em> such that <span><math><mi>g</mi><mo>(</mo><mi>B</mi><mo>)</mo><mo>=</mo><mi>B</mi></math></span> for every <span><math><mi>g</mi><mo>∈</mo><mi>G</mi></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110049"},"PeriodicalIF":1.5,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142748633","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 linear programming bounds for sphere packing via discrete reductions","authors":"Rupert Li","doi":"10.1016/j.aim.2024.110043","DOIUrl":"10.1016/j.aim.2024.110043","url":null,"abstract":"<div><div>The Cohn-Elkies linear program for sphere packing, which was used to solve the 8 and 24 dimensional cases, is conjectured to not be sharp in any other dimension <span><math><mi>d</mi><mo>&gt;</mo><mn>2</mn></math></span>. By mapping feasible points of this infinite-dimensional linear program into a finite-dimensional problem via discrete reduction, we provide a general method to obtain dual bounds on the Cohn-Elkies linear program. This reduces the number of variables to be finite, enabling computer optimization techniques to be applied. Using this method, we prove that the Cohn-Elkies bound cannot come close to the best packing densities known in dimensions <span><math><mn>3</mn><mo>≤</mo><mi>d</mi><mo>≤</mo><mn>13</mn></math></span> except for the solved case <span><math><mi>d</mi><mo>=</mo><mn>8</mn></math></span>. In particular, our dual bounds show the Cohn-Elkies bound is unable to solve the 3, 4, and 5 dimensional sphere packing problems.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"460 ","pages":"Article 110043"},"PeriodicalIF":1.5,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723196","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 profinite homotopy type of log schemes","authors":"David Carchedi ,&nbsp;Sarah Scherotzke ,&nbsp;Nicolò Sibilla ,&nbsp;Mattia Talpo","doi":"10.1016/j.aim.2024.110018","DOIUrl":"10.1016/j.aim.2024.110018","url":null,"abstract":"<div><div>We complete the program, initiated in <span><span>[8]</span></span>, to compare the many different possible definitions of the underlying homotopy type of a log scheme. We show that, up to profinite completion, they all yield the same result, and thus arrive at an unambiguous definition of the profinite homotopy type of a log scheme. Specifically, in <span><span>[8]</span></span>, we define this to be the profinite étale homotopy type of the infinite root stack, and show that, over <span><math><mi>C</mi></math></span>, this agrees up to profinite completion with the Kato-Nakayama space. Other possible candidates are the profinite shape of the Kummer étale site <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>k</mi><mover><mrow><mi>e</mi></mrow><mrow><mo>´</mo></mrow></mover><mi>t</mi></mrow></msub></math></span>, or of the representable étale site of <figure><img></figure>. Our main result is that all of these notions agree, and moreover the <em>profinite</em> étale homotopy type of the infinite root stack is not sensitive to whether or not it is viewed as a pro-system in stacks, or as an actual stack (by taking the limit of the pro-system). We furthermore show that in the log regular setting, all these notions also agree with the étale homotopy type of the classical locus <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>triv</mi></mrow></msup></math></span> (up to an appropriate completion). We deduce that, over an arbitrary locally Noetherian base, the étale homotopy type of <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>N</mi></mrow></msubsup></math></span> agrees with that of <span><math><mi>B</mi><msubsup><mrow><mi>μ</mi></mrow><mrow><mo>∞</mo></mrow><mrow><mi>N</mi></mrow></msubsup></math></span> up to completion.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"460 ","pages":"Article 110018"},"PeriodicalIF":1.5,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723192","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":"Pluripotential homotopy theory","authors":"Jonas Stelzig","doi":"10.1016/j.aim.2024.110038","DOIUrl":"10.1016/j.aim.2024.110038","url":null,"abstract":"<div><div>We build free, bigraded bidifferential algebra models for the forms on a complex manifold, with respect to a strong notion of quasi-isomorphism and compatible with the conjugation symmetry. This answers a question of Sullivan. The resulting theory naturally accomodates higher operations involving double primitives. As applications, we obtain various refinements of the homotopy groups, sensitive to the complex structure. Under a simple connectedness assumption, one obtains minimal models which are unique up to isomorphism and allow for explicit computations of the new invariants.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"460 ","pages":"Article 110038"},"PeriodicalIF":1.5,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723193","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 the isometric version of Whitney's strong embedding theorem","authors":"Wentao Cao ,&nbsp;László Székelyhidi Jr.","doi":"10.1016/j.aim.2024.110040","DOIUrl":"10.1016/j.aim.2024.110040","url":null,"abstract":"<div><div>We prove a version of Whitney's strong embedding theorem for isometric embeddings within the general setting of the Nash-Kuiper h-principle. More precisely, we show that any <em>n</em>-dimensional smooth compact manifold admits infinitely many global isometric embeddings into 2<em>n</em>-dimensional Euclidean space, of Hölder class <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>θ</mi></mrow></msup></math></span> with <span><math><mi>θ</mi><mo>&lt;</mo><mn>1</mn><mo>/</mo><mn>3</mn></math></span> for <span><math><mi>n</mi><mo>=</mo><mn>2</mn></math></span> and <span><math><mi>θ</mi><mo>&lt;</mo><msup><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>. The proof is performed by Nash-Kuiper's convex integration construction and applying the gluing technique of the authors on short embeddings with small amplitude.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"460 ","pages":"Article 110040"},"PeriodicalIF":1.5,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723194","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":"Orlov's theorem for dg-algebras","authors":"Michael K. Brown ,&nbsp;Prashanth Sridhar","doi":"10.1016/j.aim.2024.110035","DOIUrl":"10.1016/j.aim.2024.110035","url":null,"abstract":"<div><div>A landmark theorem of Orlov relates the singularity category of a graded Gorenstein algebra to the derived category of the associated noncommutative projective scheme. We generalize this theorem to the setting of differential graded algebras. As an application, we obtain new cases of the Lattice Conjecture in noncommutative Hodge theory.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"460 ","pages":"Article 110035"},"PeriodicalIF":1.5,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723191","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}