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The Σ2-Potentialist Principle
IF 1.5 1区 数学
Advances in Mathematics Pub Date : 2025-02-27 DOI: 10.1016/j.aim.2025.110182
Omer Ben Neria , Gabriel Goldberg , Eyal Kaplan
{"title":"The Σ2-Potentialist Principle","authors":"Omer Ben Neria ,&nbsp;Gabriel Goldberg ,&nbsp;Eyal Kaplan","doi":"10.1016/j.aim.2025.110182","DOIUrl":"10.1016/j.aim.2025.110182","url":null,"abstract":"<div><div>We settle a question of Woodin motivated by the philosophy of potentialism in set theory. A sentence in the language of set theory is <em>locally verifiable</em> if it asserts the existence of a level <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> of the cumulative hierarchy of sets with some first-order property; this is equivalent to being <span><math><msub><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> in the Lévy hierarchy. A sentence is <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span><em>-satisfiable</em> if it can be forced without changing <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>, and <em>V-satisfiable</em> if it is <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-satisfiable for all ordinals <em>α</em>. The <span><math><msub><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-Potentialist Principle, introduced by Woodin, asserts that every <em>V</em>-satisfiable locally verifiable sentence is true. We show in <span><span>Theorem 6.2</span></span> that the <span><math><msub><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-Potentialist Principle is consistent relative to a supercompact cardinal. We accomplish this by generalizing Gitik's method of iterating distributive forcings by embedding them into Príkry-type forcings <span><span>[6, Section 6.4]</span></span>; our generalization, <span><span>Theorem 5.2</span></span>, works for forcings that add no bounded subsets to a strongly compact cardinal, which requires a completely different proof. Finally, using the concept of mutual stationarity, we show in <span><span>Theorem 7.5</span></span> that the <span><math><msub><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-Potentialist Principle implies the consistency of a Woodin cardinal.<span><span><sup>3</sup></span></span></div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"467 ","pages":"Article 110182"},"PeriodicalIF":1.5,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509320","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}
引用次数: 0
Symmetries of the cyclic nerve
IF 1.5 1区 数学
Advances in Mathematics Pub Date : 2025-02-26 DOI: 10.1016/j.aim.2025.110170
David Ayala, Aaron Mazel-Gee, Nick Rozenblyum
{"title":"Symmetries of the cyclic nerve","authors":"David Ayala,&nbsp;Aaron Mazel-Gee,&nbsp;Nick Rozenblyum","doi":"10.1016/j.aim.2025.110170","DOIUrl":"10.1016/j.aim.2025.110170","url":null,"abstract":"<div><div>We undertake a systematic study of the Hochschild homology, i.e. (the geometric realization of) the cyclic nerve, of <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-categories (and more generally of category-objects in an ∞-category), as a version of factorization homology. In order to do this, we codify <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-categories in terms of quiver representations in them. By examining a universal instance of such Hochschild homology, we explicitly identify its natural symmetries, and construct a non-stable version of the cyclotomic trace map. Along the way we give a unified account of the cyclic, paracyclic, and epicyclic categories. We also prove that this gives a combinatorial description of the <span><math><mi>n</mi><mo>=</mo><mn>1</mn></math></span> case of factorization homology as presented in <span><span>[4]</span></span>, which parametrizes <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-categories by solidly 1-framed stratified spaces.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"466 ","pages":"Article 110170"},"PeriodicalIF":1.5,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509019","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}
引用次数: 0
Large deviations of the empirical spectral measure of supercritical sparse Wigner matrices
IF 1.5 1区 数学
Advances in Mathematics Pub Date : 2025-02-24 DOI: 10.1016/j.aim.2025.110156
Fanny Augeri
{"title":"Large deviations of the empirical spectral measure of supercritical sparse Wigner matrices","authors":"Fanny Augeri","doi":"10.1016/j.aim.2025.110156","DOIUrl":"10.1016/j.aim.2025.110156","url":null,"abstract":"<div><div>Let Ξ be the adjacency matrix of an Erdős-Rényi graph on <em>n</em> vertices and with parameter <em>p</em> and consider <em>A</em> a <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> centred random symmetric matrix with bounded i.i.d. entries above the diagonal. When the mean degree <em>np</em> diverges, the empirical spectral measure of the normalized Hadamard product <span><math><mo>(</mo><mi>A</mi><mo>∘</mo><mi>Ξ</mi><mo>)</mo><mo>/</mo><msqrt><mrow><mi>n</mi><mi>p</mi></mrow></msqrt></math></span> converges weakly in probability to the semicircle law. In the regime where <span><math><mi>p</mi><mo>≪</mo><mn>1</mn></math></span> and <span><math><mi>n</mi><mi>p</mi><mo>≫</mo><mi>log</mi><mo>⁡</mo><mi>n</mi></math></span>, we prove a large deviations principle for the empirical spectral measure with speed <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>p</mi></math></span> and with a good rate function solution of a certain variational problem. The rate function reveals in particular that the only possible deviations at the exponential scale <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>p</mi></math></span> are around measures coming from Quadratic Vector Equations. As a byproduct, we obtain a large deviations principle for the empirical spectral measure of supercritical Erdős-Rényi graphs.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"466 ","pages":"Article 110156"},"PeriodicalIF":1.5,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143473995","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}
引用次数: 0
Ortho-integral surfaces
IF 1.5 1区 数学
Advances in Mathematics Pub Date : 2025-02-20 DOI: 10.1016/j.aim.2025.110162
Nhat Minh Doan
{"title":"Ortho-integral surfaces","authors":"Nhat Minh Doan","doi":"10.1016/j.aim.2025.110162","DOIUrl":"10.1016/j.aim.2025.110162","url":null,"abstract":"<div><div>This paper introduces a natural combinatorial structure of orthogeodesics on hyperbolic surfaces and presents Ptolemy relations among them. As a primary application, we propose a recursive method for computing the trace (the hyperbolic cosine of the length) of orthogeodesics and establish the existence of surfaces where the trace of each orthogeodesic is an integer. These surfaces and their orthogeodesics are closely related to integral Apollonian circle packings. Notably, we found a new type of root-flipping that transitions between roots in different quadratic equations of a certain type, with Vieta root-flipping as a special case. Finally, we provide a combinatorial proof of Basmajian's identity for hyperbolic surfaces, akin to Bowditch's combinatorial proof of the McShane identity.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"466 ","pages":"Article 110162"},"PeriodicalIF":1.5,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143446082","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}
引用次数: 0
Norm upper-semicontinuity of functions supported on open abelian isotropy in étale groupoids. Corrigendum to “Reconstruction of groupoids and C⁎-rigidity of dynamical systems” [Adv. Math. 390 (2021) 107923]
IF 1.5 1区 数学
Advances in Mathematics Pub Date : 2025-02-20 DOI: 10.1016/j.aim.2025.110150
Toke Meier Carlsen , Anna Duwenig , Efren Ruiz , Aidan Sims
{"title":"Norm upper-semicontinuity of functions supported on open abelian isotropy in étale groupoids. Corrigendum to “Reconstruction of groupoids and C⁎-rigidity of dynamical systems” [Adv. Math. 390 (2021) 107923]","authors":"Toke Meier Carlsen ,&nbsp;Anna Duwenig ,&nbsp;Efren Ruiz ,&nbsp;Aidan Sims","doi":"10.1016/j.aim.2025.110150","DOIUrl":"10.1016/j.aim.2025.110150","url":null,"abstract":"<div><div>We consider étale Hausdorff groupoids in which the interior of the isotropy is abelian. We prove that the norms of the images under regular representations, of elements of the reduced groupoid <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra whose supports are contained in the interior of the isotropy vary upper semicontinuously. This corrects an error in <span><span>[2]</span></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"466 ","pages":"Article 110150"},"PeriodicalIF":1.5,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143446083","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}
引用次数: 0
On Jacobi–Weierstrass mock modular forms
IF 1.5 1区 数学
Advances in Mathematics Pub Date : 2025-02-19 DOI: 10.1016/j.aim.2025.110147
Claudia Alfes , Jens Funke , Michael H. Mertens , Eugenia Rosu
{"title":"On Jacobi–Weierstrass mock modular forms","authors":"Claudia Alfes ,&nbsp;Jens Funke ,&nbsp;Michael H. Mertens ,&nbsp;Eugenia Rosu","doi":"10.1016/j.aim.2025.110147","DOIUrl":"10.1016/j.aim.2025.110147","url":null,"abstract":"<div><div>We construct harmonic weak Maass forms that map to cusp forms of weight <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> with rational coefficients under the <em>ξ</em>-operator. This generalizes work of the first author, Griffin, Ono, and Rolen, who constructed distinguished preimages under this differential operator of weight 2 newforms associated to rational elliptic curves using the classical Weierstrass theory of elliptic functions. We extend this theory and construct a vector-valued Jacobi–Weierstrass <em>ζ</em>-function which is a generalization of the classical Weierstrass <em>ζ</em>-function.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"465 ","pages":"Article 110147"},"PeriodicalIF":1.5,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143444701","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}
引用次数: 0
The length spectrum of random hyperbolic 3-manifolds
IF 1.5 1区 数学
Advances in Mathematics Pub Date : 2025-02-19 DOI: 10.1016/j.aim.2025.110158
Anna Roig-Sanchis
{"title":"The length spectrum of random hyperbolic 3-manifolds","authors":"Anna Roig-Sanchis","doi":"10.1016/j.aim.2025.110158","DOIUrl":"10.1016/j.aim.2025.110158","url":null,"abstract":"<div><div>We study the length spectrum of a model of random hyperbolic 3-manifolds introduced in <span><span>[31]</span></span>. These are compact manifolds with boundary constructed by randomly gluing truncated tetrahedra along their faces. We prove that, as the volume tends to infinity, their length spectrum converge in distribution to a Poisson point process on <span><math><msub><mrow><mi>R</mi></mrow><mrow><mo>&gt;</mo><mn>0</mn></mrow></msub></math></span>, with computable intensity <em>λ</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"466 ","pages":"Article 110158"},"PeriodicalIF":1.5,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143436516","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}
引用次数: 0
The constant term algebra of type A: The structure
IF 1.5 1区 数学
Advances in Mathematics Pub Date : 2025-02-18 DOI: 10.1016/j.aim.2025.110154
Guoce Xin , Chen Zhang , Yue Zhou , Yueming Zhong
{"title":"The constant term algebra of type A: The structure","authors":"Guoce Xin ,&nbsp;Chen Zhang ,&nbsp;Yue Zhou ,&nbsp;Yueming Zhong","doi":"10.1016/j.aim.2025.110154","DOIUrl":"10.1016/j.aim.2025.110154","url":null,"abstract":"<div><div>In this paper, we discover a new noncommutative algebra. We refer this algebra as the constant term algebra of type <em>A</em>, which is generated by certain constant term operators. We characterize a structural result of this algebra by establishing an explicit basis in terms of certain forests. This algebra arises when we apply the method of the iterated Laurent series to investigate Beck and Pixton's residue computation for the Ehrhart series of the Birkhoff polytope. This algebra seems to be the first structural result in the area of the constant term world since the discovery of the Dyson constant term identity in 1962.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"465 ","pages":"Article 110154"},"PeriodicalIF":1.5,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143427785","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}
引用次数: 0
An adjunction inequality for the Bauer–Furuta type invariants, with applications to sliceness and 4-manifold topology
IF 1.5 1区 数学
Advances in Mathematics Pub Date : 2025-02-18 DOI: 10.1016/j.aim.2025.110134
Nobuo Iida , Anubhav Mukherjee , Masaki Taniguchi
{"title":"An adjunction inequality for the Bauer–Furuta type invariants, with applications to sliceness and 4-manifold topology","authors":"Nobuo Iida ,&nbsp;Anubhav Mukherjee ,&nbsp;Masaki Taniguchi","doi":"10.1016/j.aim.2025.110134","DOIUrl":"10.1016/j.aim.2025.110134","url":null,"abstract":"<div><div>Our main result gives an adjunction inequality for embedded surfaces in certain 4-manifolds with contact boundary under a non-vanishing assumption on the Bauer–Furuta type invariants. Using this, we give infinitely many knots in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> that are not smoothly H-slice (that is, bounding a null-homologous disk) in many 4-manifolds but they are topologically H-slice. In particular, we give such knots in the boundaries of the punctured elliptic surfaces <span><math><mi>E</mi><mo>(</mo><mn>2</mn><mi>n</mi><mo>)</mo></math></span>. In addition, we give obstructions to codimension-0 orientation-reversing embedding of weak symplectic fillings with <span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mn>0</mn></math></span> into closed symplectic 4-manifolds with <span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>0</mn></math></span> and <span><math><msubsup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>+</mo></mrow></msubsup><mo>≡</mo><mn>3</mn><mi>mod</mi><mspace></mspace><mn>4</mn></math></span>. From here we prove a Bennequin type inequality for strong symplectic caps of <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><mi>s</mi><mi>t</mi><mi>d</mi></mrow></msub><mo>)</mo></math></span>. We also show that any weakly symplectically fillable 3-manifold bounds a 4-manifold with at least two smooth structures.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"466 ","pages":"Article 110134"},"PeriodicalIF":1.5,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143429200","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}
引用次数: 0
Ergodic measures of intermediate entropies for dynamical systems with the approximate product property
IF 1.5 1区 数学
Advances in Mathematics Pub Date : 2025-02-18 DOI: 10.1016/j.aim.2025.110159
Peng Sun
{"title":"Ergodic measures of intermediate entropies for dynamical systems with the approximate product property","authors":"Peng Sun","doi":"10.1016/j.aim.2025.110159","DOIUrl":"10.1016/j.aim.2025.110159","url":null,"abstract":"<div><div>For a dynamical system satisfying the approximate product property and asymptotically entropy expansiveness, we characterize a delicate structure of the space of invariant measures: The ergodic measures of intermediate entropies and the ones of intermediate pressures are generic in certain subspaces. Consequently, the conjecture of Katok that ergodic measures of arbitrary intermediate entropy exist is verified for a broad class of systems.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"465 ","pages":"Article 110159"},"PeriodicalIF":1.5,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143427784","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}
引用次数: 0
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