Advances in Mathematics最新文献

{"title":"An invariance principle for the multi-slice, with applications","authors":"Mark Braverman , Subhash Khot , Noam Lifshitz , Dor Minzer","doi":"10.1016/j.aim.2025.110460","DOIUrl":"10.1016/j.aim.2025.110460","url":null,"abstract":"<div><div>Given an alphabet size <span><math><mi>m</mi><mo>∈</mo><mi>N</mi></math></span> thought of as a constant, and <span><math><mover><mrow><mi>k</mi></mrow><mrow><mo>→</mo></mrow></mover><mo>=</mo><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></math></span> whose entries sum of up <em>n</em>, the <span><math><mover><mrow><mi>k</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span>-multi-slice is the set of vectors <span><math><mi>x</mi><mo>∈</mo><msup><mrow><mo>[</mo><mi>m</mi><mo>]</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span> in which each symbol <span><math><mi>i</mi><mo>∈</mo><mo>[</mo><mi>m</mi><mo>]</mo></math></span> appears precisely <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> times. We show an invariance principle for low-degree functions over the multi-slice, to functions over the product space <span><math><mo>(</mo><msup><mrow><mo>[</mo><mi>m</mi><mo>]</mo></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msup><mrow><mi>μ</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> in which <span><math><mi>μ</mi><mo>(</mo><mi>i</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>/</mo><mi>n</mi></math></span>. This answers a question raised by <span><span>[23]</span></span>.</div><div>As applications of the invariance principle, we show:<ul><li><span>1.</span><span><div>An analogue of the “dictatorship test implies computational hardness” paradigm for problems with perfect completeness, for a certain class of dictatorship tests. Our computational hardness is proved assuming a recent strengthening of the Unique-Games Conjecture, called the Rich 2-to-1 Games Conjecture.</div><div>Using this analogue, we show that assuming the Rich 2-to-1 Games Conjecture, (a) there is an <em>r</em>-ary CSP <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> for which it is NP-hard to distinguish satisfiable instances of the CSP and instances that are at most <span><math><mfrac><mrow><mn>2</mn><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> satisfiable, and (b) hardness of distinguishing 3-colorable graphs, and graphs that do not contain an independent set of size <span><math><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>.</div></span></li><li><span>2.</span><span><div>A reduction of the problem of studying expectations of products of functions on the multi-slice to studying expectations of products of functions on correlated, product spaces. In particular, we are able to deduce analogues of the Gaussian bounds from <span><span>[42]</span></span> for the multi-slice.</div></span></li><li><span>3.</span><span><div>In a companion paper, we show further applications of ","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"480 ","pages":"Article 110460"},"PeriodicalIF":1.5,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144809967","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":"Nilpotent Lie algebras obtained by quivers and Ricci solitons","authors":"Fumika Mizoguchi,&nbsp;Hiroshi Tamaru","doi":"10.1016/j.aim.2025.110464","DOIUrl":"10.1016/j.aim.2025.110464","url":null,"abstract":"<div><div>Nilpotent Lie groups with left-invariant metrics provide non-trivial examples of Ricci solitons. One typical example is given by the class of two-step nilpotent Lie algebras obtained from simple directed graphs. In this paper, however, we focus on the use of quivers to construct nilpotent Lie algebras. A quiver is a directed graph that allows loops and multiple arrows between two vertices. Utilizing the concept of paths within quivers, we introduce a method for constructing nilpotent Lie algebras from finite quivers without cycles. We prove that for all these Lie algebras, the corresponding simply-connected nilpotent Lie groups admit left-invariant Ricci solitons. The method we introduce constructs a broad family of Ricci soliton nilmanifolds with arbitrarily high degrees of nilpotency.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"480 ","pages":"Article 110464"},"PeriodicalIF":1.5,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144781188","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":"Brunn-Minkowski and reverse isoperimetric inequalities for dual quermassintegrals","authors":"Shay Sadovsky,&nbsp;Gaoyong Zhang","doi":"10.1016/j.aim.2025.110456","DOIUrl":"10.1016/j.aim.2025.110456","url":null,"abstract":"<div><div>This paper establishes two new geometric inequalities in the dual Brunn-Minkowski theory. The first, originally conjectured by Lutwak, is the Brunn-Minkowski inequality for dual quermassintegrals of origin-symmetric convex bodies. The second, generalizing Ball's volume ratio inequality, is a reverse isoperimetric inequality: among all origin-symmetric convex bodies in John's position, the cube maximizes the dual quermassintegrals.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"480 ","pages":"Article 110456"},"PeriodicalIF":1.5,"publicationDate":"2025-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144767100","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":"Self-similar sets and Lipschitz graphs","authors":"Blair Davey ,&nbsp;Silvia Ghinassi ,&nbsp;Bobby Wilson","doi":"10.1016/j.aim.2025.110451","DOIUrl":"10.1016/j.aim.2025.110451","url":null,"abstract":"<div><div>We investigate and quantify the distinction between rectifiable and purely unrectifiable 1-sets in the plane. That is, given that purely unrectifiable 1-sets always have null intersections with Lipschitz images, we ask whether these sets intersect with Lipschitz images at a dimension that is close to one. In an answer to this question, we show that one-dimensional attractors of iterated function systems that satisfy the open set condition have subsets of dimension arbitrarily close to one that can be covered by Lipschitz graphs. Moreover, the Lipschitz constant of such graphs depends explicitly on the difference between the dimension of the original set and the subset that intersects with the graph.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110451"},"PeriodicalIF":1.5,"publicationDate":"2025-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144738466","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":"Everywhere unbalanced configurations","authors":"David Conlon ,&nbsp;Jeck Lim","doi":"10.1016/j.aim.2025.110445","DOIUrl":"10.1016/j.aim.2025.110445","url":null,"abstract":"<div><div>An old problem in discrete geometry, originating with Kupitz, asks whether there is a fixed natural number <em>k</em> such that every finite set of points in the plane has a line through at least two of its points where the number of points on either side of this line differ by at most <em>k</em>. We give a negative answer to a natural variant of this problem, showing that for every natural number <em>k</em> there exists a finite set of points in the plane together with a pseudoline arrangement such that each pseudoline contains at least two points and there is a pseudoline through any pair of points where the number of points on either side of each pseudoline differ by at least <em>k</em>. Moreover, we may find such a configuration with at most <span><math><msup><mrow><mn>2</mn></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>c</mi><mi>k</mi></mrow></msup></mrow></msup></math></span> points, which, by a result of Pinchasi, is best possible up to the value of the constant <em>c</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"480 ","pages":"Article 110445"},"PeriodicalIF":1.5,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144723185","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 Leray-Hirsch Theorem for oriented cohomology of flag varieties","authors":"J. Matthew Douglass ,&nbsp;Changlong Zhong","doi":"10.1016/j.aim.2025.110450","DOIUrl":"10.1016/j.aim.2025.110450","url":null,"abstract":"<div><div>We construct two explicit Leray-Hirsch isomorphisms for torus equivariant oriented cohomology of flag varieties and give several applications. One isomorphism is geometric, based on Bott-Samelson classes. The other is algebraic, based on the description of the torus equivariant oriented cohomology of a flag variety as the dual of a formal affine Demazure algebra.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110450"},"PeriodicalIF":1.5,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144724414","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":"Statistical properties for mixing Markov chains with applications to dynamical systems","authors":"Ao Cai ,&nbsp;Pedro Duarte ,&nbsp;Silvius Klein","doi":"10.1016/j.aim.2025.110454","DOIUrl":"10.1016/j.aim.2025.110454","url":null,"abstract":"<div><div>We establish an abstract, effective, exponential large deviations type estimate for Markov systems satisfying a weaker form of mixing. We employ this result to derive such estimates, as well as a central limit theorem, for the skew product encoding a random torus translation, a model we call a mixed random-quasiperiodic dynamical system. This abstract scheme is applicable to many other types of skew product dynamics, including systems for which the spectral gap property for the transition or the transfer operator does not hold.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110454"},"PeriodicalIF":1.5,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144724419","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":"Doubles of Gluck twists: A five-dimensional approach","authors":"David Gabai,&nbsp;Patrick Naylor,&nbsp;Hannah Schwartz","doi":"10.1016/j.aim.2025.110455","DOIUrl":"10.1016/j.aim.2025.110455","url":null,"abstract":"<div><div>Using a 5-dimensional perspective, we balance algebraic and geometric handle cancellation to show that doubles of Gluck twists of certain 2-spheres embedded in the 4-sphere with two minima are standard. This includes all 2-spheres which are unions of ribbon discs, one of which has undisking number one. As an application, we produce new examples of Schoenflies balls not known to be standard.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"480 ","pages":"Article 110455"},"PeriodicalIF":1.5,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144738812","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":"Energy cascade and Sobolev norms inflation for the quantum Euler equations on tori","authors":"Filippo Giuliani ,&nbsp;Raffaele Scandone","doi":"10.1016/j.aim.2025.110453","DOIUrl":"10.1016/j.aim.2025.110453","url":null,"abstract":"<div><div>In this paper we prove the existence of solutions to the quantum Euler equations on <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, <span><math><mi>d</mi><mo>⩾</mo><mn>2</mn></math></span>, with almost constant mass density, displaying energy transfers to high Fourier modes and polynomially fast-in-time growth of Sobolev norms above the finite-energy level. These solutions are uniformly far from vacuum, suggesting that weak turbulence in quantum hydrodynamics is not necessarily related to the occurrence of vortex structures.</div><div>In view of possible connections with instability mechanisms for the classical compressible Euler equations, we also keep track of the dependence on the semiclassical parameter, showing that, at high regularity, the time at which the Sobolev norm inflations occur is uniform when approaching the semiclassical limit.</div><div>Our construction relies on a novel result of Sobolev instability for the plane waves of the cubic nonlinear Schrödinger equation (NLS), which is connected to the quantum Euler equations through the Madelung transform. More precisely, we show the existence of smooth solutions to NLS, which are small-amplitude perturbations of a plane wave and undergo a polynomially fast <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span>-norm inflation for <span><math><mi>s</mi><mo>&gt;</mo><mn>1</mn></math></span>. The proof is based on a partial Birkhoff normal form procedure, involving the normalization of non-homogeneous Hamiltonian terms.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110453"},"PeriodicalIF":1.5,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144724316","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":"p-typical curves on p-adic Tate twists and de Rham–Witt forms","authors":"Sanath K. Devalapurkar,&nbsp;Shubhodip Mondal","doi":"10.1016/j.aim.2025.110448","DOIUrl":"10.1016/j.aim.2025.110448","url":null,"abstract":"<div><div>We show that de Rham–Witt forms are naturally isomorphic to <em>p</em>-typical curves on <em>p</em>-adic Tate twists, which revisits a question of Artin–Mazur from 1977 pursued in the earlier work of Bloch and Kato. We show this more generally by refining a result of Hesselholt on topological cyclic homology with the motivic filtrations introduced by Bhatt–Morrow–Scholze.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110448"},"PeriodicalIF":1.5,"publicationDate":"2025-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144703188","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}