Geometric and Functional Analysis最新文献

Unit and Distinct Distances in Typical Norms

IF 2.2 1区数学

Geometric and Functional Analysis Pub Date : 2025-01-24 DOI: 10.1007/s00039-025-00698-x

Noga Alon, Matija Bucić, Lisa Sauermann

{"title":"Unit and Distinct Distances in Typical Norms","authors":"Noga Alon, Matija Bucić, Lisa Sauermann","doi":"10.1007/s00039-025-00698-x","DOIUrl":"https://doi.org/10.1007/s00039-025-00698-x","url":null,"abstract":"Erdős’ unit distance problem and Erdős’ distinct distances problem are among the most classical and well-known open problems in discrete mathematics. They ask for the maximum number of unit distances, or the minimum number of distinct distances, respectively, determined by n points in the Euclidean plane. The question of what happens in these problems if one considers normed spaces other than the Euclidean plane has been raised in the 1980s by Ulam and Erdős and attracted a lot of attention over the years. We give an essentially tight answer to both questions for almost all norms on (mathbb{R}^{d}), in a certain Baire categoric sense.For the unit distance problem we prove that for almost all norms ∥.∥ on (mathbb{R}^{d}), any set of n points defines at most (frac{1}{2} d cdot n log _{2} n) unit distances according to ∥.∥. We also show that this is essentially tight, by proving that for every norm ∥.∥ on (mathbb{R}^{d}), for any large n, we can find n points defining at least (frac{1}{2}(d-1-o(1))cdot n log _{2} n) unit distances according to ∥.∥.For the distinct distances problem, we prove that for almost all norms ∥.∥ on (mathbb{R}^{d}) any set of n points defines at least (1−o(1))n distinct distances according to ∥.∥. This is clearly tight up to the o(1) term.We also answer the famous Hadwiger–Nelson problem for almost all norms on (mathbb{R}^{2}), showing that their unit distance graph has chromatic number 4.Our results settle, in a strong and somewhat surprising form, problems and conjectures of Brass, Matoušek, Brass–Moser–Pach, Chilakamarri, and Robertson. The proofs combine combinatorial and geometric ideas with tools from Linear Algebra, Topology and Algebraic Geometry.","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"30 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143026658","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

Lagrangian Subvarieties of Hyperspherical Varieties 超球变种的拉格朗日子变种

IF 2.2 1区数学

Geometric and Functional Analysis Pub Date : 2025-01-22 DOI: 10.1007/s00039-025-00703-3

Michael Finkelberg, Victor Ginzburg, Roman Travkin

引用次数: 0

Optimal Transport Between Algebraic Hypersurfaces 代数超曲面间的最优传输

IF 2.2 1区数学

Geometric and Functional Analysis Pub Date : 2025-01-22 DOI: 10.1007/s00039-025-00699-w

Paolo Antonini, Fabio Cavalletti, Antonio Lerario

{"title":"Optimal Transport Between Algebraic Hypersurfaces","authors":"Paolo Antonini, Fabio Cavalletti, Antonio Lerario","doi":"10.1007/s00039-025-00699-w","DOIUrl":"https://doi.org/10.1007/s00039-025-00699-w","url":null,"abstract":"What is the optimal way to deform a projective hypersurface into another one? In this paper we will answer this question adopting the point of view of measure theory, introducing the optimal transport problem between complex algebraic projective hypersurfaces.First, a natural topological embedding of the space of hypersurfaces of a given degree into the space of measures on the projective space is constructed. Then, the optimal transport problem between hypersurfaces is defined through a constrained dynamical formulation, minimizing the energy of absolutely continuous curves which lie on the image of this embedding. In this way an inner Wasserstein distance on the projective space of homogeneous polynomials is introduced. This distance is finer than the Fubini–Study one.The innner Wasserstein distance is complete and geodesic: geodesics corresponds to optimal deformations of one algebraic hypersurface into another one. Outside the discriminant this distance is induced by a smooth Riemannian metric, which is the real part of an explicit Hermitian structure. Moreover, this Hermitian structure is Kähler and the corresponding metric is of Weil–Petersson type.To prove these results we develop new techniques, which combine complex and symplectic geometry with optimal transport, and which we expect to be relevant on their own.We discuss applications on the regularity of the zeroes of a family of multivariate polynomials and on the condition number of polynomial systems solving.","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"27 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142991957","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

On the Distance Sets Spanned by Sets of Dimension d/2 in $mathbb{R}^{d}$ $mathbb{R}^{d}$中d/2维集张成的距离集

IF 2.2 1区数学

Geometric and Functional Analysis Pub Date : 2025-01-09 DOI: 10.1007/s00039-024-00696-5

Pablo Shmerkin, Hong Wang

引用次数: 0

The Hadwiger Theorem on Convex Functions, I 凸函数的哈德维格定理，I

IF 2.2 1区数学

Geometric and Functional Analysis Pub Date : 2024-10-16 DOI: 10.1007/s00039-024-00693-8

Andrea Colesanti, Monika Ludwig, Fabian Mussnig

引用次数: 0

Geometric Regularity of Blow-up Limits of the Kähler-Ricci Flow 凯勒-里奇流膨胀极限的几何正则性

IF 2.2 1区数学

Geometric and Functional Analysis Pub Date : 2024-10-16 DOI: 10.1007/s00039-024-00694-7

Max Hallgren, Wangjian Jian, Jian Song, Gang Tian

引用次数: 0

Universality and Sharp Matrix Concentration Inequalities 普遍性与尖锐矩阵集中不等式

IF 2.2 1区数学

Geometric and Functional Analysis Pub Date : 2024-10-10 DOI: 10.1007/s00039-024-00692-9

Tatiana Brailovskaya, Ramon van Handel

引用次数: 0

Birkhoff Conjecture for Nearly Centrally Symmetric Domains 近中心对称域的伯克霍夫猜想

IF 2.2 1区数学

Geometric and Functional Analysis Pub Date : 2024-10-10 DOI: 10.1007/s00039-024-00695-6

V. Kaloshin, C. E. Koudjinan, Ke Zhang

引用次数: 0

Gromov-Witten Invariants in Complex and Morava-Local K-Theories 复杂和莫拉瓦局域 K 理论中的格罗莫夫-维滕不变式

IF 2.2 1区数学

Geometric and Functional Analysis Pub Date : 2024-10-07 DOI: 10.1007/s00039-024-00697-4

Mohammed Abouzaid, Mark McLean, Ivan Smith

{"title":"Gromov-Witten Invariants in Complex and Morava-Local K-Theories","authors":"Mohammed Abouzaid, Mark McLean, Ivan Smith","doi":"10.1007/s00039-024-00697-4","DOIUrl":"https://doi.org/10.1007/s00039-024-00697-4","url":null,"abstract":"Given a closed symplectic manifold X, we construct Gromov-Witten-type invariants valued both in (complex) K-theory and in any complex-oriented cohomology theory (mathbb{K}) which is Kp(n)-local for some Morava K-theory Kp(n). We show that these invariants satisfy a version of the Kontsevich-Manin axioms, extending Givental and Lee’s work for the quantum K-theory of complex projective algebraic varieties. In particular, we prove a Gromov-Witten type splitting axiom, and hence define quantum K-theory and quantum (mathbb{K})-theory as commutative deformations of the corresponding (generalised) cohomology rings of X; the definition of the quantum product involves the formal group of the underlying cohomology theory. The key geometric input of these results is a construction of global Kuranishi charts for moduli spaces of stable maps of arbitrary genus to X. On the algebraic side, in order to establish a common framework covering both ordinary K-theory and Kp(n)-local theories, we introduce a formalism of ‘counting theories’ for enumerative invariants on a category of global Kuranishi charts.","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"205 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142383955","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

Direct Products of Free Groups in Aut(FN) Aut(FN) 中自由基的直积

IF 2.2 1区数学

Geometric and Functional Analysis Pub Date : 2024-08-05 DOI: 10.1007/s00039-024-00688-5

Martin R. Bridson, Richard D. Wade

引用次数: 0