{"title":"A Geometric Approach to Polynomial and Rational Approximation","authors":"Christopher J Bishop, Kirill Lazebnik","doi":"10.1093/imrn/rnae082","DOIUrl":"https://doi.org/10.1093/imrn/rnae082","url":null,"abstract":"We strengthen the classical approximation theorems of Weierstrass, Runge, and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure. In particular, when approximating a function $f$ on a compact set $K$, the critical points of our approximants may be taken to lie in any given domain containing $K$, and all the critical values in any given neighborhood of the polynomially convex hull of $f(K)$.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"76 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140831495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Extremal Numbers and Sidorenko’s Conjecture","authors":"David Conlon, Joonkyung Lee, Alexander Sidorenko","doi":"10.1093/imrn/rnae071","DOIUrl":"https://doi.org/10.1093/imrn/rnae071","url":null,"abstract":"Sidorenko’s conjecture states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge density. While still open for graphs, the analogous statement is known to be false for hypergraphs. We show that there is some advantage in this, in that if Sidorenko’s conjecture does not hold for a particular $r$-partite $r$-uniform hypergraph $H$, then it is possible to improve the standard lower bound, coming from the probabilistic deletion method, for its extremal number $textrm {ex}(n,H)$, the maximum number of edges in an $n$-vertex $H$-free $r$-uniform hypergraph. With this application in mind, we find a range of new counterexamples to the conjecture for hypergraphs, including all linear hypergraphs containing a loose triangle and all $3$-partite $3$-uniform tight cycles.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"54 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140804371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Elena S Hafner, Karola Mészáros, Alexander Vidinas
{"title":"Log-Concavity of the Alexander Polynomial","authors":"Elena S Hafner, Karola Mészáros, Alexander Vidinas","doi":"10.1093/imrn/rnae058","DOIUrl":"https://doi.org/10.1093/imrn/rnae058","url":null,"abstract":"The central question of knot theory is that of distinguishing links up to isotopy. The first polynomial invariant of links devised to help answer this question was the Alexander polynomial (1928). Almost a century after its introduction, it still presents us with tantalizing questions, such as Fox’s conjecture (1962) that the absolute values of the coefficients of the Alexander polynomial $Delta _{L}(t)$ of an alternating link $L$ are unimodal. Fox’s conjecture remains open in general with special cases settled by Hartley (1979) for two-bridge knots, by Murasugi (1985) for a family of alternating algebraic links, and by Ozsváth and Szabó (2003) for the case of genus $2$ alternating knots, among others. We settle Fox’s conjecture for special alternating links. We do so by proving that a certain multivariate generalization of the Alexander polynomial of special alternating links is Lorentzian. As a consequence, we obtain that the absolute values of the coefficients of $Delta _{L}(t)$, where $L$ is a special alternating link, form a log-concave sequence with no internal zeros. In particular, they are unimodal.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"42 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140804359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Emily Clader, Chiara Damiolini, Christopher Eur, Daoji Huang, Shiyue Li
{"title":"Multimatroids and Rational Curves with Cyclic Action","authors":"Emily Clader, Chiara Damiolini, Christopher Eur, Daoji Huang, Shiyue Li","doi":"10.1093/imrn/rnae069","DOIUrl":"https://doi.org/10.1093/imrn/rnae069","url":null,"abstract":"We study the connection between multimatroids and moduli spaces of rational curves with cyclic action. Multimatroids are generalizations of matroids and delta-matroids that naturally arise in topological graph theory. The perspective of moduli of curves provides a tropical framework for studying multimatroids, generalizing the previous connection between type-$A$ permutohedral varieties (Losev–Manin moduli spaces) and matroids, and the connection between type-$B$ permutohedral varieties and delta-matroids. Specifically, we equate a combinatorial nef cone of the moduli space with the space of ${mathbb {R}}$-multimatroids, a generalization of multimatroids, and we introduce the independence polytopal complex of a multimatroid, whose volume is identified with an intersection number on the moduli space. As an application, we give a combinatorial formula for a natural class of intersection numbers on the moduli space by relating to the volumes of independence polytopal complexes of multimatroids.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"16 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140806493","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Construction of Deformations to General Algebras","authors":"David Bowman, Dora Puljić, Agata Smoktunowicz","doi":"10.1093/imrn/rnae077","DOIUrl":"https://doi.org/10.1093/imrn/rnae077","url":null,"abstract":"One of the questions investigated in deformation theory is to determine to which algebras can a given associative algebra be deformed. In this paper we investigate a different but related question, namely: for a given associative finite-dimensional ${mathbb{C}}$-algebra $A$, find algebras $N$, which can be deformed to $A$. We develop a simple method that produces associative and flat deformations to investigate this question. As an application of this method we answer a question of Michael Wemyss about deformations of contraction algebras.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"14 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140804358","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Surfaces of General Type with Maximal Picard Number Near the Noether Line","authors":"Nguyen Bin, Vicente Lorenzo","doi":"10.1093/imrn/rnae075","DOIUrl":"https://doi.org/10.1093/imrn/rnae075","url":null,"abstract":"The first published non-trivial examples of algebraic surfaces of general type with maximal Picard number are due to Persson, who constructed surfaces with maximal Picard number on the Noether line $K^{2}=2chi -6$ for every admissible pair $(K^{2},chi )$ such that $chi not equiv 0 text {mod} 6$. In this note, given a non-negative integer $k$, algebraic surfaces of general type with maximal Picard number lying on the line $K^{2}=2chi -6+k$ are constructed for every admissible pair $(K^{2},chi )$ such that $chi geq 2k+10$. These constructions, obtained as bidouble covers of rational surfaces, not only allow to fill in Persson’s gap on the Noether line, but also provide infinitely many new examples of algebraic surfaces of general type with maximal Picard number above the Noether line.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"27 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140804361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Facial Order for Torsion Classes","authors":"Eric J Hanson","doi":"10.1093/imrn/rnae078","DOIUrl":"https://doi.org/10.1093/imrn/rnae078","url":null,"abstract":"We generalize the “facial weak order” of a finite Coxeter group to a partial order on a set of intervals in a complete lattice. We apply our construction to the lattice of torsion classes of a finite-dimensional algebra and consider its restriction to intervals coming from stability conditions. We give two additional interpretations of the resulting “facial semistable order”: one using cover relations, and one using Bongartz completions of 2-term presilting objects. For $tau $-tilting finite algebras, this allows us to prove that the facial semistable order is a semidistributive lattice. We then show that, in any abelian length category, our new partial order can be partitioned into a set of completely semidistributive lattices, one of which is the original lattice of torsion classes.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"72 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140804372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On Chow Rings of Quiver Moduli","authors":"Pieter Belmans, Hans Franzen","doi":"10.1093/imrn/rnad306","DOIUrl":"https://doi.org/10.1093/imrn/rnad306","url":null,"abstract":"We describe the point class and Todd class in the Chow ring of a moduli space of quiver representations, building on a result of Ellingsrud–Strømme. This, together with the presentation of the Chow ring by the second author, makes it possible to compute integrals on quiver moduli. To do so, we construct a canonical morphism of universal representations in great generality, and along the way point out its relation to the Kodaira–Spencer morphism. We illustrate the results by computing some invariants of some “small” Kronecker moduli spaces. We also prove that the first non-trivial (6-dimensional) Kronecker moduli space is isomorphic to the zero locus of a general section of $mathcal{Q}^{vee }(1)$ on $textrm{Gr}(2,8)$.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"50 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140623212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Transitive Centralizer and Fibered Partially Hyperbolic Systems","authors":"Danijela Damjanović, Amie Wilkinson, Disheng Xu","doi":"10.1093/imrn/rnae064","DOIUrl":"https://doi.org/10.1093/imrn/rnae064","url":null,"abstract":"We prove several rigidity results about the centralizer of a smooth diffeomorphism, concentrating on two families of examples: diffeomorphisms with transitive centralizer, and perturbations of isometric extensions of Anosov diffeomorphisms of nilmanifolds. We classify all smooth diffeomorphisms with transitive centralizer: they are exactly the maps that preserve a principal fiber bundle structure, acting minimally on the fibers and trivially on the base. We also show that for any smooth, accessible isometric extension $f_{0}colon Mto M$ of an Anosov diffeomorphism of a nilmanifold, subject to a spectral bunching condition, any $fin textrm{Diff}^{infty }(M)$ sufficiently $C^{1}$-close to $f_{0}$ has centralizer a Lie group. If the dimension of this Lie group equals the dimension of the fiber, then $f$ is a principal fiber bundle morphism covering an Anosov diffeomorphism. Using the results of this paper, we classify the centralizer of any partially hyperbolic diffeomorphism of a $3$-dimensional, nontoral nilmanifold: either the centralizer is virtually trivial, or the diffeomorphism is an isometric extension of an Anosov diffeomorphism, and the centralizer is virtually ${{mathbb{Z}}}times{{mathbb{T}}}$.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140628726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Duality in the Directed Landscape and Its Applications to Fractal Geometry","authors":"Manan Bhatia","doi":"10.1093/imrn/rnae051","DOIUrl":"https://doi.org/10.1093/imrn/rnae051","url":null,"abstract":"Geodesic coalescence, or the tendency of geodesics to merge together, is a hallmark phenomenon observed in a variety of planar random geometries involving a random distortion of the Euclidean metric. As a result of this, the union of interiors of all geodesics going to a fixed point tends to form a tree-like structure that is supported on a vanishing fraction of the space. Such geodesic trees exhibit intricate fractal behaviour; for instance, while almost every point in the space has only one geodesic going to the fixed point, there exist atypical points that admit two such geodesics. In this paper, we consider the directed landscape, the recently constructed [ 18] scaling limit of exponential last passage percolation (LPP), with the aim of developing tools to analyse the fractal aspects of the tree of semi-infinite geodesics in a given direction. We use the duality [ 39] between the geodesic tree and the interleaving competition interfaces in exponential LPP to obtain a duality between the geodesic tree and the corresponding dual tree in the landscape. Using this, we show that problems concerning the fractal behaviour of sets of atypical points for the geodesic tree can be transformed into corresponding problems for the dual tree, which might turn out to be easier. In particular, we use this method to show that the set of points admitting two semi-infinite geodesics in a fixed direction a.s. has Hausdorff dimension $4/3$, thereby answering a question posed in [ 12]. We also show that the set of points admitting three semi-infinite geodesics in a fixed direction is a.s. countable.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"102 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575504","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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