abstract:We develop a general noncommutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (M$Delta$Cs). Insight from noncommutative ring theory is used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of an M$Delta$C, ${bf K}$, and then to associate to ${bf K}$ a topological space--the Balmer spectrum ${rm Spc}{bf K}$. We develop a general framework for (noncommutative) support data, coming in three different flavors, and show that ${rm Spc}{bf K}$ is a universal terminal object for the first two notions (support and weak support). The first two types of support data are then used in a theorem that gives a method for the explicit classification of the thick (two-sided) ideals and the Balmer spectrum of an M$Delta$C. The third type (quasi support) is used in another theorem that provides a method for the explicit classification of the thick right ideals of ${bf K}$, which in turn can be applied to classify the thick two-sided ideals and ${rm Spc}{bf K}$.As a special case, our approach can be applied to the stable module categories of arbitrary finite dimensional Hopf algebras that are not necessarily cocommutative (or quasitriangular). We illustrate the general theorems with classifications of the Balmer spectra and thick two-sided/right ideals for the stable module categories of all small quantum groups for Borel subalgebras, and classifications of the Balmer spectra and thick two-sided ideals of Hopf algebras studied by Benson and Witherspoon.
{"title":"Noncommutative tensor triangular geometry","authors":"D. Nakano, Kent B. Vashaw, M. Yakimov","doi":"10.1353/ajm.2022.0041","DOIUrl":"https://doi.org/10.1353/ajm.2022.0041","url":null,"abstract":"abstract:We develop a general noncommutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (M$Delta$Cs). Insight from noncommutative ring theory is used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of an M$Delta$C, ${bf K}$, and then to associate to ${bf K}$ a topological space--the Balmer spectrum ${rm Spc}{bf K}$. We develop a general framework for (noncommutative) support data, coming in three different flavors, and show that ${rm Spc}{bf K}$ is a universal terminal object for the first two notions (support and weak support). The first two types of support data are then used in a theorem that gives a method for the explicit classification of the thick (two-sided) ideals and the Balmer spectrum of an M$Delta$C. The third type (quasi support) is used in another theorem that provides a method for the explicit classification of the thick right ideals of ${bf K}$, which in turn can be applied to classify the thick two-sided ideals and ${rm Spc}{bf K}$.As a special case, our approach can be applied to the stable module categories of arbitrary finite dimensional Hopf algebras that are not necessarily cocommutative (or quasitriangular). We illustrate the general theorems with classifications of the Balmer spectra and thick two-sided/right ideals for the stable module categories of all small quantum groups for Borel subalgebras, and classifications of the Balmer spectra and thick two-sided ideals of Hopf algebras studied by Benson and Witherspoon.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"144 1","pages":"1681 - 1724"},"PeriodicalIF":1.7,"publicationDate":"2019-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48118546","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}
Pub Date : 2019-09-08DOI: 10.2140/pjm.2020.306.203
Huabin Ge, B. Hua, Ze‐hua Zhou
Abstract:This paper investigates circle patterns with obtuse exterior intersection angles on surfaces of finite topological type. We characterise the images of the curvature maps and establish several equivalent conditions regarding long time behaviors of Chow-Luo's combinatorial Ricci flows for these patterns. As consequences, several generalizations of circle pattern theorem are obtained. Moreover, our approach suggests a computational method to find the desired circle patterns.
{"title":"Circle patterns on surfaces of finite topological type","authors":"Huabin Ge, B. Hua, Ze‐hua Zhou","doi":"10.2140/pjm.2020.306.203","DOIUrl":"https://doi.org/10.2140/pjm.2020.306.203","url":null,"abstract":"Abstract:This paper investigates circle patterns with obtuse exterior intersection angles on surfaces of finite topological type. We characterise the images of the curvature maps and establish several equivalent conditions regarding long time behaviors of Chow-Luo's combinatorial Ricci flows for these patterns. As consequences, several generalizations of circle pattern theorem are obtained. Moreover, our approach suggests a computational method to find the desired circle patterns.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"143 1","pages":"1397 - 1430"},"PeriodicalIF":1.7,"publicationDate":"2019-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/pjm.2020.306.203","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47712611","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}
abstract:In this article, we study the sum of additively twisted Fourier coefficients of an irreducible cuspidal automorphic representation of ${rm GL}_2$ or ${rm GL}_3$ over an arbitrary number field. When the representation is unramified at all non-archimedean places, we prove the Wilton type bound for ${rm GL}_2$ and the Miller type bound for ${rm GL}_3$ which are uniform in terms of the additive character.
{"title":"Cancellation in the additive twists of Fourier coefficients for GL2 and GL3 over number fields","authors":"Zhi Qi","doi":"10.1353/ajm.2019.0034","DOIUrl":"https://doi.org/10.1353/ajm.2019.0034","url":null,"abstract":"abstract:In this article, we study the sum of additively twisted Fourier coefficients of an irreducible cuspidal automorphic representation of ${rm GL}_2$ or ${rm GL}_3$ over an arbitrary number field. When the representation is unramified at all non-archimedean places, we prove the Wilton type bound for ${rm GL}_2$ and the Miller type bound for ${rm GL}_3$ which are uniform in terms of the additive character.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"141 1","pages":"1317 - 1345"},"PeriodicalIF":1.7,"publicationDate":"2019-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/ajm.2019.0034","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45876879","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}
abstract:For almost all Riemannian metrics (in the $C^infty$ Baire sense) on a compact manifold with boundary $(M^{n+1},breakpartial M)$, $3leq (n+1)leq 7$, we prove that, for any open subset $V$ of $partial M$, there exists a compact, properly embedded free boundary minimal hypersurface intersecting $V$.
{"title":"Existence of minimal hypersurfaces with non-empty free Boundary for generic metrics","authors":"Zhichao Wang","doi":"10.1353/ajm.2022.0012","DOIUrl":"https://doi.org/10.1353/ajm.2022.0012","url":null,"abstract":"abstract:For almost all Riemannian metrics (in the $C^infty$ Baire sense) on a compact manifold with boundary $(M^{n+1},breakpartial M)$, $3leq (n+1)leq 7$, we prove that, for any open subset $V$ of $partial M$, there exists a compact, properly embedded free boundary minimal hypersurface intersecting $V$.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"144 1","pages":"599 - 606"},"PeriodicalIF":1.7,"publicationDate":"2019-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44915975","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}
abstract:We solve the Dirichlet problem for $k$-Hessian equations on compact complex manifolds with boundary, given the existence of a subsolution. Our method is based on a second order a priori estimate of the solution on the boundary with a particular gradient scale. The scale allows us to apply a blow-up argument to obtain control on all necessary norms of the solution.
{"title":"The Dirichlet problem for the k-Hessian equation on a complex manifold","authors":"Tristan C. Collins, Sebastien Picard","doi":"10.1353/ajm.2022.0040","DOIUrl":"https://doi.org/10.1353/ajm.2022.0040","url":null,"abstract":"abstract:We solve the Dirichlet problem for $k$-Hessian equations on compact complex manifolds with boundary, given the existence of a subsolution. Our method is based on a second order a priori estimate of the solution on the boundary with a particular gradient scale. The scale allows us to apply a blow-up argument to obtain control on all necessary norms of the solution.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"144 1","pages":"1641 - 1680"},"PeriodicalIF":1.7,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42115387","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}
abstract:We prove the existence of rigid compact complex surfaces of general type whose Chern slopes are arbitrarily close to the Bogomolov--Miyaoka--Yau bound of $3$. In addition, each of these surfaces has first Betti number equal to $4$.
{"title":"Rigid surfaces arbitrarily close to the Bogomolov–Miyaoka–Yau line","authors":"Matthew Stover, G. Urz'ua","doi":"10.1353/ajm.2022.0044","DOIUrl":"https://doi.org/10.1353/ajm.2022.0044","url":null,"abstract":"abstract:We prove the existence of rigid compact complex surfaces of general type whose Chern slopes are arbitrarily close to the Bogomolov--Miyaoka--Yau bound of $3$. In addition, each of these surfaces has first Betti number equal to $4$.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"144 1","pages":"1783 - 1804"},"PeriodicalIF":1.7,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48868305","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}
abstract:We prove that the spectral radius of an i.i.d. random walk on ${rm GL}_d(Bbb{C})$ satisfies a strong law of large numbers under finite second moment assumption and a weak law of large numbers under finite first moment. No irreducibility assumption is supposed.
{"title":"Law of large numbers for the spectral radius of random matrix products","authors":"Richard Aoun, Cagri Sert","doi":"10.1353/AJM.2021.0025","DOIUrl":"https://doi.org/10.1353/AJM.2021.0025","url":null,"abstract":"abstract:We prove that the spectral radius of an i.i.d. random walk on ${rm GL}_d(Bbb{C})$ satisfies a strong law of large numbers under finite second moment assumption and a weak law of large numbers under finite first moment. No irreducibility assumption is supposed.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"143 1","pages":"1010 - 995"},"PeriodicalIF":1.7,"publicationDate":"2019-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/AJM.2021.0025","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44503386","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}
abstract:We adapt Guth's polynomial partitioning argument for the Fourier restriction problem to the context of the Kakeya problem. By writing out the induction argument as a recursive algorithm, additional multiscale geometric information is made available. To take advantage of this, we prove that direction-separated tubes satisfy a multiscale version of the polynomial Wolff axioms. Altogether, this yields improved bounds for the Kakeya maximal conjecture in~$Bbb{R}^n$ with $n=5$ or $nge 7$ and improved bounds for the Kakeya set conjecture for an infinite sequence of dimensions.
{"title":"Improved bounds for the Kakeya maximal conjecture in higher dimensions","authors":"J. Hickman, K. Rogers, Ruixiang Zhang","doi":"10.1353/ajm.2022.0037","DOIUrl":"https://doi.org/10.1353/ajm.2022.0037","url":null,"abstract":"abstract:We adapt Guth's polynomial partitioning argument for the Fourier restriction problem to the context of the Kakeya problem. By writing out the induction argument as a recursive algorithm, additional multiscale geometric information is made available. To take advantage of this, we prove that direction-separated tubes satisfy a multiscale version of the polynomial Wolff axioms. Altogether, this yields improved bounds for the Kakeya maximal conjecture in~$Bbb{R}^n$ with $n=5$ or $nge 7$ and improved bounds for the Kakeya set conjecture for an infinite sequence of dimensions.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"144 1","pages":"1511 - 1560"},"PeriodicalIF":1.7,"publicationDate":"2019-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46979672","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}
abstract:Arthur's conjectures predict the existence of some very interesting unitary representations occurring in spaces of automorphic forms. We prove the unitarity of the ``Langlands element'' (i.e., the representation specified by Arthur) of all unipotent Arthur packets for split real exceptional groups. The proof uses Eisenstein series, Langlands' constant term formula and square integrability criterion, analytic properties of intertwining operators, and some mild arithmetic input from the theory of Dirichlet $L$-functions, to reduce to a more combinatorial problem about intertwining operators.
{"title":"On Arthur's unitarity conjecture for split real groups","authors":"Joseph Hundley, S. Miller","doi":"10.1353/ajm.2022.0038","DOIUrl":"https://doi.org/10.1353/ajm.2022.0038","url":null,"abstract":"abstract:Arthur's conjectures predict the existence of some very interesting unitary representations occurring in spaces of automorphic forms. We prove the unitarity of the ``Langlands element'' (i.e., the representation specified by Arthur) of all unipotent Arthur packets for split real exceptional groups. The proof uses Eisenstein series, Langlands' constant term formula and square integrability criterion, analytic properties of intertwining operators, and some mild arithmetic input from the theory of Dirichlet $L$-functions, to reduce to a more combinatorial problem about intertwining operators.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"144 1","pages":"1561 - 1600"},"PeriodicalIF":1.7,"publicationDate":"2019-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45163971","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}
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