The paper provides an overarching framework for the study of some of the intrinsic geometries that a topological group may carry. An initial analysis is based on geometric nonlinear functional analysis, that is, the study of Banach spaces as metric spaces up to various notions of isomorphism, such as bi-Lipschitz equivalence, uniform homeomorphism, and coarse equivalence. This motivates the introduction of the various geometric categories applicable to all topological groups, namely, their uniform and coarse structure, along with those applicable to a more select class, that is, (local) Lipschitz and quasimetric structure. Our study touches on Lie theory, geometric group theory, and geometric nonlinear functional analysis and makes evident that these can all be seen as instances of a single coherent theory.
{"title":"Geometries of topological groups","authors":"Christian Rosendal","doi":"10.1090/bull/1807","DOIUrl":"https://doi.org/10.1090/bull/1807","url":null,"abstract":"The paper provides an overarching framework for the study of some of the intrinsic geometries that a topological group may carry. An initial analysis is based on geometric nonlinear functional analysis, that is, the study of Banach spaces as metric spaces up to various notions of isomorphism, such as bi-Lipschitz equivalence, uniform homeomorphism, and coarse equivalence. This motivates the introduction of the various geometric categories applicable to all topological groups, namely, their uniform and coarse structure, along with those applicable to a more select class, that is, (local) Lipschitz and quasimetric structure. Our study touches on Lie theory, geometric group theory, and geometric nonlinear functional analysis and makes evident that these can all be seen as instances of a single coherent theory.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46119547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yuri Manin not only made seminal contributions to a broad array of abstract mathematics and theoretical physics, he also brought his intellectual prowess to bear on a wide range of humanistic endeavors. In the following review W.T. Gowers discusses Manin’s book, Mathematics as Metaphor.
{"title":"Yuri Ivanovich Manin, An extraordinary mathematician","authors":"","doi":"10.1090/bull/1801","DOIUrl":"https://doi.org/10.1090/bull/1801","url":null,"abstract":"Yuri Manin not only made seminal contributions to a broad array of abstract mathematics and theoretical physics, he also brought his intellectual prowess to bear on a wide range of humanistic endeavors. In the following review W.T. Gowers discusses Manin’s book, Mathematics as Metaphor.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47537760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Francis Sowerby Macaulay began his career working on Brill and Noether’s theory of algebraic plane curves and their interpretation of the Riemann–Roch and Cayley–Bacharach theorems; in fact it is Macaulay who first stated and proved the modern form of the Cayley–Bacharach theorem. Later in his career Macaulay developed ideas and results that have become important in modern commutative algebra, such as the notions of unmixedness, perfection (the Cohen–Macaulay property), and super-perfection (the Gorenstein property), work that was appreciated by Emmy Noether and the people around her. He also discovered results that are now fundamental in the theory of linkage, but this work was forgotten and independently rediscovered much later. The name of a computer algebra program (now Macaulay2) recognizes that much of his work is based on examples created by refined computation. Though he never spoke of the connection, the threads of Macaulay’s work lead directly from the problems on plane curves to his later theorems. In this paper we will explain what Macaulay did, and how his results are connected.
Francis Sowerby Macaulay的职业生涯始于研究Brill和Noether的代数平面曲线理论以及他们对Riemann-Roch定理和Cayley-Bacharach定理的解释;事实上，是麦考利首先陈述并证明了现代形式的凯莱-巴沙拉克定理。在他职业生涯的后期，麦考利提出了一些在现代交换代数中变得重要的思想和结果，比如无混合、完美(科恩-麦考利性质)和超完美(戈伦斯坦性质)的概念，这些工作得到了埃米·诺特和她周围的人的赞赏。他还发现了一些结果，这些结果现在是连杆理论的基础，但这项工作被遗忘了，很久以后才被独立地重新发现。计算机代数程序(现在的Macaulay2)的名字认识到他的大部分工作都是基于精确计算创建的示例。虽然他从来没有提到这种联系，但麦考利的工作线索直接从平面曲线的问题引出了他后来的定理。在本文中，我们将解释麦考利做了什么，以及他的结果是如何联系起来的。
{"title":"F. S. Macaulay: From plane curves to Gorenstein rings","authors":"D. Eisenbud, J. Gray","doi":"10.1090/bull/1787","DOIUrl":"https://doi.org/10.1090/bull/1787","url":null,"abstract":"Francis Sowerby Macaulay began his career working on Brill and Noether’s theory of algebraic plane curves and their interpretation of the Riemann–Roch and Cayley–Bacharach theorems; in fact it is Macaulay who first stated and proved the modern form of the Cayley–Bacharach theorem. Later in his career Macaulay developed ideas and results that have become important in modern commutative algebra, such as the notions of unmixedness, perfection (the Cohen–Macaulay property), and super-perfection (the Gorenstein property), work that was appreciated by Emmy Noether and the people around her. He also discovered results that are now fundamental in the theory of linkage, but this work was forgotten and independently rediscovered much later. The name of a computer algebra program (now Macaulay2) recognizes that much of his work is based on examples created by refined computation.\u0000\u0000Though he never spoke of the connection, the threads of Macaulay’s work lead directly from the problems on plane curves to his later theorems. In this paper we will explain what Macaulay did, and how his results are connected.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45761355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Book Review: Elements of $infty $-categories","authors":"C. Weibel","doi":"10.1090/bull/1798","DOIUrl":"https://doi.org/10.1090/bull/1798","url":null,"abstract":"","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49212748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This note is a corrigendum to the paper by Elena Giorgi [Bull. Amer. Math. Soc. 60 (2023), no. 1, 1–27] pointing out a misrepresentaton of the “Collapse conjecture”, which was proved by Christodoulou [The formation of black holes in general relativity, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2009].
{"title":"Corrigendum to “Stable black holes: In vacuum and beyond”","authors":"Elena Giorgi","doi":"10.1090/bull/1797","DOIUrl":"https://doi.org/10.1090/bull/1797","url":null,"abstract":"This note is a corrigendum to the paper by Elena Giorgi [Bull. Amer. Math. Soc. 60 (2023), no. 1, 1–27] pointing out a misrepresentaton of the “Collapse conjecture”, which was proved by Christodoulou [The formation of black holes in general relativity, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2009].","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49484634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper surveys the long-standing connections and impact between Vaughan Jones’s theory of subfactors and various topics in mathematical physics, namely statistical mechanics, quantum field theory, quantum information, and two-dimensional conformal field theory.
{"title":"Subfactors and mathematical physics","authors":"David E. Evans, Yasuyuki Kawahigashi","doi":"10.1090/bull/1799","DOIUrl":"https://doi.org/10.1090/bull/1799","url":null,"abstract":"This paper surveys the long-standing connections and impact between Vaughan Jones’s theory of subfactors and various topics in mathematical physics, namely statistical mechanics, quantum field theory, quantum information, and two-dimensional conformal field theory.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46743555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article is a survey of Louis Nirenberg’s contributions to linear partial differential equations, focusing on his groundbreaking work on pseudo-differential operators and solvability.
{"title":"Nirenberg’s contributions to linear partial differential equations: Pseudo-differential operators and solvability","authors":"N. Dencker","doi":"10.1090/bull/1791","DOIUrl":"https://doi.org/10.1090/bull/1791","url":null,"abstract":"This article is a survey of Louis Nirenberg’s contributions to linear partial differential equations, focusing on his groundbreaking work on pseudo-differential operators and solvability.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43947008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Book Review: Topological methods in hydrodynamics","authors":"G. Misiołek","doi":"10.1090/bull/1794","DOIUrl":"https://doi.org/10.1090/bull/1794","url":null,"abstract":"","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45934075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A Teichmüller curve V ⊂ M g V subset mathcal {M}_g is an isometrically immersed algebraic curve in the moduli space of Riemann surfaces. These rare, extremal objects are related to billiards in polygons, Hodge theory, algebraic geometry and surface topology. This paper presents the six known families of primitive Teichmüller curves that have been discovered over the past 30 years, and a selection of open problems.
Teichmüller曲线V⊂M g Vsubetmathcal{M}_g是黎曼曲面模空间中的等距浸入代数曲线。这些罕见的极端物体与多边形中的台球、霍奇理论、代数几何和表面拓扑有关。本文介绍了在过去30年中发现的六个已知的原始Teichmüller曲线族，以及一些悬而未决的问题。
{"title":"Billiards and Teichmüller curves","authors":"C. McMullen","doi":"10.1090/bull/1782","DOIUrl":"https://doi.org/10.1090/bull/1782","url":null,"abstract":"A Teichmüller curve \u0000\u0000 \u0000 \u0000 V\u0000 ⊂\u0000 \u0000 \u0000 M\u0000 \u0000 g\u0000 \u0000 \u0000 V subset mathcal {M}_g\u0000 \u0000\u0000 is an isometrically immersed algebraic curve in the moduli space of Riemann surfaces. These rare, extremal objects are related to billiards in polygons, Hodge theory, algebraic geometry and surface topology. This paper presents the six known families of primitive Teichmüller curves that have been discovered over the past 30 years, and a selection of open problems.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43727613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Matroids are combinatorial abstractions of independence, a ubiquitous notion that pervades many branches of mathematics. June Huh and his collaborators recently made spectacular breakthroughs by developing a Hodge theory of matroids that resolved several long-standing conjectures in matroid theory. We survey the main results in this development and ideas behind them.
{"title":"Essence of independence: Hodge theory of matroids since June Huh","authors":"C. Eur","doi":"10.1090/bull/1803","DOIUrl":"https://doi.org/10.1090/bull/1803","url":null,"abstract":"Matroids are combinatorial abstractions of independence, a ubiquitous notion that pervades many branches of mathematics. June Huh and his collaborators recently made spectacular breakthroughs by developing a Hodge theory of matroids that resolved several long-standing conjectures in matroid theory. We survey the main results in this development and ideas behind them.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41704786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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