This article is a survey of Cathleen Morawetz’s contributions to the mathematical theory of transonic flows, shock waves, and partial differential equations of mixed elliptic-hyperbolic type. The main focus is on Morawetz’s fundamental work on the nonexistence of continuous transonic flows past profiles, Morawetz’s program regarding the construction of global steady weak transonic flow solutions past profiles via compensated compactness, and a potential theory for regular and Mach reflection of a shock at a wedge. The profound impact of Morawetz’s work on recent developments and breakthroughs in these research directions and related areas in pure and applied mathematics are also discussed.
{"title":"Morawetz’s contributions to the mathematical theory of transonic flows, shock waves, and partial differential equations of mixed type","authors":"Gui-Qiang Chen","doi":"10.1090/bull/1816","DOIUrl":"https://doi.org/10.1090/bull/1816","url":null,"abstract":"This article is a survey of Cathleen Morawetz’s contributions to the mathematical theory of transonic flows, shock waves, and partial differential equations of mixed elliptic-hyperbolic type. The main focus is on Morawetz’s fundamental work on the nonexistence of continuous transonic flows past profiles, Morawetz’s program regarding the construction of global steady weak transonic flow solutions past profiles via compensated compactness, and a potential theory for regular and Mach reflection of a shock at a wedge. The profound impact of Morawetz’s work on recent developments and breakthroughs in these research directions and related areas in pure and applied mathematics are also discussed.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135728944","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":"Commentary","authors":"Susan Friedlander","doi":"10.1090/bull/1817","DOIUrl":"https://doi.org/10.1090/bull/1817","url":null,"abstract":"","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135994515","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}
James Maynard has taken the analytic number theory world by storm in the last decade, proving several important and surprising theorems, resolving questions that had seemed far out of reach. He is perhaps best known for his work on small and large gaps between primes (which were discussed, hot off the press, in our 2015 Bulletin of the AMS article). In this article we will discuss two other Maynard breakthroughs: — Mersenne numbers take the form 2n−12^n-1 and so appear as 111…111111dots 111 in base 2, having no digit “00”. It is a famous conjecture that there are infinitely many such primes. More generally it was, until Maynard’s work, an open question as to whether there are infinitely many primes that miss any given digit, in any given base. We will discuss Maynard’s beautiful ideas that went into his 2019 partial resolution of this question. — In 1926, Khinchin gave remarkable conditions for when real numbers can usually be “well approximated” by infinitely many rationals. However Khinchin’s theorem regarded 1/2, 2/4, 3/6 as distinct rationals and so could not be easily modified to cope, say, with approximations by fractions with prime denominators. In 1941 Duffin and Schaeffer proposed an appropriate but significantly more general analogy involving approximation only by reduced fractions (which is much more useful). We will discuss its 2020 resolution by Maynard and Dimitris Koukoulopoulos.
{"title":"Missing digits and good approximations","authors":"Andrew Granville","doi":"10.1090/bull/1811","DOIUrl":"https://doi.org/10.1090/bull/1811","url":null,"abstract":"James Maynard has taken the analytic number theory world by storm in the last decade, proving several important and surprising theorems, resolving questions that had seemed far out of reach. He is perhaps best known for his work on small and large gaps between primes (which were discussed, hot off the press, in our 2015 <italic>Bulletin of the AMS</italic> article). In this article we will discuss two other Maynard breakthroughs: — Mersenne numbers take the form <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2 Superscript n Baseline minus 1\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">2^n-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and so appear as <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"111 ellipsis 111\"> <mml:semantics> <mml:mrow> <mml:mn>111</mml:mn> <mml:mo>…<!-- … --></mml:mo> <mml:mn>111</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">111dots 111</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in base 2, having no digit “<inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0\"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding=\"application/x-tex\">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>”. It is a famous conjecture that there are infinitely many such primes. More generally it was, until Maynard’s work, an open question as to whether there are infinitely many primes that miss any given digit, in any given base. We will discuss Maynard’s beautiful ideas that went into his 2019 partial resolution of this question. — In 1926, Khinchin gave remarkable conditions for when real numbers can usually be “well approximated” by infinitely many rationals. However Khinchin’s theorem regarded 1/2, 2/4, 3/6 as distinct rationals and so could not be easily modified to cope, say, with approximations by fractions with prime denominators. In 1941 Duffin and Schaeffer proposed an appropriate but significantly more general analogy involving approximation only by reduced fractions (which is much more useful). We will discuss its 2020 resolution by Maynard and Dimitris Koukoulopoulos.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136115971","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}
In this survey, we present the most recent highlights from the study of the homology cobordism group, with particular emphasis on its long-standing and rich history in the context of smooth manifolds. Further, we list various results on its algebraic structure and discuss its crucial role in the development of low-dimensional topology. Also, we share a series of open problems about the behavior of homology 33-spheres and the structure of ΘZ3Theta _{mathbb {Z}}^3. Finally, we briefly discuss the knot concordance group Cmathcal {C} and the rational homology cobordism group ΘQ3Theta _{mathbb {Q}}^3, focusing on their algebraic structures, relating them to ΘZ3Theta _{mathbb {Z}}^3, and highlighting several open problems. The appendix is a compilation of sev
{"title":"A survey of the homology cobordism group","authors":"Oğuz Şavk","doi":"10.1090/bull/1806","DOIUrl":"https://doi.org/10.1090/bull/1806","url":null,"abstract":"In this survey, we present the most recent highlights from the study of the homology cobordism group, with particular emphasis on its long-standing and rich history in the context of smooth manifolds. Further, we list various results on its algebraic structure and discuss its crucial role in the development of low-dimensional topology. Also, we share a series of open problems about the behavior of homology <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=\"application/x-tex\">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-spheres and the structure of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Theta Subscript double-struck upper Z Superscript 3\"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant=\"normal\">Θ<!-- Θ --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>3</mml:mn> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">Theta _{mathbb {Z}}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Finally, we briefly discuss the knot concordance group <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper C\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathcal {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the rational homology cobordism group <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Theta Subscript double-struck upper Q Superscript 3\"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant=\"normal\">Θ<!-- Θ --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>3</mml:mn> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">Theta _{mathbb {Q}}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, focusing on their algebraic structures, relating them to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Theta Subscript double-struck upper Z Superscript 3\"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant=\"normal\">Θ<!-- Θ --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>3</mml:mn> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">Theta _{mathbb {Z}}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and highlighting several open problems. The appendix is a compilation of sev","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135345434","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}
Viazovska’s solution of the sphere packing problem in eight dimensions is based on a remarkable construction of certain special functions using modular forms. Great mathematics has consequences far beyond the problems that originally inspired it, and Viazovska’s work is no exception. In this article, we’ll examine how it has led to new interpolation theorems in Fourier analysis, specifically a theorem of Radchenko and Viazovska.
{"title":"From sphere packing to Fourier interpolation","authors":"Henry Cohn","doi":"10.1090/bull/1813","DOIUrl":"https://doi.org/10.1090/bull/1813","url":null,"abstract":"Viazovska’s solution of the sphere packing problem in eight dimensions is based on a remarkable construction of certain special functions using modular forms. Great mathematics has consequences far beyond the problems that originally inspired it, and Viazovska’s work is no exception. In this article, we’ll examine how it has led to new interpolation theorems in Fourier analysis, specifically a theorem of Radchenko and Viazovska.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135346343","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: Invitation to nonlinear algebra","authors":"Alicia Dickenstein, Giorgio Ottaviani","doi":"10.1090/bull/1814","DOIUrl":"https://doi.org/10.1090/bull/1814","url":null,"abstract":"","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139322292","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}
Over the past decade or so, a broad research programme spearheaded by H. Duminil-Copin and his collaborators has vastly increased our understanding of a number of critical or near-critical statistical mechanics models. Most prominently, these include the q q -state Potts models and, essentially equivalently, the FK cluster models. In this short review, we present a small selection of recent results from this research area.
{"title":"A stroll around the critical Potts model","authors":"Martin Hairer","doi":"10.1090/bull/1802","DOIUrl":"https://doi.org/10.1090/bull/1802","url":null,"abstract":"Over the past decade or so, a broad research programme spearheaded by H. Duminil-Copin and his collaborators has vastly increased our understanding of a number of critical or near-critical statistical mechanics models. Most prominently, these include the \u0000\u0000 \u0000 q\u0000 q\u0000 \u0000\u0000-state Potts models and, essentially equivalently, the FK cluster models. In this short review, we present a small selection of recent results from this research area.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45978205","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: Amenability of discrete groups by examples","authors":"N. Matte Bon","doi":"10.1090/bull/1809","DOIUrl":"https://doi.org/10.1090/bull/1809","url":null,"abstract":"","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139358601","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}
We describe Vaughan Jones’s ground-breaking discovery that symmetries of II1mathrm {II}_1 factors, as encoded by their subfactors, are quantized and have a natural index that can be non-integral. We then comment on the impact his revolutionary work had in the study of II1mathrm {II}_1 factors.
我们描述了Vaughan Jones的突破性发现，即I I 1mathrm的对称性{II}_1因子，由它们的子因子编码，是量化的，并且具有可以是非积分的自然索引。然后，我们评论了他的革命工作对I I 1mathrm研究的影响{II}_1因素。
{"title":"The legacy of Vaughan Jones in 𝐼𝐼₁ factors","authors":"S. Popa","doi":"10.1090/bull/1805","DOIUrl":"https://doi.org/10.1090/bull/1805","url":null,"abstract":"<p>We describe Vaughan Jones’s ground-breaking discovery that symmetries of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper I normal upper I Subscript 1\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">I</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">I</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathrm {II}_1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> factors, as encoded by their subfactors, are quantized and have a natural index that can be non-integral. We then comment on the impact his revolutionary work had in the study of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper I normal upper I Subscript 1\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">I</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">I</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathrm {II}_1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> factors.</p>","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41924491","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 essay is a remembrance of Vaughan Jones and a diagrammatic exposition of the remarkable breakthroughs in knot theory and low-dimensional topology that were catalyzed by his work.
这篇文章是对沃恩·琼斯的纪念，也是对他在结理论和低维拓扑方面取得的显著突破的图解阐述。
{"title":"The Jones polynomial, Knots, diagrams, and categories","authors":"L. Kauffman","doi":"10.1090/bull/1792","DOIUrl":"https://doi.org/10.1090/bull/1792","url":null,"abstract":"This essay is a remembrance of Vaughan Jones and a diagrammatic exposition of the remarkable breakthroughs in knot theory and low-dimensional topology that were catalyzed by his work.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41805544","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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