{"title":"Book Review: Alice and Bob meet Banach: The interface of asymptotic geometric analysis and quantum information theory","authors":"Michael Brannan","doi":"10.1090/bull/1706","DOIUrl":"https://doi.org/10.1090/bull/1706","url":null,"abstract":"","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":"1 1","pages":"1"},"PeriodicalIF":1.3,"publicationDate":"2020-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49484404","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 these notes we aim to give a friendly introduction to the theory of overconvergent modular forms and some examples of recent arithmetic applications. The emphasis is on explicit examples and computations.
{"title":"Overconvergent modular forms and their explicit arithmetic","authors":"Jan Vonk","doi":"10.1090/bull/1700","DOIUrl":"https://doi.org/10.1090/bull/1700","url":null,"abstract":". In these notes we aim to give a friendly introduction to the theory of overconvergent modular forms and some examples of recent arithmetic applications. The emphasis is on explicit examples and computations.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2020-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44783201","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}
The Sullivan dictionary provides a beautiful correspondence between Kleinian groups acting on hyperbolic space and rational maps of the extended complex plane. We focus on the setting of geometrically finite Kleinian groups with parabolic elements and parabolic rational maps. In this context an especially direct correspondence exists concerning the dimension theory of the associated limit sets and Julia sets. In recent work we established formulae for the Assouad type dimensions and spectra for these fractal sets and certain conformal measures they support. This allows a rather more nuanced comparison of the two families in the context of dimension. In this expository article we discuss how these results provide new entries in the Sullivan dictionary, as well as revealing striking differences between the two families.
{"title":"A new perspective on the Sullivan dictionary via Assouad type dimensions and spectra","authors":"J. Fraser, Liam Stuart","doi":"10.1090/bull/1796","DOIUrl":"https://doi.org/10.1090/bull/1796","url":null,"abstract":"The Sullivan dictionary provides a beautiful correspondence between Kleinian groups acting on hyperbolic space and rational maps of the extended complex plane. We focus on the setting of geometrically finite Kleinian groups with parabolic elements and parabolic rational maps. In this context an especially direct correspondence exists concerning the dimension theory of the associated limit sets and Julia sets. In recent work we established formulae for the Assouad type dimensions and spectra for these fractal sets and certain conformal measures they support. This allows a rather more nuanced comparison of the two families in the context of dimension. In this expository article we discuss how these results provide new entries in the Sullivan dictionary, as well as revealing striking differences between the two families.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2020-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47767864","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}
. Moving boundary problems are ubiquitous in nature, technology, and engineering. Examples include the human heart and heart valves inter- acting with blood flow, biodegradable microbeads swimming in water to clean up water pollution, a micro camera in the human intestine used for early colon cancer detection, and the design of next-generation vascular stents to prop open clogged arteries and to prevent heart attacks. These are time-dependent, dynamic processes, which involve the interaction between fluids and various structures. Analysis and numerical simulation of fluid-structure interaction (FSI) problems can provide insight into the “invisible” properties of flows and structures, and can be used to advance design of novel technologies and im-prove the understanding of many physical and biological phenomena. Math- ematical analysis of FSI models is at the core of this understanding. In this paper we give a brief survey of recent progress in the area of mathematical well-posedness for moving boundary problems describing fluid-structure interaction between incompressible, viscous fluids and elastic, viscoelastic, and rigid solids.
{"title":"Moving boundary problems","authors":"S. Čanić","doi":"10.1090/bull/1703","DOIUrl":"https://doi.org/10.1090/bull/1703","url":null,"abstract":". Moving boundary problems are ubiquitous in nature, technology, and engineering. Examples include the human heart and heart valves inter- acting with blood flow, biodegradable microbeads swimming in water to clean up water pollution, a micro camera in the human intestine used for early colon cancer detection, and the design of next-generation vascular stents to prop open clogged arteries and to prevent heart attacks. These are time-dependent, dynamic processes, which involve the interaction between fluids and various structures. Analysis and numerical simulation of fluid-structure interaction (FSI) problems can provide insight into the “invisible” properties of flows and structures, and can be used to advance design of novel technologies and im-prove the understanding of many physical and biological phenomena. Math- ematical analysis of FSI models is at the core of this understanding. In this paper we give a brief survey of recent progress in the area of mathematical well-posedness for moving boundary problems describing fluid-structure interaction between incompressible, viscous fluids and elastic, viscoelastic, and rigid solids.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2020-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/bull/1703","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44338642","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 begins with a provocative question: Are there any genuine continuous multivariate real-valued functions? This may seem to be a silly question, but it is in essence what David Hilbert asked as one of the 23 problems he posed at the second International Congress of Mathematicians, held in Paris in 1900. These problems guided a large portion of the research in mathematics of the 20th century. Hilbert’s 13th problem conjectured that there exists a continuous function f : I 3 → R , where I = [0 , 1], which cannot be expressed in terms of composition and addition of continuous functions from R 2 → R , that is, as composition and addition of continuous real-valued functions of two variables. It took over 50 years to prove that Hilbert’s conjecture is false. This article discusses the solution.
{"title":"Hilbert 13: Are there any genuine continuous multivariate real-valued functions?","authors":"S. Morris","doi":"10.1090/bull/1698","DOIUrl":"https://doi.org/10.1090/bull/1698","url":null,"abstract":". This article begins with a provocative question: Are there any genuine continuous multivariate real-valued functions? This may seem to be a silly question, but it is in essence what David Hilbert asked as one of the 23 problems he posed at the second International Congress of Mathematicians, held in Paris in 1900. These problems guided a large portion of the research in mathematics of the 20th century. Hilbert’s 13th problem conjectured that there exists a continuous function f : I 3 → R , where I = [0 , 1], which cannot be expressed in terms of composition and addition of continuous functions from R 2 → R , that is, as composition and addition of continuous real-valued functions of two variables. It took over 50 years to prove that Hilbert’s conjecture is false. This article discusses the solution.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2020-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/bull/1698","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44854798","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":"About the cover: Tribute to Elias Stein","authors":"C. Fefferman","doi":"10.1090/bull/1704","DOIUrl":"https://doi.org/10.1090/bull/1704","url":null,"abstract":"","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":"57 1","pages":"639-640"},"PeriodicalIF":1.3,"publicationDate":"2020-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46917214","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: Boundary value problems, Weyl functions, and differential operators","authors":"F. Gesztesy","doi":"10.1090/bull/1705","DOIUrl":"https://doi.org/10.1090/bull/1705","url":null,"abstract":"","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":"58 1","pages":"129-136"},"PeriodicalIF":1.3,"publicationDate":"2020-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48859891","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}
Decisions about how the population of the United States should be divided into legislative districts have powerful and not fully understood effects on the outcomes of elections. The problem of understanding what we might mean by "fair districting" intertwines mathematical, political, and legal reasoning; but only in recent years has the academic mathematical community gotten directly involved in the process. I'll report on recent progress in this area, how newly developed mathematical tools have affected real political decisions, and what remains to be done. This survey represents the content of a lecture presented by the author in the Current Events Bulletin session of the Joint Mathematics Meetings in January 2020.
{"title":"Geometry, inference, complexity, and democracy","authors":"J. Ellenberg","doi":"10.1090/bull/1708","DOIUrl":"https://doi.org/10.1090/bull/1708","url":null,"abstract":"Decisions about how the population of the United States should be divided into legislative districts have powerful and not fully understood effects on the outcomes of elections. The problem of understanding what we might mean by \"fair districting\" intertwines mathematical, political, and legal reasoning; but only in recent years has the academic mathematical community gotten directly involved in the process. I'll report on recent progress in this area, how newly developed mathematical tools have affected real political decisions, and what remains to be done. This survey represents the content of a lecture presented by the author in the Current Events Bulletin session of the Joint Mathematics Meetings in January 2020.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2020-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42705511","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: A comprehensive introduction to sub-Riemannian geometry. From the Hamiltonian viewpoint","authors":"R. Montgomery","doi":"10.1090/bull/1701","DOIUrl":"https://doi.org/10.1090/bull/1701","url":null,"abstract":"","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2020-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60549696","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 1922, Mordell conjectured the striking statement that for a polynomial equation $f(x,y)=0$, if the topology of the set of complex number solutions is complicated enough, then the set of rational number solutions is finite. This was proved by Faltings in 1983, and again by a different method by Vojta in 1991, but neither proof provided a way to provably find all the rational solutions, so the search for other proofs has continued. Recently, Lawrence and Venkatesh found a third proof, relying on variation in families of $p$-adic Galois representations; this is the subject of the present exposition.
{"title":"A $p$-adic approach to rational points on curves","authors":"B. Poonen","doi":"10.1090/bull/1707","DOIUrl":"https://doi.org/10.1090/bull/1707","url":null,"abstract":"In 1922, Mordell conjectured the striking statement that for a polynomial equation $f(x,y)=0$, if the topology of the set of complex number solutions is complicated enough, then the set of rational number solutions is finite. This was proved by Faltings in 1983, and again by a different method by Vojta in 1991, but neither proof provided a way to provably find all the rational solutions, so the search for other proofs has continued. Recently, Lawrence and Venkatesh found a third proof, relying on variation in families of $p$-adic Galois representations; this is the subject of the present exposition.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2020-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45656508","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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