Pub Date : 2021-01-03DOI: 10.4007/annals.2023.197.1.6
Mikolaj Fraczyk, T. Gelander
We prove the following conjecture of Margulis. Let G be a higher rank simple Lie group and let Λ ≤ G be a discrete subgroup of infinite covolume. Then, the locally symmetric space ΛG/K admits injected balls of any radius. This can be considered as a geometric interpretation of the celebrated Margulis normal subgroup theorem. However, it applies to general discrete subgroups not necessarily associated to lattices. Yet, the result is new even for subgroups of infinite index of lattices. We establish similar results for higher rank semisimple groups with Kazhdan’s property (T). We prove a stiffness result for discrete stationary random subgroups in higher rank semisimple groups and a stationary variant of the Stück–Zimmer theorem for higher rank semisimple groups with property (T). We also show that a stationary limit of a measure supported on discrete subgroups is almost surely discrete.
我们证明了马古利斯的下列猜想。设G为高阶单李群，设Λ≤G为无穷余体积的离散子群。然后，局部对称空间ΛG/K允许任意半径的注入球。这可以看作是著名的马古利正规子群定理的一个几何解释。然而，它适用于不一定与格相关联的一般离散子群。然而，即使对于无限索引格的子群，这个结果也是新的。我们在具有Kazhdan性质(T)的高阶半单群上建立了类似的结果。我们证明了高阶半单群中离散平稳随机子群的一个刚度结果和具有性质(T)的高阶半单群的st ck - zimmer定理的一个平稳变式。我们还证明了在离散子群上支持的测度的平稳极限几乎肯定是离散的。
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Pub Date : 2021-01-01DOI: 10.4007/ANNALS.2021.193.2.4
David Harvey, J. Hoeven
We present an algorithm that computes the product of two n-bit integers in O(n log n) bit operations, thus confirming a conjecture of Schonhage and Strassen from 1971. Our complexity analysis takes place in the multitape Turing machine model, with integers encoded in the usual binary representa- tion. Central to the new algorithm is a novel “Gaussian resampling” technique that enables us to reduce the integer multiplication problem to a collection of multidimensional discrete Fourier transforms over the complex numbers, whose dimensions are all powers of two. These transforms may then be evaluated rapidly by means of Nussbaumer’s fast polynomial transforms.
{"title":"Integer multiplication in time O(n log n)","authors":"David Harvey, J. Hoeven","doi":"10.4007/ANNALS.2021.193.2.4","DOIUrl":"https://doi.org/10.4007/ANNALS.2021.193.2.4","url":null,"abstract":"We present an algorithm that computes the product of two n-bit integers in O(n log n) bit operations, thus confirming a conjecture of Schonhage and Strassen from 1971. Our complexity analysis takes place in the multitape Turing machine model, with integers encoded in the usual binary representa- tion. Central to the new algorithm is a novel “Gaussian resampling” technique that enables us to reduce the integer multiplication problem to a collection of multidimensional discrete Fourier transforms over the complex numbers, whose dimensions are all powers of two. These transforms may then be evaluated rapidly by means of Nussbaumer’s fast polynomial transforms.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":4.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70186827","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}
{"title":"Coefficients of Maass Forms and the Siegel Zero","authors":"Jeffrey Hoffstein And Paul Lockhart","doi":"10.2307/2118543","DOIUrl":"https://doi.org/10.2307/2118543","url":null,"abstract":"","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":4.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2307/2118543","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69188289","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 : 2021-01-01DOI: 10.4007/annals.2021.194.2.1
Oh, Tataru
{"title":"The threshold conjecture for the energy critical hyperbolic\u0000 Yang–Mills equation","authors":"Oh, Tataru","doi":"10.4007/annals.2021.194.2.1","DOIUrl":"https://doi.org/10.4007/annals.2021.194.2.1","url":null,"abstract":"","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":4.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70186889","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 : 2021-01-01DOI: 10.4007/ANNALS.2021.193.3.3
D. Asperó, R. Schindler
We show that Martin’s Maximum++ implies Woodin’s Pmax axiom (∗). This answers a question from the 1990’s and amalgamates two prominent axioms of set theory which were both known to imply that there are א2 many real numbers.
{"title":"Martin's Maximum++ implies Woodin's axiom\u0000 (*)","authors":"D. Asperó, R. Schindler","doi":"10.4007/ANNALS.2021.193.3.3","DOIUrl":"https://doi.org/10.4007/ANNALS.2021.193.3.3","url":null,"abstract":"We show that Martin’s Maximum++ implies Woodin’s Pmax axiom (∗). This answers a question from the 1990’s and amalgamates two prominent axioms of set theory which were both known to imply that there are א2 many real numbers.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"193 1","pages":"793-835"},"PeriodicalIF":4.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70186839","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 : 2021-01-01DOI: 10.4007/annals.2021.193.3.1
Neeman
{"title":"Strong generators in D<sup>perf</sup>(<em>X</em>) and\u0000 D<em><sup>b</sup></em><sub>coh</sub>(<em>X</em>)","authors":"Neeman","doi":"10.4007/annals.2021.193.3.1","DOIUrl":"https://doi.org/10.4007/annals.2021.193.3.1","url":null,"abstract":"","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":4.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70186752","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 : 2021-01-01DOI: 10.4007/annals.2021.193.3.0979
{"title":"Index","authors":"","doi":"10.4007/annals.2021.193.3.0979","DOIUrl":"https://doi.org/10.4007/annals.2021.193.3.0979","url":null,"abstract":"","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":4.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70186925","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 : 2020-12-20DOI: 10.4007/annals.2021.194.3.7
J. Fres'an, P. Jossen
We answer in the negative Siegel’s question whether all E-functions are polynomial expressions in hypergeometric E-functions. Namely, we show that if an irreducible differential operator of order three annihilates an E-function in the hypergeometric class, then the singularities of its Fourier transform are constrained to satisfy a symmetry property that generically does not hold. The proof relies on André’s theory of E-operators and Katz’s computation of the Galois group of hypergeometric differential equations.
{"title":"A non-hypergeometric $E$-function","authors":"J. Fres'an, P. Jossen","doi":"10.4007/annals.2021.194.3.7","DOIUrl":"https://doi.org/10.4007/annals.2021.194.3.7","url":null,"abstract":"We answer in the negative Siegel’s question whether all E-functions are polynomial expressions in hypergeometric E-functions. Namely, we show that if an irreducible differential operator of order three annihilates an E-function in the hypergeometric class, then the singularities of its Fourier transform are constrained to satisfy a symmetry property that generically does not hold. The proof relies on André’s theory of E-operators and Katz’s computation of the Galois group of hypergeometric differential equations.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44995606","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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