{"title":"Stationary $C^*$-dynamical systems (with an appendix by Uri Bader, Yair Hartman, and Mehrdad Kalantar)","authors":"Yair Hartman, Mehrdad Kalantar","doi":"10.4171/jems/1225","DOIUrl":"https://doi.org/10.4171/jems/1225","url":null,"abstract":"","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2022-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89832133","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":"The Riemannian quantitative isoperimetric inequality","authors":"Otis Chodosh,Max Engelstein,Luca Spolaor","doi":"10.4171/jems/1223","DOIUrl":"https://doi.org/10.4171/jems/1223","url":null,"abstract":"","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2022-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138503877","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":"Integral $p$-adic Hodge filtrations in low dimension and ramification","authors":"Shizhang Li","doi":"10.4171/jems/1237","DOIUrl":"https://doi.org/10.4171/jems/1237","url":null,"abstract":"","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2022-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138503876","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":"Instabilities of invariant quasi-periodic tori","authors":"Gerard Farré,Bassam Fayad","doi":"10.4171/jems/1206","DOIUrl":"https://doi.org/10.4171/jems/1206","url":null,"abstract":"","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138503875","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}
In a smoothly bounded convex domain Ω ⊂ R, we consider the chemotaxis-Navier-Stokes model nt + u · ∇n = ∆n−∇ · (n∇c), x ∈ Ω, t > 0, ct + u · ∇c = ∆c− nc, x ∈ Ω, t > 0, ut + (u · ∇)u = ∆u+∇P + n∇Φ, ∇ · u = 0, x ∈ Ω, t > 0, (⋆) proposed by Goldstein et al. to describe pattern formation in populations of aerobic bacteria interacting with their liquid environment via transport and buoyancy. Known results have asserted that under appropriate regularity assumptions on Φ and the initial data, a corresponding no-flux/noflux/Dirichlet initial-boundary value problem is globally solvable in a framework of so-called weak energy solutions, and that any such solution eventually becomes smooth and classical. Going beyond this, the present work focuses on the possible extent of unboundedness phenomena also on short timescales, and hence investigates in more detail the set of times in (0,∞) at which solutions may develop singularities. The main results in this direction reveal the existence of a global weak energy solution which coincides with a smooth function throughout Ω × E, where E denotes a countable union of open intervals which is such that |(0,∞) E| = 0. In particular, this indicates that a similar feature of the unperturbed Navier-Stokes equations, known as Leray’s structure theorem, persists even in the presence of the coupling to the attractive and hence potentially destabilizing cross-diffusive mechanism in the full system (⋆).
{"title":"Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid interaction?","authors":"M. Winkler","doi":"10.4171/jems/1226","DOIUrl":"https://doi.org/10.4171/jems/1226","url":null,"abstract":"In a smoothly bounded convex domain Ω ⊂ R, we consider the chemotaxis-Navier-Stokes model nt + u · ∇n = ∆n−∇ · (n∇c), x ∈ Ω, t > 0, ct + u · ∇c = ∆c− nc, x ∈ Ω, t > 0, ut + (u · ∇)u = ∆u+∇P + n∇Φ, ∇ · u = 0, x ∈ Ω, t > 0, (⋆) proposed by Goldstein et al. to describe pattern formation in populations of aerobic bacteria interacting with their liquid environment via transport and buoyancy. Known results have asserted that under appropriate regularity assumptions on Φ and the initial data, a corresponding no-flux/noflux/Dirichlet initial-boundary value problem is globally solvable in a framework of so-called weak energy solutions, and that any such solution eventually becomes smooth and classical. Going beyond this, the present work focuses on the possible extent of unboundedness phenomena also on short timescales, and hence investigates in more detail the set of times in (0,∞) at which solutions may develop singularities. The main results in this direction reveal the existence of a global weak energy solution which coincides with a smooth function throughout Ω × E, where E denotes a countable union of open intervals which is such that |(0,∞) E| = 0. In particular, this indicates that a similar feature of the unperturbed Navier-Stokes equations, known as Leray’s structure theorem, persists even in the presence of the coupling to the attractive and hence potentially destabilizing cross-diffusive mechanism in the full system (⋆).","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84934888","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}
We show that the largest prime factor of $n^2+1$ is infinitely often greater than $n^{1.279}$. This improves the result of de la Bret`eche and Drappeau (2019) who obtained this with $1.2182$ in place of $1.279.$ The main new ingredients in the proof are a new Type II estimate and using this estimate by applying Harman's sieve method. To prove the Type II estimate we use the bounds of Dehouillers and Iwaniec on linear forms of Kloosterman sums. We also show that conditionally on Selberg's eigenvalue conjecture the exponent $1.279$ may be increased to $1.312.$
我们证明了n^2+1$的最大素数因子通常无限大于n^{1.279}$。这改善了de la Bret ' eche和Drappeau(2019)的结果，他们用1.2182美元代替1.279美元获得了这个结果。证明中的主要新成分是一种新的II型估计，并通过哈曼筛子法使用这种估计。为了证明II型估计，我们使用了Kloosterman和的线性形式上的Dehouillers和Iwaniec的界。我们还证明了在Selberg特征值猜想的条件下，指数$1.279$可以增加到$1.312.$
{"title":"On the largest prime factor of $n^2+1$","authors":"Jori Merikoski","doi":"10.4171/jems/1216","DOIUrl":"https://doi.org/10.4171/jems/1216","url":null,"abstract":"We show that the largest prime factor of $n^2+1$ is infinitely often greater than $n^{1.279}$. This improves the result of de la Bret`eche and Drappeau (2019) who obtained this with $1.2182$ in place of $1.279.$ The main new ingredients in the proof are a new Type II estimate and using this estimate by applying Harman's sieve method. To prove the Type II estimate we use the bounds of Dehouillers and Iwaniec on linear forms of Kloosterman sums. We also show that conditionally on Selberg's eigenvalue conjecture the exponent $1.279$ may be increased to $1.312.$","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2022-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138503874","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}
We prove the last open case of the conjecture on the possible values of the first isotropy index of an anisotropic quadratic form over a field. It was initially stated by Detlev Hoffmann for fields of characteristic ̸= 2 and then extended to arbitrary characteristic by Burt Totaro. The initial statement was proven by the author in 2002. In characteristic 2, the case of a totally singular quadratic form was done by Stephen Scully in 2015 and the nonsingular case by Eric Primozic in early 2019.
{"title":"An ultimate proof of Hoffmann–Totaro’s conjecture","authors":"N. Karpenko","doi":"10.4171/jems/1211","DOIUrl":"https://doi.org/10.4171/jems/1211","url":null,"abstract":"We prove the last open case of the conjecture on the possible values of the first isotropy index of an anisotropic quadratic form over a field. It was initially stated by Detlev Hoffmann for fields of characteristic ̸= 2 and then extended to arbitrary characteristic by Burt Totaro. The initial statement was proven by the author in 2002. In characteristic 2, the case of a totally singular quadratic form was done by Stephen Scully in 2015 and the nonsingular case by Eric Primozic in early 2019.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2022-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74741293","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":"Erratum to “Uniform K-stability and asymptotics of energy functionals in Kähler geometry”","authors":"S. Boucksom, Tomoyuki Hisamoto, Mattias Jonsson","doi":"10.4171/jems/1215","DOIUrl":"https://doi.org/10.4171/jems/1215","url":null,"abstract":"","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2022-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81451536","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":"Local convergence of random planar graphs","authors":"Benedikt Stufler","doi":"10.4171/jems/1174","DOIUrl":"https://doi.org/10.4171/jems/1174","url":null,"abstract":"","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138503873","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}
We classify the irreducible representations of smooth, connected affine algebraic groups over a field, by tackling the case of pseudo-reductive groups. We reduce the problem of calculating the dimension for pseudo-split pseudo-reductive groups to the split reductive case and the pseudo-split pseudo-reductive commutative case. Moreover, we give the first results on the latter, including a rather complete description of the rank one case.
{"title":"Irreducible modules for pseudo-reductive groups","authors":"Michael Bate, David I. Stewart","doi":"10.4171/jems/1153","DOIUrl":"https://doi.org/10.4171/jems/1153","url":null,"abstract":"We classify the irreducible representations of smooth, connected affine algebraic groups over a field, by tackling the case of pseudo-reductive groups. We reduce the problem of calculating the dimension for pseudo-split pseudo-reductive groups to the split reductive case and the pseudo-split pseudo-reductive commutative case. Moreover, we give the first results on the latter, including a rather complete description of the rank one case.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2021-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138503872","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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