For (n+1)-webs by curves in an ambiant n-dimensional manifold, we first define a generalization of the well known Blaschke curvature of the dimension two, which vanishes iff the web has the maximum possible rank which is one. But, contrary to the dimension two where all 3-webs of rank one are locally isomorphic, we prove that there are infinitely many classes of isomorphism for germs of 4-webs by curves of rank one in the dimension three: we provide a procedure for building all of them, and give examples of invariants of these classes allowing in particular to distinguish the so-called quadrilateral webs among them.
{"title":"Étude des (n+1)-tissus de courbes en dimension n","authors":"Jean-Paul Dufour, Daniel Lehmann","doi":"10.5802/crmath.500","DOIUrl":"https://doi.org/10.5802/crmath.500","url":null,"abstract":"For (n+1)-webs by curves in an ambiant n-dimensional manifold, we first define a generalization of the well known Blaschke curvature of the dimension two, which vanishes iff the web has the maximum possible rank which is one. But, contrary to the dimension two where all 3-webs of rank one are locally isomorphic, we prove that there are infinitely many classes of isomorphism for germs of 4-webs by curves of rank one in the dimension three: we provide a procedure for building all of them, and give examples of invariants of these classes allowing in particular to distinguish the so-called quadrilateral webs among them.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":" 1292","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135186698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let G be a connected reductive group over a number field F, and let S be a set (finite or infinite) of places of F. We give a necessary and sufficient condition for the surjectivity of the localization map from H 1 (F,G) to the “direct sum” of the sets H 1 (F v ,G) where v runs over S. In the appendices, we give a new construction of the abelian Galois cohomology of a reductive group over a field of arbitrary characteristic.
让G是一个连接还原组数域F,并让年代是一组(有限或无限)的地方F .我们给一个充要条件的surjectivity定位地图从H 1 (F, G)的“直接求和”集H 1 v (v F, G)运行在附录中,我们给一个新的建设一个还原的阿贝耳伽罗瓦上同调群在任意领域的特点。
{"title":"Criterion for surjectivity of localization in Galois cohomology of a reductive group over a number field","authors":"Mikhail Borovoi","doi":"10.5802/crmath.455","DOIUrl":"https://doi.org/10.5802/crmath.455","url":null,"abstract":"Let G be a connected reductive group over a number field F, and let S be a set (finite or infinite) of places of F. We give a necessary and sufficient condition for the surjectivity of the localization map from H 1 (F,G) to the “direct sum” of the sets H 1 (F v ,G) where v runs over S. In the appendices, we give a new construction of the abelian Galois cohomology of a reductive group over a field of arbitrary characteristic.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":" 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135141651","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop a point-free approach for constructing the Nakano–Vashaw–Yakimov–Balmer spectrum of a noncommutative tensor triangulated category under certain assumptions. In particular, we provide a conceptual way of classifying radical thick tensor ideals of a noncommutative tensor triangulated category using frame theoretic methods, recovering the universal support data in the process. We further show that there is a homeomorphism between the spectral space of radical thick tensor ideals of a noncommutative tensor triangulated category and the collection of open subsets of its spectrum in the Hochster dual topology.
{"title":"Noncommutative tensor triangulated categories and coherent frames","authors":"Vivek Mohan Mallick, Samarpita Ray","doi":"10.5802/crmath.461","DOIUrl":"https://doi.org/10.5802/crmath.461","url":null,"abstract":"We develop a point-free approach for constructing the Nakano–Vashaw–Yakimov–Balmer spectrum of a noncommutative tensor triangulated category under certain assumptions. In particular, we provide a conceptual way of classifying radical thick tensor ideals of a noncommutative tensor triangulated category using frame theoretic methods, recovering the universal support data in the process. We further show that there is a homeomorphism between the spectral space of radical thick tensor ideals of a noncommutative tensor triangulated category and the collection of open subsets of its spectrum in the Hochster dual topology.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":" 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135141499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish a Lipschitz stability inequality for the problem of determining the nonlinear term in a quasilinear elliptic equation by boundary measurements. We give a proof based on a linearization procedure together with special solutions constructed from the fundamental solution of the linearized problem.
{"title":"Stable determination of the nonlinear term in a quasilinear elliptic equation by boundary measurements","authors":"Mourad Choulli","doi":"10.5802/crmath.484","DOIUrl":"https://doi.org/10.5802/crmath.484","url":null,"abstract":"We establish a Lipschitz stability inequality for the problem of determining the nonlinear term in a quasilinear elliptic equation by boundary measurements. We give a proof based on a linearization procedure together with special solutions constructed from the fundamental solution of the linearized problem.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":" 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135141920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let G n,2n be the Grassmannian parameterizing the n-dimensional subspaces of ℂ 2n . The Picard group of G n,2n is generated by a unique ample line bundle 𝒪(1). Let T be a maximal torus of SL(2n,ℂ) which acts on G n,2n and 𝒪(1). By [10, Theorem 3.10, p. 764], 2 is the minimal integer k such that 𝒪(k) descends to the GIT quotient. In this article, we prove that the GIT quotient of G n,2n (n≥3) by T with respect to 𝒪(2)=𝒪(1) ⊗2 is not projectively normal when polarized with the descent of 𝒪(2).
{"title":"Torus quotient of the Grassmannian G n,2n ","authors":"Arpita Nayek, Pinakinath Saha","doi":"10.5802/crmath.501","DOIUrl":"https://doi.org/10.5802/crmath.501","url":null,"abstract":"Let G n,2n be the Grassmannian parameterizing the n-dimensional subspaces of ℂ 2n . The Picard group of G n,2n is generated by a unique ample line bundle 𝒪(1). Let T be a maximal torus of SL(2n,ℂ) which acts on G n,2n and 𝒪(1). By [10, Theorem 3.10, p. 764], 2 is the minimal integer k such that 𝒪(k) descends to the GIT quotient. In this article, we prove that the GIT quotient of G n,2n (n≥3) by T with respect to 𝒪(2)=𝒪(1) ⊗2 is not projectively normal when polarized with the descent of 𝒪(2).","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"119 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135137178","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 Levi-Civita tensors are semistable in the sense of Geometric Invariant Theory, which is equivalent to an analogue of the Alon–Tarsi conjecture on Latin squares. The proof uses the connection of Tao’s slice rank with semistable tensors. We also show an application to an asymptotic saturation-type version of Rota’s basis conjecture.
{"title":"Stability of the Levi-Civita tensors and an Alon–Tarsi type theorem","authors":"Damir Yeliussizov","doi":"10.5802/crmath.505","DOIUrl":"https://doi.org/10.5802/crmath.505","url":null,"abstract":"We show that the Levi-Civita tensors are semistable in the sense of Geometric Invariant Theory, which is equivalent to an analogue of the Alon–Tarsi conjecture on Latin squares. The proof uses the connection of Tao’s slice rank with semistable tensors. We also show an application to an asymptotic saturation-type version of Rota’s basis conjecture.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135870023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that finite-type surfaces are characterized by a topological analogue of the Hopf property. Namely, an oriented surface Σ is of finite-type if and only if every proper map f:Σ→Σ of degree one is homotopic to a homeomorphism.
{"title":"Surfaces of infinite-type are non-Hopfian","authors":"Sumanta Das, Siddhartha Gadgil","doi":"10.5802/crmath.504","DOIUrl":"https://doi.org/10.5802/crmath.504","url":null,"abstract":"We show that finite-type surfaces are characterized by a topological analogue of the Hopf property. Namely, an oriented surface Σ is of finite-type if and only if every proper map f:Σ→Σ of degree one is homotopic to a homeomorphism.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"101 24","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135870193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A conjecture by the second author, proven by Bonnaf'e-Rouquier, says that the multiplicity matrix for baby Verma modules over the restricted rational Cherednik algebra has rank one over $mathbb{Q}$ when restricted to each block of the algebra. In this paper, we show that if $H$ is a prime algebra that is a free Frobenius extension over a regular central subalgebra $R$, and the centre of $H$ is normal Gorenstein, then each central quotient $A$ of $H$ by a maximal ideal $mathfrak{m}$ of $R$ satisfies the rank one property with respect to the Cartan matrix of $A$. Examples where the result is applicable include graded Hecke algebras, extended affine Hecke algebras, quantized enveloping algebras at roots of unity, non-commutative crepant resolutions of Gorenstein domains and 3 and 4 dimensional PI Skylanin algebras. In particular, since the multiplicity matrix for restricted rational Cherednik algebras has the rank one property if and only if its Cartan matrix does, our result provides a different proof of the original conjecture.
{"title":"The Rank-One property for free Frobenius extensions","authors":"Gwyn Bellamy, Ulrich Thiel","doi":"10.5802/crmath.502","DOIUrl":"https://doi.org/10.5802/crmath.502","url":null,"abstract":"A conjecture by the second author, proven by Bonnaf'e-Rouquier, says that the multiplicity matrix for baby Verma modules over the restricted rational Cherednik algebra has rank one over $mathbb{Q}$ when restricted to each block of the algebra. In this paper, we show that if $H$ is a prime algebra that is a free Frobenius extension over a regular central subalgebra $R$, and the centre of $H$ is normal Gorenstein, then each central quotient $A$ of $H$ by a maximal ideal $mathfrak{m}$ of $R$ satisfies the rank one property with respect to the Cartan matrix of $A$. Examples where the result is applicable include graded Hecke algebras, extended affine Hecke algebras, quantized enveloping algebras at roots of unity, non-commutative crepant resolutions of Gorenstein domains and 3 and 4 dimensional PI Skylanin algebras. In particular, since the multiplicity matrix for restricted rational Cherednik algebras has the rank one property if and only if its Cartan matrix does, our result provides a different proof of the original conjecture.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"1 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135870826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article revolves around a recent numerical framework for shape and topology optimization, which features an exact mesh of the shape at each iteration of the process, while still leaving the room for an arbitrary evolution of the latter (including changes in its topology). In a nutshell, two complementary representations of the shape are combined: on the one hand, it is meshed exactly, which allows for precise mechanical calculations based on the finite element method; on the other hand, it is described implicitly, using the level set method, which makes it possible to track its evolution in a robust way. In the first part of this work, we overview the main aspects of this numerical strategy. After a brief presentation of some necessary background material – related to shape optimization and meshing, among others – we describe the numerical schemes involved, notably when it comes to the practice of the level set method, the remeshing algorithms, and the considered optimization solver. This strategy is illustrated with 2d and 3d numerical examples in various physical contexts. In the second part of this article, we propose a simple albeit efficient python-based implementation of this framework. The code is described with a fair amount of details, and it is expected that the reader can easily elaborate upon the presented examples to tackle his own problems.
{"title":"Shape optimization using a level set based mesh evolution method: an overview and tutorial","authors":"Charles Dapogny, Florian Feppon","doi":"10.5802/crmath.498","DOIUrl":"https://doi.org/10.5802/crmath.498","url":null,"abstract":"This article revolves around a recent numerical framework for shape and topology optimization, which features an exact mesh of the shape at each iteration of the process, while still leaving the room for an arbitrary evolution of the latter (including changes in its topology). In a nutshell, two complementary representations of the shape are combined: on the one hand, it is meshed exactly, which allows for precise mechanical calculations based on the finite element method; on the other hand, it is described implicitly, using the level set method, which makes it possible to track its evolution in a robust way. In the first part of this work, we overview the main aspects of this numerical strategy. After a brief presentation of some necessary background material – related to shape optimization and meshing, among others – we describe the numerical schemes involved, notably when it comes to the practice of the level set method, the remeshing algorithms, and the considered optimization solver. This strategy is illustrated with 2d and 3d numerical examples in various physical contexts. In the second part of this article, we propose a simple albeit efficient python-based implementation of this framework. The code is described with a fair amount of details, and it is expected that the reader can easily elaborate upon the presented examples to tackle his own problems.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"64 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135765685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tomohiro Asano, Stéphane Guillermou, Vincent Humilière, Yuichi Ike, Claude Viterbo
We prove that for any element L in the completion of the space of smooth compact exact Lagrangian submanifolds of a cotangent bundle equipped with the spectral distance, the γ-support of L coincides with the reduced micro-support of its sheaf quantization. As an application, we give a characterization of the Vichery subdifferential in terms of γ-support.
{"title":"The γ-support as a micro-support","authors":"Tomohiro Asano, Stéphane Guillermou, Vincent Humilière, Yuichi Ike, Claude Viterbo","doi":"10.5802/crmath.499","DOIUrl":"https://doi.org/10.5802/crmath.499","url":null,"abstract":"We prove that for any element L in the completion of the space of smooth compact exact Lagrangian submanifolds of a cotangent bundle equipped with the spectral distance, the γ-support of L coincides with the reduced micro-support of its sheaf quantization. As an application, we give a characterization of the Vichery subdifferential in terms of γ-support.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"2012 15","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135813793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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