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A Torelli type theorem for exp-algebraic curves 指数代数曲线的一个Torelli型定理
Annales de la Faculté des sciences de Toulouse : Mathématiques Pub Date : 2016-06-21 DOI: 10.5802/afst.1634
I. Biswas, K. Biswas
{"title":"A Torelli type theorem for exp-algebraic curves","authors":"I. Biswas, K. Biswas","doi":"10.5802/afst.1634","DOIUrl":"https://doi.org/10.5802/afst.1634","url":null,"abstract":"An exp-algebraic curve consists of a compact Riemann surface $S$ together with $n$ equivalence classes of germs of meromorphic functions modulo germs of holomorphic functions, $HH = { [h_1], cdots, [h_n] }$, with poles of orders $d_1, cdots, d_n geq 1$ at points $p_1, cdots, p_n$. This data determines a space of functions $OO_{HH}$ (respectively, a space of $1$-forms $Omega^0_{HH}$) holomorphic on the punctured surface $S' = S - {p_1, cdots, p_n}$ with exponential singularities at the points $p_1, cdots, p_n$ of types $[h_1], cdots, [h_n]$, i.e., near $p_i$ any $f in OO_{HH}$ is of the form $f = ge^{h_i}$ for some germ of meromorphic function $g$ (respectively, any $omega in Omega^0_{HH}$ is of the form $omega = alpha e^{h_i}$ for some germ of meromorphic $1$-form). \u0000For any $omega in Omega^0_{HH}$ the completion of $S'$ with respect to the flat metric $|omega|$ gives a space $S^* = S' cup RR$ obtained by adding a finite set $RR$ of $sum_i d_i$ points, and it is known that integration along curves produces a nondegenerate pairing of the relative homology $H_1(S^*, RR ; C)$ with the deRham cohomology group defined by $H^1_{dR}(S, HH) := Omega^0_{HH}/dOO_{HH}$. \u0000There is a degree zero line bundle $L_{HH}$ associated to an exp-algebraic curve, with a natural isomorphism between $Omega^0_{HH}$ and the space $W_{HH}$ of meromorphic $L_{HH}$-valued $1$-forms which are holomorphic on $S'$, so that $H_1(S^*, RR ; C)$ maps to a subspace $K_{HH} subset W^*_{HH}$. We show that the exp-algebraic curve $(S, HH)$ is determined uniquely by the pair $(L_{HH},, K_{HH} subset W^*_{HH})$.","PeriodicalId":169800,"journal":{"name":"Annales de la Faculté des sciences de Toulouse : Mathématiques","volume":"495 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131636202","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Heat kernel asymptotics on sub-Riemannian manifolds with symmetries and applications to the bi-Heisenberg group 对称子黎曼流形的热核渐近性及其在双海森堡群中的应用
Annales de la Faculté des sciences de Toulouse : Mathématiques Pub Date : 2016-06-03 DOI: 10.5802/afst.1613
D. Barilari, U. Boscain, R. Neel
{"title":"Heat kernel asymptotics on sub-Riemannian manifolds with symmetries and applications to the bi-Heisenberg group","authors":"D. Barilari, U. Boscain, R. Neel","doi":"10.5802/afst.1613","DOIUrl":"https://doi.org/10.5802/afst.1613","url":null,"abstract":"By adapting a technique of Molchanov, we obtain the heat kernel asymptotics at the sub-Riemannian cut locus, when the cut points are reached by an $r$-dimensional parametric family of optimal geodesics. We apply these results to the bi-Heisenberg group, that is, a nilpotent left-invariant sub-Rieman-nian structure on $mathbb{R}^{5}$ depending on two real parameters $alpha_{1}$ and $alpha_{2}$. We develop some results about its geodesics and heat kernel associated to its sub-Laplacian and we illuminate some interesting geometric and analytic features appearing when one compares the isotropic ($alpha_{1}=alpha_{2}$) and the non-isotropic cases ($alpha_{1}neq alpha_{2}$). In particular, we give the exact structure of the cut locus, and we get the complete small-time asymptotics for its heat kernel.","PeriodicalId":169800,"journal":{"name":"Annales de la Faculté des sciences de Toulouse : Mathématiques","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116934678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 31
A new definition of rough paths on manifolds 流形上粗糙路径的新定义
Annales de la Faculté des sciences de Toulouse : Mathématiques Pub Date : 2015-10-27 DOI: 10.5802/afst.1717
Y. Boutaib, Terry Lyons
{"title":"A new definition of rough paths on manifolds","authors":"Y. Boutaib, Terry Lyons","doi":"10.5802/afst.1717","DOIUrl":"https://doi.org/10.5802/afst.1717","url":null,"abstract":"Smooth manifolds are not the suitable context for trying to generalize the concept of rough paths as quantitative estimates -which will be lost is this case- are key in this matter. Moreover, even with a definition of rough paths in smooth manifolds, rough differential equations can only be expected to be solved locally in such a case. In this paper, we first recall the foundations of the Lipschitz geometry, introduced in \"Rough Paths on Manifolds\" (Cass, T., Litterer, C. & Lyons, T.), along with the main findings that encompass the classical theory of rough paths in Banach spaces. Then we give what we believe to be a minimal framework for defining rough paths on a manifold that is both less rigid than the classical one and emphasized on the local behaviour of rough paths. We end by explaining how this same idea can be used to define any notion of coloured paths on a manifold.","PeriodicalId":169800,"journal":{"name":"Annales de la Faculté des sciences de Toulouse : Mathématiques","volume":"145 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124631756","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes 位移辛导出格式的拉格朗日邻域定理
Annales de la Faculté des sciences de Toulouse : Mathématiques Pub Date : 2015-06-12 DOI: 10.5802/afst.1616
D. Joyce, P. Safronov
{"title":"A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes","authors":"D. Joyce, P. Safronov","doi":"10.5802/afst.1616","DOIUrl":"https://doi.org/10.5802/afst.1616","url":null,"abstract":"Pantev, Toen, Vaqui'e and Vezzosi arXiv:1111.3209 defined $k$-shifted symplectic derived schemes and stacks ${bf X}$ for $kinmathbb Z$, and Lagrangians ${bf f}:{bf L}to{bf X}$ in them. They have important applications to Calabi-Yau geometry and quantization. Bussi, Brav and Joyce arXiv:1305.6302 proved a 'Darboux Theorem' giving explicit Zariski or 'etale local models for $k$-shifted symplectic derived schemes ${bf X}$ for $k<0$ presenting them as twisted shifted cotangent bundles. \u0000We prove a 'Lagrangian Neighbourhood Theorem' giving explicit Zariski or etale local models for Lagrangians ${bf f}:{bf L}to{bf X}$ in $k$-shifted symplectic derived schemes ${bf X}$ for $k<0$, relative to the Bussi-Brav-Joyce 'Darboux form' local models for ${bf X}$. That is, locally such Lagrangians can be presented as twisted shifted conormal bundles. We also give a partial result when $k=0$. \u0000We expect our results will have future applications to $k$-shifted Poisson geometry (see arXiv:1506.03699), to defining 'Fukaya categories' of complex or algebraic symplectic manifolds, and to categorifying Donaldson-Thomas theory of Calabi-Yau 3-folds and 'Cohomological Hall algebras'.","PeriodicalId":169800,"journal":{"name":"Annales de la Faculté des sciences de Toulouse : Mathématiques","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131039850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 12
Toward quantization of Galois theory 伽罗瓦理论的量子化
Annales de la Faculté des sciences de Toulouse : Mathématiques Pub Date : 2015-01-27 DOI: 10.5802/AFST.1663
A. Masuoka, Katsunori Saito, H. Umemura
{"title":"Toward quantization of Galois theory","authors":"A. Masuoka, Katsunori Saito, H. Umemura","doi":"10.5802/AFST.1663","DOIUrl":"https://doi.org/10.5802/AFST.1663","url":null,"abstract":"This note is a development of our two previous papers, arXiv:1212.3392v1 and 1306.3660v1. \u0000The fundamental question is whether there exists a Galois theory, in which the Galois group is a quantum group. \u0000For a linear equations with respect to a Hopf algebra, we arrived at a final form if the base field consists of constants. In this case, we have non-commutative Picard-Vessiot rings and asymmetric Tannaka theory. \u0000For non-linear equations there are examples that might make us optimistic.","PeriodicalId":169800,"journal":{"name":"Annales de la Faculté des sciences de Toulouse : Mathématiques","volume":"102 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116300076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
The lexicographic degree of the first two-bridge knots 前两桥结的词典编纂程度
Annales de la Faculté des sciences de Toulouse : Mathématiques Pub Date : 2015-01-23 DOI: 10.5802/afst.1645
Erwan Brugall'e, P. Koseleff, D. Pecker
{"title":"The lexicographic degree of the first two-bridge knots","authors":"Erwan Brugall'e, P. Koseleff, D. Pecker","doi":"10.5802/afst.1645","DOIUrl":"https://doi.org/10.5802/afst.1645","url":null,"abstract":"We study the degree of polynomial representations of knots. We give the lexicographic degree of all two-bridge knots with 11 or fewer crossings. First, we estimate the total degree of a lexicographic parametrisation of such a knot. This allows us to transform this problem into a study of real algebraic trigonal plane curves, and in particular to use the braid theoretical method developed by Orevkov.","PeriodicalId":169800,"journal":{"name":"Annales de la Faculté des sciences de Toulouse : Mathématiques","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127501398","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Levels of Distribution and the Affine Sieve 分布水平和仿射筛
Annales de la Faculté des sciences de Toulouse : Mathématiques Pub Date : 2014-06-01 DOI: 10.5802/AFST.1432
Alex Kontorovich
{"title":"Levels of Distribution and the Affine Sieve","authors":"Alex Kontorovich","doi":"10.5802/AFST.1432","DOIUrl":"https://doi.org/10.5802/AFST.1432","url":null,"abstract":"This article is an expanded version of the author's lecture in the Basic Notions Seminar at Harvard, September 2013. Our goal is a brief and introductory exposition of aspects of two topics in sieve theory which have received attention recently: (1) the spectacular work of Yitang Zhang, under the title \"Level of Distribution,\" and (2) the so-called \"Affine Sieve,\" introduced by Bourgain-Gamburd-Sarnak.","PeriodicalId":169800,"journal":{"name":"Annales de la Faculté des sciences de Toulouse : Mathématiques","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127542107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 17
Skeleta in non-Archimedean and tropical geometry 非阿基米德和热带几何的骨架
Annales de la Faculté des sciences de Toulouse : Mathématiques Pub Date : 2013-11-03 DOI: 10.25560/25016
Andrew W. Macpherson
{"title":"Skeleta in non-Archimedean and tropical geometry","authors":"Andrew W. Macpherson","doi":"10.25560/25016","DOIUrl":"https://doi.org/10.25560/25016","url":null,"abstract":"I describe an algebro-geometric theory of skeleta, which provides a unified setting for the study of tropical varieties, skeleta of non-Archimedean analytic spaces, and affine manifolds with singularities. Skeleta are spaces equipped with a structure sheaf of topological semirings, and are locally modelled on the spectra of the same. The primary result of this paper is that the topological space underlying a non-Archimedean analytic space may locally be recovered from the sheaf of `pointwise valuations' of its analytic functions. \u0000In a sequel, I will apply the theory of skeleta to address a question of constructing affine manifolds from maximally degenerating Calabi-Yau manifolds, as predicted by the SYZ conjecture.","PeriodicalId":169800,"journal":{"name":"Annales de la Faculté des sciences de Toulouse : Mathématiques","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123549223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
On the conductor of cohomological transforms 关于上同调变换的导体
Annales de la Faculté des sciences de Toulouse : Mathématiques Pub Date : 2013-10-14 DOI: 10.5802/AFST.1671
'Etienne Fouvry, E. Kowalski, P. Michel
{"title":"On the conductor of cohomological transforms","authors":"'Etienne Fouvry, E. Kowalski, P. Michel","doi":"10.5802/AFST.1671","DOIUrl":"https://doi.org/10.5802/AFST.1671","url":null,"abstract":"In the analytic study of trace functions of $ell$-adic sheaves over finite fields, a crucial issue is to control the conductor of sheaves constructed in various ways. We consider cohomological transforms on the affine line over a finite field which have trace functions given by linear operators with an additive character of a rational function in two variables as a kernel. We prove that the conductor of such a transform is bounded in terms of the complexity of the input sheaf and of the rational function defining the kernel, and discuss applications of this result, including motivating examples arising from the Polymath8 project.","PeriodicalId":169800,"journal":{"name":"Annales de la Faculté des sciences de Toulouse : Mathématiques","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125619058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 15
Orthogonal polynomials and diffusion operators 正交多项式与扩散算子
Annales de la Faculté des sciences de Toulouse : Mathématiques Pub Date : 2013-09-22 DOI: 10.5802/afst.1693
D. Bakry, S. Orevkov, M. Zani
{"title":"Orthogonal polynomials and diffusion operators","authors":"D. Bakry, S. Orevkov, M. Zani","doi":"10.5802/afst.1693","DOIUrl":"https://doi.org/10.5802/afst.1693","url":null,"abstract":"We want to describe the triplets (Omega, (g), mu) where (g) is the (co)metric associated to some symmetric second order differential operator L defined on the domain Omega of R^d and such that L is expandable on a basis of orthogonal polynomials of L_2(mu), and mu is some admissible measure. Up to affine transformation, we find 11 compact domains in dimension 2, and also give some non--compact cases in this dimension.","PeriodicalId":169800,"journal":{"name":"Annales de la Faculté des sciences de Toulouse : Mathématiques","volume":"86 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127654199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 10
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