Annales Mathematiques du Quebec最新文献

{"title":"Statistical fluctuation of infinitesimal spaces","authors":"Damien Tageddine,&nbsp;Jean-Christophe Nave","doi":"10.1007/s40316-026-00268-z","DOIUrl":"10.1007/s40316-026-00268-z","url":null,"abstract":"<div><p>This paper is a follow-up on the <i>Noncommutative Differential Geometry on Infinitesimal Spaces</i> [16]. In the present work, we extend the algebraic convergence established in [16] to the Riemannian geometric setting. On the one hand, we reformulate the definition of finite dimensional compatible Dirac operators using Clifford algebras. This definition also leads to a new construction of a Laplace operator. On the other hand, after choosing a family of Green functions to define the coefficients of the Dirac matrices, we show that these matrices can be interpreted as stochastic matrices. The sequence of operators <span>((D_n)_{nin mathbb {N}})</span> converges then in average to the local expression of the classical Dirac operator on an oriented compact Riemann manifold. The same result is drawn for the Laplace operator.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"50 2","pages":"405 - 429"},"PeriodicalIF":0.8,"publicationDate":"2026-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149125563","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}

{"title":"Symplectification of rank 2 distributions, normal Cartan connections, and Cartan prolongations","authors":"Nicklas Day,&nbsp;Boris Doubrov,&nbsp;Igor Zelenko","doi":"10.1007/s40316-026-00265-2","DOIUrl":"10.1007/s40316-026-00265-2","url":null,"abstract":"<div><p>We study the Doubrov–Zelenko symplectification procedure for rank 2 distributions with 5-dimensional cube—originally motivated by optimal control theory—through the lens of Tanaka–Morimoto theory for normal Cartan connections. In this way, for ambient manifolds of dimension <span>( n ge 5 )</span>, we prove the existence of the normal Cartan connection associated with the symplectified distribution. Furthermore, we show that this symplectification can be interpreted as the <span>((n-4))</span>th iterated Cartan prolongation at a generic point. This interpretation naturally leads to two questions for an arbitrary rank 2 distribution with 5-dimensional cube: (1) Is the <span>((n-4))</span>th iterated Cartan prolongation the minimal iteration where the Tanaka symbols become unified at generic points? (2) Is the <span>((n-4))</span>th iterated Cartan prolongation the minimal iteration admitting a normal Cartan connection via Tanaka–Morimoto theory? Our main results demonstrate that: (a) For <span>(n &gt; 5)</span>, the answer to the second question is positive (in contrast to the classical <span>(n = 5)</span> case from <span>(G_2)</span>-parabolic geometries); (b) For <span>(n ge 5)</span>, the answer to the first question is negative: unification occurs already at the <span>((n-5))</span>th iterated Cartan prolongation.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"50 2","pages":"323 - 349"},"PeriodicalIF":0.8,"publicationDate":"2026-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149124609","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}

{"title":"Paramètres dans les corps algébriquement clos","authors":"Bruno Poizat","doi":"10.1007/s40316-025-00242-1","DOIUrl":"10.1007/s40316-025-00242-1","url":null,"abstract":"&lt;p&gt; Ce papier est une incursion de la Théorie des Modèles dans la Géométrie Algébrique, et son auteur attache un plus grand prix aux méthodes employées pour parvenir aux résultats obtenus qu'à la valeur intrinsèque de ces résultats-mêmes. En témoigne la nature des questions réparties dans le texte. On y étudie les influences réciproques du groupe des automorphismes d'un corps algébriquement clos K sur le groupe des automorphismes d'une structure S définissable dans K. Les paramètres nécessaires aux définitions vont y jouer un rôle de premier plan, ainsi que les propriétés très particulières de la Théorie des Modèles des corps algébriquement clos. Il commence par un commentaire d’un résultat de A.V. Borovik, qui a été la source de son inspiration, mettant en évidence sa dépendance au célèbre Théorème de Borel et Tits sur les isomorphismes abstraits des groupes algébriques simples, considéré d’un point de vue modèle-théorique. Ce théorème conduit finalement à la description des automorphismes d'ordre fini d'un groupe algébrique simple (sur un corps de base algébriquement clos), et de ses groupes superstables d’automorphismes, qui est basée sur des arguments généraux de Théorie des Modèles, ne demandant qu’une inspection minimale de la structure du groupe; pour en tirer des conséquences, nous devons affermir un argument elliptique d’Altinel, Borovik et Cherlin, dans une démonstration qui est pourtant cruciale dans leur contexte inductif. En fait, notre version du Théorème de Borel et Tits ne dépend pas de la présence d’une loi de groupe; il est valable plus généralement pour ce que nous appelons les &lt;i&gt;structures constructives autonomes,&lt;/i&gt; qui sont les structures infinies S, définissables dans un corps algébriquement clos K, pour lesquelles tout ce qui est définissable sur S dans le langage du corps K est définissable (avec paramètres) dans le langage de S. Un exemple significatif de telles structures est donné par les &lt;i&gt;multicorps,&lt;/i&gt; dont la banale définition cache une subtile théorie de Galois en caractéristique p; en effet, dans n’importe quelle structure constructible autonome S on peut définir sans paramètres un multicorps qui contrôle les automorphismes de S au sens suivant: ceux d’entre eux dont l’action sur ce multicorps est d’ordre fini forment un groupe, noté Aut&lt;sub&gt;max&lt;/sub&gt;(S), qui est à la fois le plus grand groupe définissable d’automorphismes de S, et le plus grand groupe superstable d’automorphismes de S. En caractéristique nulle, ce résultat est facile à établir, car on peut alors définir sans paramètres dans S un unicorps, c’est-à-dire une copie L du corps de base K. Quand S est un groupe algébrique simple G, dans les cas ordinaires une telle copie L existe même en caractéristique p; mais il y a des cas spéciaux, dus à la présence d'endogénies exceptionnelles, ou seulement un bicorps (L&lt;sub&gt;1&lt;/sub&gt;,L&lt;sub&gt;2&lt;/sub&gt;) est définissable sans paramètres; Aut&lt;sub&gt;max&lt;/sub&gt;(G) est le noyau de l’action des automorphismes de G sur le cor","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"50 1","pages":"295 - 321"},"PeriodicalIF":0.4,"publicationDate":"2025-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147865690","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}

{"title":"The de Rham period map for punctured elliptic curves and the KZB equation","authors":"Ben Moore","doi":"10.1007/s40316-025-00246-x","DOIUrl":"10.1007/s40316-025-00246-x","url":null,"abstract":"<div><p>We demonstrate that the algebraic KZB connection of Levin–Racinet and Luo on a once-punctured elliptic curve represents Kim’s universal unipotent connection, and we observe that the Hodge filtration on the KZB connection has a particularly simple form. This allows us to generalise previous work of Beacom by writing down explicitly the maximal metabelian quotient of Kim’s de Rham period map in terms of elliptic polylogarithms. As far as we are aware this is the first time that the de Rham period map has been written out for an infinite dimensional quotient of the de Rham fundamental group on any curve of positive genus.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"50 1","pages":"167 - 184"},"PeriodicalIF":0.4,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147865601","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}

{"title":"Iwasawa theory for weighted graphs","authors":"Taiga Adachi,&nbsp;Kosuke Mizuno,&nbsp;Sohei Tateno","doi":"10.1007/s40316-025-00247-w","DOIUrl":"10.1007/s40316-025-00247-w","url":null,"abstract":"<div><p>Let <i>p</i> be a prime number and let <i>d</i> be a positive integer. In this paper, we generalize Iwasawa theory for graphs initiated by Gonet and Vallières to weighted graphs. In particular, we prove an analogue of Iwasawa’s class number formula and that of Kida’s formula for compatible systems of <span>((mathbb {Z}/p^nmathbb {Z})^d)</span>-covers of weighted graphs. We also provide numerical examples of characteristic elements and Iwasawa invariants. At the end of this paper, we give an application of the ideas of Iwasawa theory to the theory of discrete-time quantum walks in graphs.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"50 1","pages":"231 - 265"},"PeriodicalIF":0.4,"publicationDate":"2025-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147865565","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}

{"title":"Noncommutative differential geometry on infinitesimal spaces","authors":"Damien Tageddine,&nbsp;Jean-Christophe Nave","doi":"10.1007/s40316-025-00264-9","DOIUrl":"10.1007/s40316-025-00264-9","url":null,"abstract":"<div><p>In this paper, we use noncommutative differential geometry to formalize a framework for discrete differential calculus. We begin with a brief review of inverse limit of posets as an approximation of topological spaces. We then show how to associate a <span>(C^*)</span>-algebra over a poset, giving it a piecewise-linear structure. Furthermore, we explain how dually the algebra of continuous function <i>C</i>(<i>M</i>) over a manifold <i>M</i> can be approximated by a direct limit of <span>(C^*)</span>-algebras over posets. Finally, in the spirit of noncommutative differential geometry, we define a finite dimensional spectral triple on each poset. We show how the usual finite difference calculus is recovered as the eigenvalues of the commutator with the Dirac operator. We prove a convergence result in the case of the <i>d</i>-lattice in <span>(mathbb {R}^d)</span> and for the torus <span>(mathbb {T}^d)</span>.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"50 1","pages":"1 - 33"},"PeriodicalIF":0.4,"publicationDate":"2025-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147865600","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}

{"title":"Poisson structure on character varieties, II","authors":"Indranil Biswas,&nbsp;Lisa C. Jeffrey","doi":"10.1007/s40316-025-00256-9","DOIUrl":"10.1007/s40316-025-00256-9","url":null,"abstract":"<div><p>Let <i>G</i> be a complex reductive group and <span>(D, subset , X)</span> a finite subset of a compact Riemann surface <i>X</i>. It was shown in Biswas and Jeffrey (Ann Math Québec 45: 213–219, 2021) that the moduli space of <i>G</i>–characters of <span>(pi _1(X{setminus } D))</span> has a natural Poisson structure. We show that the moduli space of logarithmic <i>G</i>–connections on <i>X</i> singular over <i>D</i> has a Poisson structure. It is proved that the monodromy map from the moduli space of logarithmic <i>G</i>–connections to the moduli space of <i>G</i>–characters is Poisson structure preserving.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"50 1","pages":"267 - 276"},"PeriodicalIF":0.4,"publicationDate":"2025-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147865566","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}

{"title":"Symplectically self-polar polytopes of minimal capacity","authors":"Mark Berezovik","doi":"10.1007/s40316-025-00251-0","DOIUrl":"10.1007/s40316-025-00251-0","url":null,"abstract":"<div><p>In this paper we continue the study of symplectically self-polar convex bodies started in [3]. We construct symplectically self-polar convex bodies of the minimal Ekeland–Hofer–Zehnder capacity. This in turn proves that the lower bound for the Ekeland–Hofer–Zehnder capacity for centrally symmetric convex bodies obtained in [1] cannot be improved. We also make some numerical experiments and speculations regarding the minimal volume of symplectically self-polar convex bodies.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"49 2","pages":"335 - 353"},"PeriodicalIF":0.4,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40316-025-00251-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The multiplicative formula of langlands for orbital integrals in GL(2)","authors":"Malors Espinosa","doi":"10.1007/s40316-025-00259-6","DOIUrl":"10.1007/s40316-025-00259-6","url":null,"abstract":"<div><p>In Langlands (Contributions to automorphic forms, geometry, and number theory. Johns Hopkins University Press, Baltimore, pp. 611–697, 2004.), Langlands introduces a formula for a specific product of orbital integrals in <span>(text{ GL }(2, mathbb {Q}))</span>. Altuğ (Compos Math 151(10), 1791–1820, 2015), employs this formula to manipulate the regular elliptic part of the trace formula, with the aim of eliminating the contribution of the trivial representation from the spectral side. In (Arthur, Adv Math 327:425–469, 2018), Arthur predicts that this formula coincides with a product of polynomials associated with zeta functions of orders developed by Yun (Ramanujan Math Soc Lect Notes 20:399–420, 2013). In (Espinosa, J Number Theory 252:379–404, 2023), we determined the explicit polynomials for the relevant quadratic orders. This paper demonstrates how these polynomials can effectively generalize Langlands’ formula to <i>GL</i>(2, <i>K</i>), for general algebraic number fields <i>K</i>. Furthermore, we also use this formula to extend a well-known formula of Zagier to any algebraic number field. <b>Résumé.</b> Langlands a trouvé une formule pour un produit d’integrales orbitales sur <span>(text{ GL }(2, mathbb {Q}))</span> [8]. Altuğ a employé cette formule pour gérer la partie régulière elliptique de la formule de trace, avec l’objectif d’annuler la contribution de la représentation triviale de la composante spectrale [1]. Arthur a conjecturé [2] que cette formule est le produit des polynômes associés aux fonctions zêta d’ordres developé par Zhiwei Yun dans [10]. Dans [5], on a calculé ces polynômes pour les ordres quadratiques pertinents. Dans cet article, on prouve une généralization de la formule de Langlands pour GL(2,K), où K est un corps de nombres quelconque, en utilisant les polynômes. De plus, on utilise cette formule pour vérifier la formule de Zagier dans un corps de nombres quelconque.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"50 1","pages":"201 - 229"},"PeriodicalIF":0.4,"publicationDate":"2025-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147865599","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}

{"title":"Eigenvalue bounds for the Steklov problem on differential forms in warped product manifolds","authors":"Tirumala Chakradhar","doi":"10.1007/s40316-025-00255-w","DOIUrl":"10.1007/s40316-025-00255-w","url":null,"abstract":"<div><p>We consider the Steklov problem on differential <span>(ptext {-})</span>forms defined by Karpukhin and present geometric eigenvalue bounds in the setting of warped product manifolds in various scenarios. In particular, we obtain Escobar type lower bounds for warped product manifolds with non-negative Ricci curvature and strictly convex boundary, and certain sharp bounds for hypersurfaces of revolution, among others. We compare and contrast the behaviour with known results in the case of functions (i.e., <span>(0text {-})</span>forms), highlighting the influence of the underlying topology on the spectrum for <span>(ptext {-})</span>forms in general.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"49 2","pages":"421 - 443"},"PeriodicalIF":0.4,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40316-025-00255-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}