We introduce an $ell$-adic analogue of Gauss's hypergeometric function arising from the Galois action on the fundamental torsor of the projective line minus three points. Its definition is motivated by a relation between the KZ-equation and the hypergeometric differential equation in the complex case. We show two basic properties, analogues of Gauss's hypergeometric theorem and of Euler's transformation formula for our $ell$-adic function. We prove them by detecting a connection of a certain two-by-two matrix specialization of even unitary associators with the associated gamma function, which extends the result of Ohno and Zagier.
{"title":"The ℓ-adic hypergeometric function and\u0000associators","authors":"H. Furusho","doi":"10.2140/tunis.2023.5.1","DOIUrl":"https://doi.org/10.2140/tunis.2023.5.1","url":null,"abstract":"We introduce an $ell$-adic analogue of Gauss's hypergeometric function arising from the Galois action on the fundamental torsor of the projective line minus three points. Its definition is motivated by a relation between the KZ-equation and the hypergeometric differential equation in the complex case. We show two basic properties, analogues of Gauss's hypergeometric theorem and of Euler's transformation formula for our $ell$-adic function. We prove them by detecting a connection of a certain two-by-two matrix specialization of even unitary associators with the associated gamma function, which extends the result of Ohno and Zagier.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49511170","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}
Pub Date : 2021-05-11DOI: 10.2140/tunis.2022.4.443
Yajnaseni Dutta, D. Huybrechts
We prove that curves in a non-primitive, base point free, ample linear system on a K3 surface have maximal variation. The result is deduced from general restriction theorems applied to the tangent bundle. We also show how to use specialisation to spectral curves to deduce information about the variation of curves contained in a K3 surface more directly. The situation for primitive linear systems is not clear at the moment. However, the maximal variation holds in genus two and can, in many cases, be deduced from a recent result of van Geemen and Voisin confirming a conjecture due to Matsushita.
{"title":"Maximal variation of curves on K3 surfaces","authors":"Yajnaseni Dutta, D. Huybrechts","doi":"10.2140/tunis.2022.4.443","DOIUrl":"https://doi.org/10.2140/tunis.2022.4.443","url":null,"abstract":"We prove that curves in a non-primitive, base point free, ample linear system on a K3 surface have maximal variation. The result is deduced from general restriction theorems applied to the tangent bundle. We also show how to use specialisation to spectral curves to deduce information about the variation of curves contained in a K3 surface more directly. The situation for primitive linear systems is not clear at the moment. However, the maximal variation holds in genus two and can, in many cases, be deduced from a recent result of van Geemen and Voisin confirming a conjecture due to Matsushita.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42006400","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}
A typical large complex-structure limit for mirror symmetry consists of toric varieties glued to each other along their toric boundaries. Here we construct the mirror large volume limit space as a Weinstein symplectic manifold. We prove homological mirror symmetry: the category of coherent sheaves on the first space is equivalent to the Fukaya category of the second. Our equivalence intertwines the Viterbo restriction maps for a generalized pair-of-pants cover of the symplectic manifold with the restriction of coherent sheaves for a certain affine cover of the algebraic variety. We deduce a posteriori a local-to-global principle conjectured by Seidel -- certain diagrams of Viterbo restrictions are cartesian -- by passing Zariski descent through our mirror symmetry result.
{"title":"Homological mirror symmetry at large volume","authors":"Benjamin Gammage, V. Shende","doi":"10.2140/tunis.2023.5.31","DOIUrl":"https://doi.org/10.2140/tunis.2023.5.31","url":null,"abstract":"A typical large complex-structure limit for mirror symmetry consists of toric varieties glued to each other along their toric boundaries. Here we construct the mirror large volume limit space as a Weinstein symplectic manifold. We prove homological mirror symmetry: the category of coherent sheaves on the first space is equivalent to the Fukaya category of the second. Our equivalence intertwines the Viterbo restriction maps for a generalized pair-of-pants cover of the symplectic manifold with the restriction of coherent sheaves for a certain affine cover of the algebraic variety. We deduce a posteriori a local-to-global principle conjectured by Seidel -- certain diagrams of Viterbo restrictions are cartesian -- by passing Zariski descent through our mirror symmetry result.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44053971","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}
Pub Date : 2021-03-29DOI: 10.2140/tunis.2022.4.673
Y. Bonthonneau, C. Guillarmou, Thibault de Poyferr'e
. We develop a paradifferential approach for studying non-smooth hyperbolic dynamics on manifolds and related non-linear PDE from a microlocal point of view. As an application, we describe the microlocal regularity, i.e the H s wave-front set for all s , of the unstable bundle E u for an Anosov flow. We also recover rigidity results of Hurder-Katok and Hasselblatt in the Sobolev class rather than H¨older: there is s 0 > 0 such that if E u has H s regularity for s > s 0 then it is smooth (with s 0 = 2 for volume preserving 3-dimensional Anosov flows). It is also shown in the Appendix that it can be applied to deal with non-smooth flows and potentials. This work could serve as a toolbox for other applications.
{"title":"A paradifferential approach for hyperbolic dynamical systems and applications","authors":"Y. Bonthonneau, C. Guillarmou, Thibault de Poyferr'e","doi":"10.2140/tunis.2022.4.673","DOIUrl":"https://doi.org/10.2140/tunis.2022.4.673","url":null,"abstract":". We develop a paradifferential approach for studying non-smooth hyperbolic dynamics on manifolds and related non-linear PDE from a microlocal point of view. As an application, we describe the microlocal regularity, i.e the H s wave-front set for all s , of the unstable bundle E u for an Anosov flow. We also recover rigidity results of Hurder-Katok and Hasselblatt in the Sobolev class rather than H¨older: there is s 0 > 0 such that if E u has H s regularity for s > s 0 then it is smooth (with s 0 = 2 for volume preserving 3-dimensional Anosov flows). It is also shown in the Appendix that it can be applied to deal with non-smooth flows and potentials. This work could serve as a toolbox for other applications.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48331778","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}
Pub Date : 2021-03-01DOI: 10.2140/tunis.2023.5.273
Andrea Gagna, Viktoriya Ozornova, M. Rovelli
We show that the complicial nerve construction is homotopically compatible with two flavors of cone constructions when starting with an $omega$-category that is suitably free and loop-free. An instance of the result recovers the fact that the standard $m$-simplex is equivalent to the complicial nerve of the $m$-oriental.
{"title":"Nerves and cones of free loop-free\u0000ω-categories","authors":"Andrea Gagna, Viktoriya Ozornova, M. Rovelli","doi":"10.2140/tunis.2023.5.273","DOIUrl":"https://doi.org/10.2140/tunis.2023.5.273","url":null,"abstract":"We show that the complicial nerve construction is homotopically compatible with two flavors of cone constructions when starting with an $omega$-category that is suitably free and loop-free. An instance of the result recovers the fact that the standard $m$-simplex is equivalent to the complicial nerve of the $m$-oriental.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43957246","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}
Pub Date : 2021-02-02DOI: 10.2140/tunis.2022.4.635
S. Gouezel
We consider nonelementary random walks on general hyperbolic spaces. Without any moment condition on the walk, we show that it escapes linearly to infinity, with exponential error bounds. We even get such exponential bounds up to the rate of escape of the walk. Our proof relies on an inductive decomposition of the walk, recording times at which it could go to infinity in several independent directions, and using these times to control further backtracking.
{"title":"Exponential bounds for random walks on hyperbolic spaces without moment conditions","authors":"S. Gouezel","doi":"10.2140/tunis.2022.4.635","DOIUrl":"https://doi.org/10.2140/tunis.2022.4.635","url":null,"abstract":"We consider nonelementary random walks on general hyperbolic spaces. Without any moment condition on the walk, we show that it escapes linearly to infinity, with exponential error bounds. We even get such exponential bounds up to the rate of escape of the walk. Our proof relies on an inductive decomposition of the walk, recording times at which it could go to infinity in several independent directions, and using these times to control further backtracking.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42855799","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}
. We present criteria for the coulombian interaction energy of infinitely many points in the plane with a uniformly charged background introduced in [5] to be finite, as well as examples. We also show that in this unbounded setting, it is not always possible to project an L 2loc vector field onto the set of gradients in a way that reduces its average L 2 norm on large balls.
{"title":"Lattices with finite renormalized coulombian interaction energy in the plane","authors":"Yuxin Ge, E. Sandier","doi":"10.2140/tunis.2021.3.93","DOIUrl":"https://doi.org/10.2140/tunis.2021.3.93","url":null,"abstract":". We present criteria for the coulombian interaction energy of infinitely many points in the plane with a uniformly charged background introduced in [5] to be finite, as well as examples. We also show that in this unbounded setting, it is not always possible to project an L 2loc vector field onto the set of gradients in a way that reduces its average L 2 norm on large balls.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46700678","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}
Pub Date : 2021-01-01DOI: 10.2140/tunis.2021.3.395
T. Goudon
We investigate the behavior of solutions of simple linear kinetic equations, subjected to a strong and fast time-oscillating force field, in the diffusion regime. We consider quasiperiodic oscillations. The asymptotic behavior is governed by a convection-diffusion equation, for which we can identify the effective coefficients.
{"title":"Diffusion regime with a high and oscillating field","authors":"T. Goudon","doi":"10.2140/tunis.2021.3.395","DOIUrl":"https://doi.org/10.2140/tunis.2021.3.395","url":null,"abstract":"We investigate the behavior of solutions of simple linear kinetic equations, subjected to a strong and fast time-oscillating force field, in the diffusion regime. We consider quasiperiodic oscillations. The asymptotic behavior is governed by a convection-diffusion equation, for which we can identify the effective coefficients.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68572165","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}
Pub Date : 2020-12-15DOI: 10.2140/tunis.2022.4.307
Y. Benoist, D. Hulin
We prove that a harmonic quasi-isometric map between pinched Hadamard surfaces is a quasi-conformal diffeomorphism.
我们证明了压缩Hadamard曲面之间的调和拟等距映射是一个拟共形微分同胚。
{"title":"Harmonic quasi-isometries of pinched Hadamard surfaces are injective","authors":"Y. Benoist, D. Hulin","doi":"10.2140/tunis.2022.4.307","DOIUrl":"https://doi.org/10.2140/tunis.2022.4.307","url":null,"abstract":"We prove that a harmonic quasi-isometric map between pinched Hadamard surfaces is a quasi-conformal diffeomorphism.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41684490","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}
Pub Date : 2020-10-30DOI: 10.2140/tunis.2022.4.559
H. Furusho
We introduce a method of analytic continuation of Carlitz multiple (star) polylogarithms to the whole space and present a treatment of their branches. As applications of this method, we obtain (1) a method of continuation of the logarithms of higher tensor powers of Carlitz module, (2) the orthogonal property (Chang-Mishiba functional relations), (3) a branch independency of the Eulerian property.
{"title":"Analytic continuation of multiple polylogarithms in positive characteristic","authors":"H. Furusho","doi":"10.2140/tunis.2022.4.559","DOIUrl":"https://doi.org/10.2140/tunis.2022.4.559","url":null,"abstract":"We introduce a method of analytic continuation of Carlitz multiple (star) polylogarithms to the whole space and present a treatment of their branches. As applications of this method, we obtain (1) a method of continuation of the logarithms of higher tensor powers of Carlitz module, (2) the orthogonal property (Chang-Mishiba functional relations), (3) a branch independency of the Eulerian property.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43322654","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}
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