Pub Date : 2019-01-01DOI: 10.2140/TUNIS.2019.1.585
J. Faraut
{"title":"Horn’s problem and Fourier analysis","authors":"J. Faraut","doi":"10.2140/TUNIS.2019.1.585","DOIUrl":"https://doi.org/10.2140/TUNIS.2019.1.585","url":null,"abstract":"","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/TUNIS.2019.1.585","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68571721","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 : 2018-09-15DOI: 10.2140/tunis.2020.2.337
C. Bushnell, G. Henniart
Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$. Let $sigma$ be an irreducible smooth representation of the absolute Weil group $Cal W_F$ of $F$ and $sw(sigma)$ the Swan exponent of $sigma$. Assume $sw(sigma) ge1$. Let $Cal I_F$ be the inertia subgroup of $Cal W_F$ and $Cal P_F$ the wild inertia subgroup. There is an essentially unique, finite, cyclic group $varSigma$, of order prime to $p$, so that $sigma(Cal I_F) = sigma(Cal P_F)varSigma$. In response to a query of Mark Reeder, we show that the multiplicity in $sigma$ of any character of $varSigma$ is bounded by $sw(sigma)$.
{"title":"Tame multiplicity and conductor for local Galois representations","authors":"C. Bushnell, G. Henniart","doi":"10.2140/tunis.2020.2.337","DOIUrl":"https://doi.org/10.2140/tunis.2020.2.337","url":null,"abstract":"Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$. Let $sigma$ be an irreducible smooth representation of the absolute Weil group $Cal W_F$ of $F$ and $sw(sigma)$ the Swan exponent of $sigma$. Assume $sw(sigma) ge1$. Let $Cal I_F$ be the inertia subgroup of $Cal W_F$ and $Cal P_F$ the wild inertia subgroup. There is an essentially unique, finite, cyclic group $varSigma$, of order prime to $p$, so that $sigma(Cal I_F) = sigma(Cal P_F)varSigma$. In response to a query of Mark Reeder, we show that the multiplicity in $sigma$ of any character of $varSigma$ is bounded by $sw(sigma)$.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2020.2.337","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42856046","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 : 2018-09-11DOI: 10.2140/tunis.2022.4.387
Alexander Adam, V. Baladi
We study semigroups of weighted transfer operators for Anosov flows of regularity C^r, r>1, on compact manifolds without boundary. We construct an anisotropic Banach space on which the resolvent of the generator is quasi-compact and where the upper bound on the essential spectral radius depends continuously on the regularity. We apply this result to the ergodic average of the horocycle flow for C^3 contact Anosov flows in dimension three.
{"title":"Horocycle averages on closed manifolds and transfer operators","authors":"Alexander Adam, V. Baladi","doi":"10.2140/tunis.2022.4.387","DOIUrl":"https://doi.org/10.2140/tunis.2022.4.387","url":null,"abstract":"We study semigroups of weighted transfer operators for Anosov flows of regularity C^r, r>1, on compact manifolds without boundary. We construct an anisotropic Banach space on which the resolvent of the generator is quasi-compact and where the upper bound on the essential spectral radius depends continuously on the regularity. We apply this result to the ergodic average of the horocycle flow for C^3 contact Anosov flows in dimension three.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42557705","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 : 2018-08-09DOI: 10.2140/tunis.2020.2.791
G. Link
Let $X$ be a proper, geodesically complete Hadamard space, and $ Gamma
设$X$是一个固有的、测地完备的Hadamard空间，$ Gamma
{"title":"Equidistribution and counting of orbit points for discrete rank one isometry groups of Hadamard spaces","authors":"G. Link","doi":"10.2140/tunis.2020.2.791","DOIUrl":"https://doi.org/10.2140/tunis.2020.2.791","url":null,"abstract":"Let $X$ be a proper, geodesically complete Hadamard space, and $ Gamma<mbox{Is}(X)$ a discrete subgroup of isometries of $X$ with the fixed point of a rank one isometry of $X$ in its infinite limit set. In this paper we prove that if $Gamma$ has non-arithmetic length spectrum, then the Ricks' Bowen-Margulis measure -- which generalizes the well-known Bowen-Margulis measure in the CAT$(-1)$ setting -- is mixing. If in addition the Ricks' Bowen-Margulis measure is finite, then we also have equidistribution of $Gamma$-orbit points in $X$, which in particular yields an asymptotic estimate for the orbit counting function of $Gamma$. This generalizes well-known facts for non-elementary discrete isometry groups of Hadamard manifolds with pinched negative curvature and proper CAT$(-1)$-spaces.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2020.2.791","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43588462","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 : 2018-07-30DOI: 10.2140/tunis.2020.2.309
S. Saito, Kanetomo Sato
In this paper, we will show that the sheaf of p-adic vanishing cycles of a log smooth family over a DVR of mixed characteristic is generated by Milnor symboles. A key ingredient is a computation (due to K. Kato) on the graded quotients of a multi-indexed filtration on the sheaf concerned, which has been used in several papers of the first author.
{"title":"On p-adic vanishing cycles of log smooth\u0000families","authors":"S. Saito, Kanetomo Sato","doi":"10.2140/tunis.2020.2.309","DOIUrl":"https://doi.org/10.2140/tunis.2020.2.309","url":null,"abstract":"In this paper, we will show that the sheaf of p-adic vanishing cycles of a log smooth family over a DVR of mixed characteristic is generated by Milnor symboles. A key ingredient is a computation (due to K. Kato) on the graded quotients of a multi-indexed filtration on the sheaf concerned, which has been used in several papers of the first author.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2020.2.309","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41352703","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 : 2018-05-08DOI: 10.2140/tunis.2020.2.667
J. Fontaine
Let $K$ be a finite extension of $mathbb{Q}_p$ and $G_K$ the absolute Galois group. Then $G_K$ acts on the fundamental curve $X$ of $p$-adic Hodge theory and we may consider the abelian category $mathcal{M}(G_K)$ of coherent $mathcal{O}_X$-modules equipped with a continuous and semi-linear action of $G_K$. An almost $C_p$-representation of $G_K$ is a $p$-adic Banach space $V$ equipped with a linear and continuous action of $G_K$ such that there exists $dinmathbb{N}$, two $G_K$-stable finite dimensional sub-$mathbb{Q}_p$-vector spaces $U_+$ of $V$, $U_-$ of $C_p^d$, and a $G_K$-equivariant isomorphism $V/U_+to C_p^d/U_-$. These representations form an abelian category $mathcal{C}(G_K)$. The main purpose of this paper is to prove that $mathcal{C}(G_K)$ can be recovered from $mathcal{M}(G_K)$ by a simple construction (and conversely) inducing, in particular, an equivalence of triangulated categories $D^b(mathcal{M}(G_K))to D^b(mathcal{C}(G_K))$.
{"title":"Almost ℂp Galois representations and vector\u0000bundles","authors":"J. Fontaine","doi":"10.2140/tunis.2020.2.667","DOIUrl":"https://doi.org/10.2140/tunis.2020.2.667","url":null,"abstract":"Let $K$ be a finite extension of $mathbb{Q}_p$ and $G_K$ the absolute Galois group. Then $G_K$ acts on the fundamental curve $X$ of $p$-adic Hodge theory and we may consider the abelian category $mathcal{M}(G_K)$ of coherent $mathcal{O}_X$-modules equipped with a continuous and semi-linear action of $G_K$. An almost $C_p$-representation of $G_K$ is a $p$-adic Banach space $V$ equipped with a linear and continuous action of $G_K$ such that there exists $dinmathbb{N}$, two $G_K$-stable finite dimensional sub-$mathbb{Q}_p$-vector spaces $U_+$ of $V$, $U_-$ of $C_p^d$, and a $G_K$-equivariant isomorphism $V/U_+to C_p^d/U_-$. These representations form an abelian category $mathcal{C}(G_K)$. The main purpose of this paper is to prove that $mathcal{C}(G_K)$ can be recovered from $mathcal{M}(G_K)$ by a simple construction (and conversely) inducing, in particular, an equivalence of triangulated categories $D^b(mathcal{M}(G_K))to D^b(mathcal{C}(G_K))$.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2020.2.667","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45798138","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 : 2018-02-24DOI: 10.2140/tunis.2020.2.147
R. Cohen, Inbar Klang
In this paper, we import the theory of "Calabi-Yau" algebras and categories from symplectic topology and topological field theories to the setting of spectra in stable homotopy theory. Twistings in this theory will be particularly important. There will be two types of Calabi-Yau structures in the setting of ring spectra: one that applies to compact algebras and one that applies to smooth algebras. The main application of twisted compact Calabi-Yau ring spectra that we will study is to describe, prove, and explain a certain duality phenomenon in string topology. This is a duality between the manifold string topology of Chas-Sullivan and the Lie group string topology of Chataur-Menichi. This will extend and generalize work of Gruher. Then, generalizing work of the first author and Jones, we show how the gauge group of the principal bundle acts on this compact Calabi-Yau structure, and compute some explicit examples. We then extend the notion of the Calabi-Yau structure to smooth ring spectra, and prove that Thom ring spectra of (virtual) bundles over the loop space, $Omega M$, have this structure. In the case when $M$ is a sphere we will use these twisted smooth Calabi-Yau ring spectra to study Lagrangian immersions of the sphere into its cotangent bundle. We recast the work of Abouzaid-Kragh to show that the topological Hochschild homology of the Thom ring spectrum induced by the $h$-principle classifying map of the Lagrangian immersion, detects whether that immersion can be Lagrangian isotopic to an embedding. We then compute some examples. Finally, we interpret these Calabi-Yau structures directly in terms of topological Hochschild homology and cohomology.
{"title":"Twisted Calabi–Yau ring spectra, string\u0000topology, and gauge symmetry","authors":"R. Cohen, Inbar Klang","doi":"10.2140/tunis.2020.2.147","DOIUrl":"https://doi.org/10.2140/tunis.2020.2.147","url":null,"abstract":"In this paper, we import the theory of \"Calabi-Yau\" algebras and categories from symplectic topology and topological field theories to the setting of spectra in stable homotopy theory. Twistings in this theory will be particularly important. There will be two types of Calabi-Yau structures in the setting of ring spectra: one that applies to compact algebras and one that applies to smooth algebras. The main application of twisted compact Calabi-Yau ring spectra that we will study is to describe, prove, and explain a certain duality phenomenon in string topology. This is a duality between the manifold string topology of Chas-Sullivan and the Lie group string topology of Chataur-Menichi. This will extend and generalize work of Gruher. Then, generalizing work of the first author and Jones, we show how the gauge group of the principal bundle acts on this compact Calabi-Yau structure, and compute some explicit examples. We then extend the notion of the Calabi-Yau structure to smooth ring spectra, and prove that Thom ring spectra of (virtual) bundles over the loop space, $Omega M$, have this structure. In the case when $M$ is a sphere we will use these twisted smooth Calabi-Yau ring spectra to study Lagrangian immersions of the sphere into its cotangent bundle. We recast the work of Abouzaid-Kragh to show that the topological Hochschild homology of the Thom ring spectrum induced by the $h$-principle classifying map of the Lagrangian immersion, detects whether that immersion can be Lagrangian isotopic to an embedding. We then compute some examples. Finally, we interpret these Calabi-Yau structures directly in terms of topological Hochschild homology and cohomology.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68571729","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 : 2018-02-06DOI: 10.2140/tunis.2020.2.455
Piotr Achinger, A. Ogus
A now classical construction due to Kato and Nakayama attaches a topological space (the "Betti realization") to a log scheme over $mathbf{C}$. We show that in the case of a log smooth degeneration over the standard log disc, this construction allows one to recover the topology of the germ of the family from the log special fiber alone. We go on to give combinatorial formulas for the monodromy and the $d^2$ differentials acting on the nearby cycle complex in terms of the log structures. We also provide variants of these results for the Kummer etale topology. In the case of curves, these data are essentially equivalent to those encoded by the dual graph of a semistable degeneration, including the monodromy pairing and the Picard-Lefschetz formula.
{"title":"Monodromy and log geometry","authors":"Piotr Achinger, A. Ogus","doi":"10.2140/tunis.2020.2.455","DOIUrl":"https://doi.org/10.2140/tunis.2020.2.455","url":null,"abstract":"A now classical construction due to Kato and Nakayama attaches a topological space (the \"Betti realization\") to a log scheme over $mathbf{C}$. We show that in the case of a log smooth degeneration over the standard log disc, this construction allows one to recover the topology of the germ of the family from the log special fiber alone. We go on to give combinatorial formulas for the monodromy and the $d^2$ differentials acting on the nearby cycle complex in terms of the log structures. We also provide variants of these results for the Kummer etale topology. In the case of curves, these data are essentially equivalent to those encoded by the dual graph of a semistable degeneration, including the monodromy pairing and the Picard-Lefschetz formula.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2020.2.455","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48607982","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 : 2018-01-29DOI: 10.2140/tunis.2019.1.519
Jinghao Li, A. Vasiu
Let $p$ be a prime. Let $ninmathbb N-{0}$. Let $mathcal C$ be an $F^n$-crystal over a locally noetherian $mathbb F_p$-scheme $S$. Let $(a,b)inmathbb N^2$. We show that the reduced locally closed subscheme of $S$ whose points are exactly those $xin S$ such that $(a,b)$ is a break point of the Newton polygon of the fiber $mathcal C_x$ of $mathcal C$ at $x$ is pure in $S$, i.e., it is an affine $S$-scheme. This result refines and reobtains previous results of de Jong--Oort, Vasiu, and Yang. As an application, we show that for all $min mathbb N$ the reduced locally closed subscheme of $S$ whose points are exactly those $xin S$ for which the $p$-rank of $mathcal C_x$ is $m$ is pure in $S$; the case $n=1$ was previously obtained by Deligne (unpublished) and the general case $nge 1$ refines and reobtains a result of Zink.
{"title":"Purity of crystalline strata","authors":"Jinghao Li, A. Vasiu","doi":"10.2140/tunis.2019.1.519","DOIUrl":"https://doi.org/10.2140/tunis.2019.1.519","url":null,"abstract":"Let $p$ be a prime. Let $ninmathbb N-{0}$. Let $mathcal C$ be an $F^n$-crystal over a locally noetherian $mathbb F_p$-scheme $S$. Let $(a,b)inmathbb N^2$. We show that the reduced locally closed subscheme of $S$ whose points are exactly those $xin S$ such that $(a,b)$ is a break point of the Newton polygon of the fiber $mathcal C_x$ of $mathcal C$ at $x$ is pure in $S$, i.e., it is an affine $S$-scheme. This result refines and reobtains previous results of de Jong--Oort, Vasiu, and Yang. As an application, we show that for all $min mathbb N$ the reduced locally closed subscheme of $S$ whose points are exactly those $xin S$ for which the $p$-rank of $mathcal C_x$ is $m$ is pure in $S$; the case $n=1$ was previously obtained by Deligne (unpublished) and the general case $nge 1$ refines and reobtains a result of Zink.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2019.1.519","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43885858","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 : 2018-01-23DOI: 10.2140/tunis.2019.1.273
J. Chemin, I. Gallagher
The purpose of this article is to establish bounds from below for the life span of regular solutions to the incompressible Navier-Stokes system, which involve norms not only of the initial data, but also of nonlinear functions of the initial data. We provide examples showing that those bounds are significant improvements to the one provided by the classical fixed point argument. One of the important ingredients is the use of a scale-invariant energy estimate.
{"title":"A nonlinear estimate of the life span of solutions\u0000of the three dimensional Navier–Stokes equations","authors":"J. Chemin, I. Gallagher","doi":"10.2140/tunis.2019.1.273","DOIUrl":"https://doi.org/10.2140/tunis.2019.1.273","url":null,"abstract":"The purpose of this article is to establish bounds from below for the life span of regular solutions to the incompressible Navier-Stokes system, which \u0000involve norms not only of the initial data, but also of nonlinear functions of the initial data. We provide examples showing that those bounds are significant improvements to the one provided by the classical fixed point argument. One of the important ingredients is the use of a scale-invariant energy estimate.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2019.1.273","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42200442","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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