<p>We study Bernoulli bond percolation on nonunimodular quasi-transitive graphs, and more generally graphs whose automorphism group has a nonunimodular quasi-transitive subgroup. We prove that percolation on any such graph has a nonempty phase in which there are infinite <italic>light</italic> clusters, which implies the existence of a nonempty phase in which there are <italic>infinitely many</italic> infinite clusters. That is, we show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript c Baseline greater-than p Subscript h Baseline less-than-or-equal-to p Subscript u"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>></mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>h</mml:mi> </mml:msub> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>u</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">p_c>p_h leq p_u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any such graph. This answers a question of Häggström, Peres, and Schonmann (1999), and verifies the nonunimodular case of a well-known conjecture of Benjamini and Schramm (1996). We also prove that the triangle condition holds at criticality on any such graph, which implies that various critical exponents exist and take their mean-field values.</p> <p>All our results apply, for example, to the product <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript k Baseline times double-struck upper Z Superscript d"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>×<!-- × --></mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">T_ktimes mathbb {Z}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-regular tree with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {Z}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formu
我们研究了非一致拟传递图上的Bernoulli键渗流，以及更一般的自同构群具有非一致拟转移子群的图。我们证明了任何这样的图上的渗流都有一个非空相，其中有无限个光团簇，这意味着存在一个非空相，其中存在无限多个无限团簇。也就是说，我们证明了对于任何这样的图，p c>p h≤p u p_c>p hleq p_u。这回答了Häggström、Peres和Schonmann（1999）的一个问题，并验证了Benjamini和Schramm（1996）一个著名猜想的非一致性情况。我们还证明了在任何这样的图上，三角形条件在临界时成立，这意味着存在各种临界指数，并取其平均场值。例如，我们的所有结果都适用于k≥3kgeq3和d≥1dgeq1的具有Z d mathbb｛Z｝^d的k-正则树的乘积Tk×，对于这些结果先前仅对于大的k k是已知的。此外，我们的方法还使我们能够在这类乘积上建立各向异性渗流相图的基本拓扑特征，其中树边和Z d mathbb｛Z｝^d边被赋予不同的保留概率。这些特征以前只是针对d=1 d=1，k k大而建立的。
{"title":"Nonuniqueness and mean-field criticality for percolation on nonunimodular transitive graphs","authors":"Tom Hutchcroft","doi":"10.1090/jams/953","DOIUrl":"https://doi.org/10.1090/jams/953","url":null,"abstract":"<p>We study Bernoulli bond percolation on nonunimodular quasi-transitive graphs, and more generally graphs whose automorphism group has a nonunimodular quasi-transitive subgroup. We prove that percolation on any such graph has a nonempty phase in which there are infinite <italic>light</italic> clusters, which implies the existence of a nonempty phase in which there are <italic>infinitely many</italic> infinite clusters. That is, we show that <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p Subscript c Baseline greater-than p Subscript h Baseline less-than-or-equal-to p Subscript u\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mi>c</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo>></mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mi>h</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo>≤<!-- ≤ --></mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mi>u</mml:mi>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">p_c>p_h leq p_u</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> for any such graph. This answers a question of Häggström, Peres, and Schonmann (1999), and verifies the nonunimodular case of a well-known conjecture of Benjamini and Schramm (1996). We also prove that the triangle condition holds at criticality on any such graph, which implies that various critical exponents exist and take their mean-field values.</p>\u0000\u0000<p>All our results apply, for example, to the product <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T Subscript k Baseline times double-struck upper Z Superscript d\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>T</mml:mi>\u0000 <mml:mi>k</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo>×<!-- × --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>d</mml:mi>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">T_ktimes mathbb {Z}^d</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of a <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\u0000 <mml:semantics>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-regular tree with <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Z Superscript d\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>d</mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {Z}^d</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formu","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2017-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jams/953","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48776274","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that if a normal subgroup of the extended mapping class group of a closed surface has an element of sufficiently small support, then its automorphism group and abstract commensurator group are both isomorphic to the extended mapping class group. The proof relies on another theorem we prove, which states that many simplicial complexes associated to a closed surface have automorphism group isomorphic to the extended mapping class group. These results resolve the metaconjecture of N. V. Ivanov, which asserts that any “sufficiently rich” object associated to a surface has automorphism group isomorphic to the extended mapping class group, for a broad class of such objects. As applications, we show: (1) right-angled Artin groups and surface groups cannot be isomorphic to normal subgroups of mapping class groups containing elements of small support, (2) normal subgroups of distinct mapping class groups cannot be isomorphic if they both have elements of small support, and (3) distinct normal subgroups of the mapping class group with elements of small support are not isomorphic. Our results also suggest a new framework for the classification of normal subgroups of the mapping class group.
{"title":"Normal subgroups of mapping class groups and the metaconjecture of Ivanov","authors":"Tara E. Brendle, D. Margalit","doi":"10.1090/JAMS/927","DOIUrl":"https://doi.org/10.1090/JAMS/927","url":null,"abstract":"We prove that if a normal subgroup of the extended mapping class group of a closed surface has an element of sufficiently small support, then its automorphism group and abstract commensurator group are both isomorphic to the extended mapping class group. The proof relies on another theorem we prove, which states that many simplicial complexes associated to a closed surface have automorphism group isomorphic to the extended mapping class group. These results resolve the metaconjecture of N. V. Ivanov, which asserts that any “sufficiently rich” object associated to a surface has automorphism group isomorphic to the extended mapping class group, for a broad class of such objects. As applications, we show: (1) right-angled Artin groups and surface groups cannot be isomorphic to normal subgroups of mapping class groups containing elements of small support, (2) normal subgroups of distinct mapping class groups cannot be isomorphic if they both have elements of small support, and (3) distinct normal subgroups of the mapping class group with elements of small support are not isomorphic. Our results also suggest a new framework for the classification of normal subgroups of the mapping class group.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2017-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/927","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47198217","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, it is shown that any surface automorphism of positive mapping-class entropy possesses a virtual homological eigenvalue which lies outside the unit circle of the complex plane.
本文证明了任何正映射类熵的曲面自同构在复平面的单位圆外都有一个虚同调特征值。
{"title":"Virtual homological spectral radii for automorphisms of surfaces","authors":"Yi Liu","doi":"10.1090/jams/949","DOIUrl":"https://doi.org/10.1090/jams/949","url":null,"abstract":"In this paper, it is shown that any surface automorphism of positive mapping-class entropy possesses a virtual homological eigenvalue which lies outside the unit circle of the complex plane.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2017-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jams/949","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42023908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The height fluctuations of the models in the KPZ class are expected to converge to a universal process. The spatial process at equal time is known to converge to the Airy process or its variations. However, the temporal process, or more generally the two-dimensional space-time fluctuation field, is less well understood. We consider this question for the periodic TASEP (totally asymmetric simple exclusion process). For a particular initial condition, we evaluate the multitime and multilocation distribution explicitly in terms of a multiple integral involving a Fredholm determinant. We then evaluate the large-time limit in the so-called relaxation time scale.
{"title":"Multipoint distribution of periodic TASEP","authors":"J. Baik, Zhipeng Liu","doi":"10.1090/JAMS/915","DOIUrl":"https://doi.org/10.1090/JAMS/915","url":null,"abstract":"The height fluctuations of the models in the KPZ class are expected to converge to a universal process. The spatial process at equal time is known to converge to the Airy process or its variations. However, the temporal process, or more generally the two-dimensional space-time fluctuation field, is less well understood. We consider this question for the periodic TASEP (totally asymmetric simple exclusion process). For a particular initial condition, we evaluate the multitime and multilocation distribution explicitly in terms of a multiple integral involving a Fredholm determinant. We then evaluate the large-time limit in the so-called relaxation time scale.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2017-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/915","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41978246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the symplectic topology of certain K3 surfaces (including the "mirror quartic" and "mirror double plane"), equipped with certain Kahler forms. In particular, we prove that the symplectic Torelli group may be infinitely generated, and derive new constraints on Lagrangian tori. The key input, via homological mirror symmetry, is a result of Bayer and Bridgeland on the autoequivalence group of the derived category of an algebraic K3 surface of Picard rank one.
{"title":"Symplectic topology of $K3$ surfaces via mirror symmetry","authors":"Nick Sheridan, I. Smith","doi":"10.1090/JAMS/946","DOIUrl":"https://doi.org/10.1090/JAMS/946","url":null,"abstract":"We study the symplectic topology of certain K3 surfaces (including the \"mirror quartic\" and \"mirror double plane\"), equipped with certain Kahler forms. In particular, we prove that the symplectic Torelli group may be infinitely generated, and derive new constraints on Lagrangian tori. The key input, via homological mirror symmetry, is a result of Bayer and Bridgeland on the autoequivalence group of the derived category of an algebraic K3 surface of Picard rank one.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2017-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/946","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49383546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pramod N. Achar, Shotaro Makisumi, S. Riche, G. Williamson
We establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic ℓell in terms of ℓell-Kazhdan–Lusztig polynomials, for ℓ>hell > h the Coxeter number. Using results of Andersen, one may deduce a character formula for simple modules if ℓ≥2h−2ell ge 2h-2. Our results are a consequence of an extension to modular coefficients of a monoidal Koszul duality equivalence established by Bezrukavnikov and Yun.
我们建立了一个关于特征为r well的连通约化群的不可分解倾斜模的特征公式，它是用r well -Kazhdan-Lusztig多项式表示的，对于r well - h well - > h为Coxeter数。利用Andersen的结果，我们可以推导出当r≥2h−2 well ge 2h-2时简单模的特征公式。我们的结果是将Bezrukavnikov和Yun建立的一元Koszul对偶等价推广到模系数的结果。
{"title":"Koszul duality for Kac–Moody groups and characters of tilting modules","authors":"Pramod N. Achar, Shotaro Makisumi, S. Riche, G. Williamson","doi":"10.1090/JAMS/905","DOIUrl":"https://doi.org/10.1090/JAMS/905","url":null,"abstract":"<p>We establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l\">\u0000 <mml:semantics>\u0000 <mml:mi>ℓ<!-- ℓ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">ell</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in terms of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l\">\u0000 <mml:semantics>\u0000 <mml:mi>ℓ<!-- ℓ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">ell</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-Kazhdan–Lusztig polynomials, for <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l greater-than h\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>ℓ<!-- ℓ --></mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mi>h</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">ell > h</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> the Coxeter number. Using results of Andersen, one may deduce a character formula for simple modules if <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l greater-than-or-equal-to 2 h minus 2\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>ℓ<!-- ℓ --></mml:mi>\u0000 <mml:mo>≥<!-- ≥ --></mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mi>h</mml:mi>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">ell ge 2h-2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Our results are a consequence of an extension to modular coefficients of a monoidal Koszul duality equivalence established by Bezrukavnikov and Yun.</p>","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/905","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41491690","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For embedded 2-spheres in a 4-manifold sharing the same embedded transverse sphere homotopy implies isotopy, provided the ambient 4-manifold has no $BZ_2$-torsion in the fundamental group. This gives a generalization of the classical light bulb trick to 4-dimensions, the uniqueness of spanning discs for a simple closed curve in $S^4$ and $pi_0(Diff_0(S^2times D^2)/Diff_0(B^4))=1$. In manifolds with $BZ_2$-torsion, one surface can be put into a normal form relative to the other.
{"title":"The 4-dimensional light bulb theorem","authors":"David Gabai","doi":"10.1090/JAMS/920","DOIUrl":"https://doi.org/10.1090/JAMS/920","url":null,"abstract":"For embedded 2-spheres in a 4-manifold sharing the same embedded transverse sphere homotopy implies isotopy, provided the ambient 4-manifold has no $BZ_2$-torsion in the fundamental group. This gives a generalization of the classical light bulb trick to 4-dimensions, the uniqueness of spanning discs for a simple closed curve in $S^4$ and $pi_0(Diff_0(S^2times D^2)/Diff_0(B^4))=1$. In manifolds with $BZ_2$-torsion, one surface can be put into a normal form relative to the other.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2017-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/920","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48891922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main result of this article is a rigidity result pertaining to the spreading model structure for Banach spaces coarsely embeddable into Tsirelson's original space $T^*$. Every Banach space that is coarsely embeddable into $T^*$ must be reflexive and all its spreading models must be isomorphic to $c_0$. Several important consequences follow from our rigidity result. We obtain a coarse version of an influential theorem of Tsirelson: $T^*$ does not coarsely contain $c_0$ nor $ell_p$ for $pin[1,infty)$. We show that there is no infinite dimensional Banach space that coarsely embeds into every infinite dimensional Banach space. In particular, we disprove the conjecture that the separable infinite dimensional Hilbert space coarsely embeds into every infinite dimensional Banach space. The rigidity result follows from a new concentration inequality for Lipschitz maps on the infinite Hamming graphs and taking values in $T^*$, and from the embeddability of the infinite Hamming graphs into Banach spaces that admit spreading models not isomorphic to $c_0$. Also, a purely metric characterization of finite dimensionality is obtained.
{"title":"The coarse geometry of Tsirelson’s space and applications","authors":"F. Baudier, G. Lancien, T. Schlumprecht","doi":"10.1090/jams/899","DOIUrl":"https://doi.org/10.1090/jams/899","url":null,"abstract":"The main result of this article is a rigidity result pertaining to the spreading model structure for Banach spaces coarsely embeddable into Tsirelson's original space $T^*$. Every Banach space that is coarsely embeddable into $T^*$ must be reflexive and all its spreading models must be isomorphic to $c_0$. Several important consequences follow from our rigidity result. We obtain a coarse version of an influential theorem of Tsirelson: $T^*$ does not coarsely contain $c_0$ nor $ell_p$ for $pin[1,infty)$. We show that there is no infinite dimensional Banach space that coarsely embeds into every infinite dimensional Banach space. In particular, we disprove the conjecture that the separable infinite dimensional Hilbert space coarsely embeds into every infinite dimensional Banach space. The rigidity result follows from a new concentration inequality for Lipschitz maps on the infinite Hamming graphs and taking values in $T^*$, and from the embeddability of the infinite Hamming graphs into Banach spaces that admit spreading models not isomorphic to $c_0$. Also, a purely metric characterization of finite dimensionality is obtained.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"31 1","pages":"699-717"},"PeriodicalIF":3.9,"publicationDate":"2017-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jams/899","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43521339","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that any finite-degree polynomial functor over an infinite field is topologically Noetherian. This theorem is motivated by the recent resolution, by Ananyan-Hochster, of Stillman’s conjecture; and a recent Noetherianity proof by Derksen-Eggermont-Snowden for the space of cubics. Via work by Erman-Sam-Snowden, our theorem implies Stillman’s conjecture and indeed boundedness of a wider class of invariants of ideals in polynomial rings with a fixed number of generators of prescribed degrees.
{"title":"Topological Noetherianity of polynomial functors","authors":"J. Draisma","doi":"10.1090/JAMS/923","DOIUrl":"https://doi.org/10.1090/JAMS/923","url":null,"abstract":"We prove that any finite-degree polynomial functor over an infinite field is topologically Noetherian. This theorem is motivated by the recent resolution, by Ananyan-Hochster, of Stillman’s conjecture; and a recent Noetherianity proof by Derksen-Eggermont-Snowden for the space of cubics. Via work by Erman-Sam-Snowden, our theorem implies Stillman’s conjecture and indeed boundedness of a wider class of invariants of ideals in polynomial rings with a fixed number of generators of prescribed degrees.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2017-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/923","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43051433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We compute the pp-adic geometric étale cohomology of the coverings of the Drinfeld half-plane, and we show that, if the base field is Qpmathbf {Q}_p, this cohomology encodes the pp-adic local Langlands correspondence for 22-dimensional de Rham representations (of weight 00 and 11).
我们计算了Drinfeld半平面覆盖物的p-p-adic几何étale上同调，并证明了如果基场是Q p mathbf{Q}_p，该上同调对2个二维de Rham表示（权重为0 0和1 1）的p p-adic局部Langlands对应关系进行编码。
{"title":"Cohomologie 𝑝-adique de la tour de Drinfeld: le cas de la dimension 1","authors":"P. Colmez, Gabriel Dospinescu, Wiesława Nizioł","doi":"10.1090/jams/935","DOIUrl":"https://doi.org/10.1090/jams/935","url":null,"abstract":"<p>We compute the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-adic geometric étale cohomology of the coverings of the Drinfeld half-plane, and we show that, if the base field is <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold upper Q Subscript p\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"bold\">Q</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>p</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbf {Q}_p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, this cohomology encodes the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-adic local Langlands correspondence for <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\u0000 <mml:semantics>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-dimensional de Rham representations (of weight <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0\">\u0000 <mml:semantics>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1\">\u0000 <mml:semantics>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>).</p>","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2017-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jams/935","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43585342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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