Pub Date : 2019-10-29DOI: 10.1215/00127094-2021-0045
Weikun He, Nicolas de Saxc'e
We prove a quantitative equidistribution result for linear random walks on the torus, similar to a theorem of Bourgain, Furman, Lindenstrauss and Mozes, but without any proximality assumption. An application is given to expansion in simple groups, modulo arbitrary integers.
{"title":"Linear random walks on the torus","authors":"Weikun He, Nicolas de Saxc'e","doi":"10.1215/00127094-2021-0045","DOIUrl":"https://doi.org/10.1215/00127094-2021-0045","url":null,"abstract":"We prove a quantitative equidistribution result for linear random walks on the torus, similar to a theorem of Bourgain, Furman, Lindenstrauss and Mozes, but without any proximality assumption. An application is given to expansion in simple groups, modulo arbitrary integers.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42957250","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}
Pub Date : 2019-10-17DOI: 10.1215/00127094-2021-0047
Xu Shen, Chia-Fu Yu, Chao Zhang
In this paper we study the geometry of reduction modulo $p$ of the Kisin-Pappas integral models for certain Shimura varieties of abelian type with parahoric level structure. We give some direct and geometric constructions for the EKOR strata on these Shimura varieties, using the theories of $G$-zips and mixed characteristic local $mathcal{G}$-Shtukas. We establish several basic properties of these strata, including the smoothness, dimension formula, and closure relation. Moreover, we apply our results to the study of Newton strata and central leaves on these Shimura varieties.
{"title":"EKOR strata for Shimura varieties with parahoric level structure","authors":"Xu Shen, Chia-Fu Yu, Chao Zhang","doi":"10.1215/00127094-2021-0047","DOIUrl":"https://doi.org/10.1215/00127094-2021-0047","url":null,"abstract":"In this paper we study the geometry of reduction modulo $p$ of the Kisin-Pappas integral models for certain Shimura varieties of abelian type with parahoric level structure. We give some direct and geometric constructions for the EKOR strata on these Shimura varieties, using the theories of $G$-zips and mixed characteristic local $mathcal{G}$-Shtukas. We establish several basic properties of these strata, including the smoothness, dimension formula, and closure relation. Moreover, we apply our results to the study of Newton strata and central leaves on these Shimura varieties.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48067322","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}
Pub Date : 2019-10-17DOI: 10.1215/00127094-2022-0039
Ciprian Manolescu, Marco Marengon, Sucharit Sarkar, Michael Willis
We extend the definition of Khovanov-Lee homology to links in connected sums of $S^1 times S^2$'s, and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain links in $S^1 times S^2$, we compute the invariant by reinterpreting it in terms of Hochschild homology. As applications, we prove inequalities relating the Rasmussen-type invariant to the genus of surfaces with boundary in the following four-manifolds: $B^2 times S^2$, $S^1 times B^3$, $mathbb{CP}^2$, and various connected sums and boundary sums of these. We deduce that Rasmussen's invariant also gives genus bounds for surfaces inside homotopy 4-balls obtained from $B^4$ by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4-balls are non-standard.
{"title":"A generalization of Rasmussen’s invariant, with applications to surfaces in some four-manifolds","authors":"Ciprian Manolescu, Marco Marengon, Sucharit Sarkar, Michael Willis","doi":"10.1215/00127094-2022-0039","DOIUrl":"https://doi.org/10.1215/00127094-2022-0039","url":null,"abstract":"We extend the definition of Khovanov-Lee homology to links in connected sums of $S^1 times S^2$'s, and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain links in $S^1 times S^2$, we compute the invariant by reinterpreting it in terms of Hochschild homology. As applications, we prove inequalities relating the Rasmussen-type invariant to the genus of surfaces with boundary in the following four-manifolds: $B^2 times S^2$, $S^1 times B^3$, $mathbb{CP}^2$, and various connected sums and boundary sums of these. We deduce that Rasmussen's invariant also gives genus bounds for surfaces inside homotopy 4-balls obtained from $B^4$ by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4-balls are non-standard.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49238342","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}
Pub Date : 2019-10-04DOI: 10.1215/00127094-2021-0048
Weronika Buczy'nska, Jaroslaw Buczy'nski
We introduce an elementary method to study the border rank of polynomials and tensors, analogous to the apolarity lemma. This can be used to describe the border rank of all cases uniformly, including those very special ones that resisted a systematic approach. We also define a border rank version of the variety of sums of powers and analyse how it is useful in studying tensors and polynomials with large symmetries. In particular, it can also be applied to provide lower bounds for the border rank of some very interesting tensors, such as the matrix multiplication tensor. We work in a general setting, where the base variety is not necessarily a Segre or Veronese variety, but an arbitrary smooth toric projective variety. A critical ingredient of our work is an irreducible component of a multigraded Hilbert scheme related to the toric variety in question.
{"title":"Apolarity, border rank, and multigraded Hilbert scheme","authors":"Weronika Buczy'nska, Jaroslaw Buczy'nski","doi":"10.1215/00127094-2021-0048","DOIUrl":"https://doi.org/10.1215/00127094-2021-0048","url":null,"abstract":"We introduce an elementary method to study the border rank of polynomials and tensors, analogous to the apolarity lemma. This can be used to describe the border rank of all cases uniformly, including those very special ones that resisted a systematic approach. We also define a border rank version of the variety of sums of powers and analyse how it is useful in studying tensors and polynomials with large symmetries. In particular, it can also be applied to provide lower bounds for the border rank of some very interesting tensors, such as the matrix multiplication tensor. We work in a general setting, where the base variety is not necessarily a Segre or Veronese variety, but an arbitrary smooth toric projective variety. A critical ingredient of our work is an irreducible component of a multigraded Hilbert scheme related to the toric variety in question.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49420759","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}
Pub Date : 2019-09-26DOI: 10.1215/00127094-2020-0083
Thomas Budzinski, N. Curien, Bram Petri
We prove that the minimal diameter of a hyperbolic compact orientable surface of genus $g$ is asymptotic to $log g$ as $g to infty$. The proof relies on a random construction, which we analyse using lattice point counting theory and the exploration of random trivalent graphs.
{"title":"On the minimal diameter of closed hyperbolic surfaces","authors":"Thomas Budzinski, N. Curien, Bram Petri","doi":"10.1215/00127094-2020-0083","DOIUrl":"https://doi.org/10.1215/00127094-2020-0083","url":null,"abstract":"We prove that the minimal diameter of a hyperbolic compact orientable surface of genus $g$ is asymptotic to $log g$ as $g to infty$. The proof relies on a random construction, which we analyse using lattice point counting theory and the exploration of random trivalent graphs.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42884800","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}
Pub Date : 2019-08-27DOI: 10.1215/00127094-2022-0069
Ian Petrow, M. Young
We prove a Lindelof-on-average upper bound for the fourth moment of Dirichlet $L$-functions of conductor $q$ along a coset of the subgroup of characters modulo $d$ when $q^*|d$, where $q^*$ is the least positive integer such that $q^2|(q^*)^3$. As a consequence, we finish the previous work of the authors and establish a Weyl-strength subconvex bound for all Dirichlet $L$-functions with no restrictions on the conductor.
{"title":"The fourth moment of Dirichlet L-functions along a coset and the Weyl bound","authors":"Ian Petrow, M. Young","doi":"10.1215/00127094-2022-0069","DOIUrl":"https://doi.org/10.1215/00127094-2022-0069","url":null,"abstract":"We prove a Lindelof-on-average upper bound for the fourth moment of Dirichlet $L$-functions of conductor $q$ along a coset of the subgroup of characters modulo $d$ when $q^*|d$, where $q^*$ is the least positive integer such that $q^2|(q^*)^3$. As a consequence, we finish the previous work of the authors and establish a Weyl-strength subconvex bound for all Dirichlet $L$-functions with no restrictions on the conductor.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66769902","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}
Pub Date : 2019-08-19DOI: 10.1215/00127094-2020-0094
Tobias Finis, E. Lapid
Let $X$ be a locally symmetric space defined by a simple Chevalley group $G$ and a congruence subgroup of $G(mathbb Q)$. In this generality, the Weyl law for $X$ was proved by Lindenstrauss--Venkatesh. In the case where $G$ is simply connected, we sharpen their result by giving a power saving estimate for the remainder term.
{"title":"On the remainder term of the Weyl law for congruence subgroups of Chevalley groups","authors":"Tobias Finis, E. Lapid","doi":"10.1215/00127094-2020-0094","DOIUrl":"https://doi.org/10.1215/00127094-2020-0094","url":null,"abstract":"Let $X$ be a locally symmetric space defined by a simple Chevalley group $G$ and a congruence subgroup of $G(mathbb Q)$. In this generality, the Weyl law for $X$ was proved by Lindenstrauss--Venkatesh. In the case where $G$ is simply connected, we sharpen their result by giving a power saving estimate for the remainder term.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44421892","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}
Pub Date : 2019-08-05DOI: 10.1215/00127094-2021-0002
Javier G'omez-Serrano, Jaemin Park, Jia Shi, Yao Yao
In this paper, we study the radial symmetry properties of stationary and uniformly-rotating solutions of the 2D Euler and gSQG equations, both in the smooth setting and the patch setting. For the 2D Euler equation, we show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial, without any assumptions on the connectedness of the support or the level sets. In the patch setting, for the 2D Euler equation we show that every uniformly-rotating patch $D$ with angular velocity $Omega leq 0$ or $Omegageq frac{1}{2}$ must be radial, where both bounds are sharp. For the gSQG equation we obtain a similar symmetry result for $Omegaleq 0$ or $Omegageq Omega_alpha$ (with the bounds being sharp), under the additional assumption that the patch is simply-connected. These results settle several open questions in [T. Hmidi, J. Evol. Equ., 15(4): 801-816, 2015] and [F. de la Hoz, Z. Hassainia, T. Hmidi, and J. Mateu, Anal. PDE, 9(7):1609-1670, 2016] on uniformly-rotating patches. Along the way, we close a question on overdetermined problems for the fractional Laplacian [R. Choksi, R. Neumayer, and I. Topaloglu, Arxiv preprint arXiv:1810.08304, 2018, Remark 1.4], which may be of independent interest. The main new ideas come from a calculus of variations point of view.
本文研究了二维Euler方程和gSQG方程在光滑和贴片条件下的稳态解和均匀旋转解的径向对称性。对于二维欧拉方程，我们证明了任何具有紧支撑和非负涡度的光滑平稳解必须是径向的，而不需要假设支撑或水平集的连通性。在贴片设置中，对于二维欧拉方程，我们表明每个具有角速度$Omega leq 0$或$Omegageq frac{1}{2}$的均匀旋转贴片$D$必须是径向的，其中两个边界都是尖锐的。对于gSQG方程，我们在附加假设patch是单连通的情况下，对$Omegaleq 0$或$Omegageq Omega_alpha$(边界很明显)获得了类似的对称结果。这些结果解决了[T]中的几个开放性问题。J.进化。等式。[j] .中国机械工程，2015(4):801-816。de la Hoz, Z. Hassainia, T. Hmidi和J. Mateu, Anal。地球物理学报，9(7):1609-1670,2016 [j]。在此过程中，我们结束了分数阶拉普拉斯函数的超定问题。Choksi, R. Neumayer, and I. Topaloglu, Arxiv预印本[j] . [j] .预印本[j] . vol . 11(4):1 - 2, 2018。主要的新思想来自变分演算的观点。
{"title":"Symmetry in stationary and uniformly rotating solutions of active scalar equations","authors":"Javier G'omez-Serrano, Jaemin Park, Jia Shi, Yao Yao","doi":"10.1215/00127094-2021-0002","DOIUrl":"https://doi.org/10.1215/00127094-2021-0002","url":null,"abstract":"In this paper, we study the radial symmetry properties of stationary and uniformly-rotating solutions of the 2D Euler and gSQG equations, both in the smooth setting and the patch setting. For the 2D Euler equation, we show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial, without any assumptions on the connectedness of the support or the level sets. In the patch setting, for the 2D Euler equation we show that every uniformly-rotating patch $D$ with angular velocity $Omega leq 0$ or $Omegageq frac{1}{2}$ must be radial, where both bounds are sharp. For the gSQG equation we obtain a similar symmetry result for $Omegaleq 0$ or $Omegageq Omega_alpha$ (with the bounds being sharp), under the additional assumption that the patch is simply-connected. These results settle several open questions in [T. Hmidi, J. Evol. Equ., 15(4): 801-816, 2015] and [F. de la Hoz, Z. Hassainia, T. Hmidi, and J. Mateu, Anal. PDE, 9(7):1609-1670, 2016] on uniformly-rotating patches. Along the way, we close a question on overdetermined problems for the fractional Laplacian [R. Choksi, R. Neumayer, and I. Topaloglu, Arxiv preprint arXiv:1810.08304, 2018, Remark 1.4], which may be of independent interest. The main new ideas come from a calculus of variations point of view.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47832168","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}
Pub Date : 2019-07-17DOI: 10.1215/00127094-2021-0025
Alberto Abbondandolo, Jungsoo Kang
We prove that the Floer complex that is associated with a convex Hamiltonian function on $mathbb{R}^{2n}$ is isomorphic to the Morse complex of Clarke's dual action functional that is associated with the Fenchel-dual Hamiltonian. This isomorphism preserves the action filtrations. As a corollary, we obtain that the symplectic capacity from the symplectic homology of a convex domain with smooth boundary coincides with the minimal action of closed characteristics on its boundary.
{"title":"Symplectic homology of convex domains and Clarke’s duality","authors":"Alberto Abbondandolo, Jungsoo Kang","doi":"10.1215/00127094-2021-0025","DOIUrl":"https://doi.org/10.1215/00127094-2021-0025","url":null,"abstract":"We prove that the Floer complex that is associated with a convex Hamiltonian function on $mathbb{R}^{2n}$ is isomorphic to the Morse complex of Clarke's dual action functional that is associated with the Fenchel-dual Hamiltonian. This isomorphism preserves the action filtrations. As a corollary, we obtain that the symplectic capacity from the symplectic homology of a convex domain with smooth boundary coincides with the minimal action of closed characteristics on its boundary.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44153306","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}
Pub Date : 2019-07-14DOI: 10.1215/00127094-2022-0056
Yusheng Luo
In this paper, we study the dynamics of degenerating sequences of rational maps on Riemann sphere $hat{mathbb{C}}$ using $mathbb{R}$-trees. Given a sequence of degenerating rational maps, we give two constructions for limiting dynamics on $mathbb{R}$-trees: one geometric and one algebraic. The geometric construction uses the ultralimit of rescalings of barycentric extensions of rational maps, while the algebraic construction uses the Berkovich space of complexified Robinson's field. We show the two approaches are equivalent. The limiting dynamics on the $mathbb{R}$-tree are analogues to isometric group actions on $mathbb{R}$-trees studied in Kleinian groups and Teichmuller theory. We use the limiting map to classify hyperbolic components of rational maps that admit degeneracies with bounded length spectra (multipliers).
{"title":"Trees, length spectra for rational maps via barycentric extensions, and Berkovich spaces","authors":"Yusheng Luo","doi":"10.1215/00127094-2022-0056","DOIUrl":"https://doi.org/10.1215/00127094-2022-0056","url":null,"abstract":"In this paper, we study the dynamics of degenerating sequences of rational maps on Riemann sphere $hat{mathbb{C}}$ using $mathbb{R}$-trees. Given a sequence of degenerating rational maps, we give two constructions for limiting dynamics on $mathbb{R}$-trees: one geometric and one algebraic. The geometric construction uses the ultralimit of rescalings of barycentric extensions of rational maps, while the algebraic construction uses the Berkovich space of complexified Robinson's field. We show the two approaches are equivalent. The limiting dynamics on the $mathbb{R}$-tree are analogues to isometric group actions on $mathbb{R}$-trees studied in Kleinian groups and Teichmuller theory. We use the limiting map to classify hyperbolic components of rational maps that admit degeneracies with bounded length spectra (multipliers).","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46194217","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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