Acta Mathematica最新文献_第3页

The Mirković–Vilonen basis and Duistermaat–Heckman measures Mirković-Vilonen基础和Duistermaat-Heckman测量

IF 3.7 1区数学

Acta Mathematica Pub Date : 2019-05-21 DOI: 10.4310/acta.2021.v227.n1.a1

Pierre Baumann, J. Kamnitzer, A. Knutson

{"title":"The Mirković–Vilonen basis and Duistermaat–Heckman measures","authors":"Pierre Baumann, J. Kamnitzer, A. Knutson","doi":"10.4310/acta.2021.v227.n1.a1","DOIUrl":"https://doi.org/10.4310/acta.2021.v227.n1.a1","url":null,"abstract":"Using the geometric Satake correspondence, the Mirkovic-Vilonen cycles in the affine Grasssmannian give bases for representations of a semisimple group G . We prove that these bases are \"perfect\", i.e. compatible with the action of the Chevelley generators of the positive half of the Lie algebra g. We compute this action in terms of intersection multiplicities in the affine Grassmannian. We prove that these bases stitch together to a basis for the algebra C[N] of regular functions on the unipotent subgroup. We compute the multiplication in this MV basis using intersection multiplicities in the Beilinson-Drinfeld Grassmannian, thus proving a conjecture of Anderson. In the third part of the paper, we define a map from C[N] to a convolution algebra of measures on the dual of the Cartan subalgebra of g. We characterize this map using the universal centralizer space of G. We prove that the measure associated to an MV basis element equals the Duistermaat-Heckman measure of the corresponding MV cycle. This leads to a proof of a conjecture of Muthiah. Finally, we use the map to measures to compare the MV basis and Lusztig's dual semicanonical basis. We formulate conjectures relating the algebraic invariants of preprojective algebra modules (which underlie the dual semicanonical basis) and geometric invariants of MV cycles. In the appendix, we use these ideas to prove that the MV basis and the dual semicanonical basis do not coincide in SL_6.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2019-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45929406","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}

引用次数: 19

The Fuglede conjecture for convex domains is true in all dimensions 凸域的Fuglede猜想在所有维度上都是正确的

IF 3.7 1区数学

Acta Mathematica Pub Date : 2019-04-28 DOI: 10.4310/ACTA.2022.v228.n2.a3

Nir Lev, M. Matolcsi

{"title":"The Fuglede conjecture for convex domains is true in all dimensions","authors":"Nir Lev, M. Matolcsi","doi":"10.4310/ACTA.2022.v228.n2.a3","DOIUrl":"https://doi.org/10.4310/ACTA.2022.v228.n2.a3","url":null,"abstract":"A set $Omega subset mathbb{R}^d$ is said to be spectral if the space $L^2(Omega)$ has an orthogonal basis of exponential functions. A conjecture due to Fuglede (1974) stated that $Omega$ is a spectral set if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it has long been known that for a convex body $Omega subset mathbb{R}^d$ the \"tiling implies spectral\" part of the conjecture is in fact true. \u0000To the contrary, the \"spectral implies tiling\" direction of the conjecture for convex bodies was proved only in $mathbb{R}^2$, and also in $mathbb{R}^3$ under the a priori assumption that $Omega$ is a convex polytope. In higher dimensions, this direction of the conjecture remained completely open (even in the case when $Omega$ is a polytope) and could not be treated using the previously developed techniques. \u0000In this paper we fully settle Fuglede's conjecture for convex bodies affirmatively in all dimensions, i.e. we prove that if a convex body $Omega subset mathbb{R}^d$ is a spectral set then it can tile the space by translations. To prove this we introduce a new technique, involving a construction from crystallographic diffraction theory, which allows us to establish a geometric \"weak tiling\" condition necessary for a set $Omega subset mathbb{R}^d$ to be spectral.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2019-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47687210","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}

引用次数: 44

Correction to “On topological cyclic homology” 对“On拓扑循环同调”的更正

IF 3.7 1区数学

Acta Mathematica Pub Date : 2019-03-01 DOI: 10.4310/ACTA.2019.V222.N1.A2

T. Nikolaus, P. Scholze

引用次数: 2

Ancient solutions to the Ricci flow in dimension $3$ 利玛窦流的古代解法（3美元）$

IF 3.7 1区数学

Acta Mathematica Pub Date : 2018-11-06 DOI: 10.4310/acta.2020.v225.n1.a1

S. Brendle

引用次数: 60

Systems of holomorphic multivalued projections on complex manifolds 复流形上的全纯多值投影系统

IF 3.7 1区数学

Acta Mathematica Pub Date : 2018-11-01 DOI: 10.4467/20843828am.18.001.9717

Kamil Drzyzga

引用次数: 0

Irreducibility of random polynomials of large degree 大次随机多项式的不可约性

IF 3.7 1区数学

Acta Mathematica Pub Date : 2018-10-31 DOI: 10.4310/ACTA.2019.v223.n2.a1

E. Breuillard, P. P. Varj'u

引用次数: 27

Ancient low-entropy flows, mean-convex neighborhoods, and uniqueness 古代低熵流、均值凸邻域和唯一性

IF 3.7 1区数学

Acta Mathematica Pub Date : 2018-10-19 DOI: 10.4310/acta.2022.v228.n2.a1

K. Choi, Robert Haslhofer, Or Hershkovits

{"title":"Ancient low-entropy flows, mean-convex neighborhoods, and uniqueness","authors":"K. Choi, Robert Haslhofer, Or Hershkovits","doi":"10.4310/acta.2022.v228.n2.a1","DOIUrl":"https://doi.org/10.4310/acta.2022.v228.n2.a1","url":null,"abstract":"In this article, we prove the mean convex neighborhood conjecture for the mean curvature flow of surfaces in $mathbb{R}^3$. Namely, if the flow has a spherical or cylindrical singularity at a space-time point $X=(x,t)$, then there exists a positive $varepsilon=varepsilon(X)>0$ such that the flow is mean convex in a space-time neighborhood of size $varepsilon$ around $X$. The major difficulty is to promote the infinitesimal information about the singularity to a conclusion of macroscopic size. In fact, we prove a more general classification result for all ancient low entropy flows that arise as potential limit flows near $X$. Namely, we prove that any ancient, unit-regular, cyclic, integral Brakke flow in $mathbb{R}^3$ with entropy at most $sqrt{2pi/e}+delta$ is either a flat plane, a round shrinking sphere, a round shrinking cylinder, a translating bowl soliton, or an ancient oval. As an application, we prove the uniqueness conjecture for mean curvature flow through spherical or cylindrical singularities. In particular, assuming Ilmanen's multiplicity one conjecture, we conclude that for embedded two-spheres the mean curvature flow through singularities is well-posed.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2018-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47680734","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}

引用次数: 53

The special fiber of the motivic deformation of the stable homotopy category is algebraic 稳定同伦范畴运动变形的特殊纤维是代数的

IF 3.7 1区数学

Acta Mathematica Pub Date : 2018-09-25 DOI: 10.4310/acta.2021.v226.n2.a2

Bogdan Gheorghe, Guozhen Wang, Zhouli Xu

引用次数: 32

An invariance principle for ergodic scale-free random environments 遍历无标度随机环境的一个不变性原理

IF 3.7 1区数学

Acta Mathematica Pub Date : 2018-07-19 DOI: 10.4310/acta.2022.v228.n2.a2

Ewain Gwynne, Jason Miller, S. Sheffield

{"title":"An invariance principle for ergodic scale-free random environments","authors":"Ewain Gwynne, Jason Miller, S. Sheffield","doi":"10.4310/acta.2022.v228.n2.a2","DOIUrl":"https://doi.org/10.4310/acta.2022.v228.n2.a2","url":null,"abstract":"There are many classical random walk in random environment results that apply to ergodic random planar environments. We extend some of these results to random environments in which the length scale varies from place to place, so that the law of the environment is in a certain sense only translation invariant modulo scaling. For our purposes, an \"environment\" consists of an infinite random planar map embedded in $mathbb C$, each of whose edges comes with a positive real conductance. Our main result is that under modest constraints (translation invariance modulo scaling together with the finiteness of a type of specific energy) a random walk in this kind of environment converges to Brownian motion modulo time parameterization in the quenched sense. \u0000Environments of the type considered here arise naturally in the study of random planar maps and Liouville quantum gravity. In fact, the results of this paper are used in separate works to prove that certain random planar maps (embedded in the plane via the so-called Tutte embedding) have scaling limits given by SLE-decorated Liouville quantum gravity, and also to provide a more explicit construction of Brownian motion on the Brownian map. However, the results of this paper are much more general and can be read independently of that program. \u0000One general consequence of our main result is that if a translation invariant (modulo scaling) random embedded planar map and its dual have finite energy per area, then they are close on large scales to a minimal energy embedding (the harmonic embedding). To establish Brownian motion convergence for an infinite energy embedding, it suffices to show that one can perturb it to make the energy finite.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2018-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42123975","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}

引用次数: 10

Convergence and divergence of formal CR mappings 形式CR映射的收敛性和发散性

IF 3.7 1区数学

Acta Mathematica Pub Date : 2018-06-01 DOI: 10.4310/ACTA.2018.V220.N2.A5

B. Lamel, N. Mir

{"title":"Convergence and divergence of formal CR mappings","authors":"B. Lamel, N. Mir","doi":"10.4310/ACTA.2018.V220.N2.A5","DOIUrl":"https://doi.org/10.4310/ACTA.2018.V220.N2.A5","url":null,"abstract":"A formal holomorphic map H: (M,p)!M ′ from a germ of a real-analytic submanifold M⊂C at p∈M into a real-analytic subset M ′⊂CN ′ is an N ′-tuple of formal holomorphic power series H=(H1, ...,HN ′) satisfying H(p)∈M ′ with the property that, for any germ of a real-analytic function δ(w, w) at H(p)∈C ′ which vanishes on M ′, the formal power series δ(H(z), H(z)) vanishes on M . There is an abundance of examples showing that formal maps may diverge: After the trivial example of self-maps of a complex submanifold, possibly the simplest non-trivial example is given by the formal maps of (R, 0) into R which are just given by the formal power series in z∈C with real coefficients, that is, by elements of R[[z]]. It is a surprising fact at first that, for formal mappings between real submanifolds in complex spaces, if one assumes that the trivial examples above are excluded in a suitable sense, the situation is fundamentally different. The first result of this kind was encountered by Chern and Moser in [CM], where—as a byproduct of the convergence of their normal form—it follows that every formal holomorphic invertible map between Levinon-degenerate hypersurfaces in C necessarily converges. The convergence problem, that is, deciding whether formal maps, as described above, are in fact convergent, has been studied intensively in different contexts, both for CR manifolds and for manifolds with CR singularities, for which we refer the reader to the papers [Rot], [MMZ2], [LM1], [HY1], [HY2], [HY3], [Sto], [GS] and the references therein. Solutions to the convergence problem have important applications, for example, to the biholomorphic equivalence","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"220 1","pages":"367-406"},"PeriodicalIF":3.7,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42862607","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}

引用次数: 9