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On the boundaries of highly connected, almost closed manifolds Acta Math. (IF 3.7) Pub Date : 2023-12-19 Robert Burklund, Jeremy Hahn, Andrew Senger
Building on work of Stolz, we prove for integers $0 \leqslant d \leqslant 3$ and $k \gt 232$ that the boundaries of $(k-1)$-connected, almost closed $(2k+d)$-manifolds also bound parallelizable manifolds. Away from finitely many dimensions, this settles longstanding questions of C.T.C. Wall, determines all Stein fillable homotopy spheres, and proves a conjecture of Galatius and Randal–Williams. Implications
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Every complete Pick space satisfies the column-row property Acta Math. (IF 3.7) Pub Date : 2023-12-19 Michael Hartz
In the theory of complete Pick spaces, the column-row property has appeared in a variety of contexts. We show that it is satisfied by every complete Pick space in the following strong form: each sequence of multipliers that induces a contractive column multiplication operator also induces a contractive row multiplication operator. In combination with known results, this yields a number of consequences
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Correction to “On the geometry of metric measure spaces. I” Acta Math. (IF 3.7) Pub Date : 2023-12-19 Karl-Theodor Sturm
This is a correction to $\href{https://dx.doi.org/10.1007/s11511-006-0002-8}{[11]}$ (Acta Math.), as well as to the follow-up publications $\href{https://doi.org/10.1016/j.jfa.2010.03.024}{[3]}$ and $\href{https://doi.org/10.1016/j.jfa.2011.02.026}{[5]}$ (both in J. Funct. Anal.).
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Khintchine’s theorem and Diophantine approximation on manifolds Acta Math. (IF 3.7) Pub Date : 2023-09-29 Victor Beresnevich, Lei Yang
In this paper we initiate a new approach to studying approximations by rational points to points on smooth submanifolds of $\mathbb{R}^n$. Our main result is a convergence Khintchine type theorem for arbitrary non-degenerate submanifolds of $\mathbb{R}^n$, which resolves a longstanding problem in the theory of Diophantine approximation. Furthermore, we refine this result using Hausdorff $s$-measures
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Sharp well-posedness results of the Benjamin–Ono equation in $H^s (\mathbb{T}, \mathbb{R})$ and qualitative properties of its solutions Acta Math. (IF 3.7) Pub Date : 2023-09-29 Patrick Gérard, Thomas Kappeler, Peter Topalov
$\def\HSTR{H^s (\mathbb{T}, \mathbb{R})}$We prove that the Benjamin–Ono equation on the torus is globally in time well-posed in the Sobolev space $\HSTR$ for any $s \gt -\frac{1}{2}$ and ill-posed for $s \leqslant -\frac{1}{2}$. Hence the critical Sobolev exponent $s_c = -\frac{1}{2}$ of the Benjamin–Ono equation is the threshold for wellposedness on the torus. The obtained solutions are almost periodic
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The extremals of the Alexandrov–Fenchel inequality for convex polytopes Acta Math. (IF 3.7) Pub Date : 2023-09-29 Yair Shenfeld, Ramon van Handel
The Alexandrov–Fenchel inequality, a far-reaching generalization of the classical isoperimetric inequality to arbitrary mixed volumes, lies at the heart of convex geometry. The characterization of its extremal bodies is a long-standing open problem that dates back to Alexandrov’s original 1937 paper. The known extremals already form a very rich family, and even the fundamental conjectures on their
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Pixton’s formula and Abel–Jacobi theory on the Picard stack Acta Math. (IF 3.7) Pub Date : 2023-07-18 Younghan Bae, David Holmes, Rahul Pandharipande, Johannes Schmitt, Rosa Schwarz
Let $A=(a_1,\ldots,a_n)$ be a vector of integers with $d=\sum_{i=1}^n a_i$. By partial resolution of the classical Abel–Jacobi map, we construct a universal twisted double ramification cycle $\mathsf{DR}^{{\sf op}}_{g,A}$ as an operational Chow class on the Picard stack $\mathfrak{Pic}_{g,n,d}$ of $n$-pointed genus-$g$ curves carrying a degree $d$ line bundle. The method of construction follows the
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Nonlinear inviscid damping near monotonic shear flows Acta Math. (IF 3.7) Pub Date : 2023-07-18 Alexandru D. Ionescu, Hao Jia
We prove non-linear asymptotic stability of a large class of monotonic shear flows among solutions of the 2D Euler equations in the channel $\mathbb{T} \times [0, 1]$. More precisely, we consider shear flows $(b(y), 0)$ given by a function $b$ which is Gevrey smooth, strictly increasing, and linear outside a compact subset of the interval $(0, 1)$ (to avoid boundary contributions which are incompatible
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Soliton resolution for the radial critical wave equation in all odd space dimensions Acta Math. (IF 3.7) Pub Date : 2023-03-24 Thomas Duyckaerts, Carlos Kenig, Frank Merle
Consider the energy-critical focusing wave equation in odd space dimension $N \geqslant 3$. The equation has a non-zero radial stationary solution $W$, which is unique up to scaling and sign change. In this paper we prove that any radial, bounded in the energy norm solution of the equation behaves asymptotically as a sum of modulated $W$’s, decoupled by the scaling, and a radiation term. The proof
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Convergence of uniform triangulations under the Cardy embedding Acta Math. (IF 3.7) Pub Date : 2023-03-24 Nina Holden, Xin Sun
We consider an embedding of planar maps into an equilateral triangle $\Delta$ which we call the Cardy embedding. The embedding is a discrete approximation of a conformal map based on percolation observables that are used in Smirnov’s proof of Cardy’s formula. Under the Cardy embedding, the planar map induces a metric and an area measure on $\Delta$ and a boundary measure on $\partial \Delta$. We prove
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The directed landscape Acta Math. (IF 3.7) Pub Date : 2023-02-21 Duncan Dauvergne, Janosch Ortmann, Bálint Virág
The conjectured limit of last passage percolation is a scale-invariant, independent, stationary increment process with respect to metric composition. We prove this for Brownian last passage percolation. We construct the Airy sheet and characterize it in terms of the Airy line ensemble. We also show that last passage geodesics converge to random functions with Hölder-$\frac{2}{3}^-$ continuous paths
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Mirror symmetry for very affine hypersurfaces Acta Math. (IF 3.7) Pub Date : 2023-02-21 Benjamin Gammage, Vivek Shende
We show that the category of coherent sheaves on the toric boundary divisor of a smooth quasi-projective toric DM stack is equivalent to the wrapped Fukaya category of a hypersurface in $(C^\times)^n$. Hypersurfaces with every Newton polytope can be obtained. Our proof has the following ingredients. Using recent results on localization, we may trade wrapped Fukaya categories for microlocal sheaf theory
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Sendov’s conjecture for sufficiently-high-degree polynomials Acta Math. (IF 3.7) Pub Date : 2023-02-21 Terence Tao
Sendov’s conjecture asserts that if a complex polynomial $f$ of degree $n \geqslant 2$ has all of its zeros in closed unit disk $\lbrace z: \vert z \vert \leqslant 1 \rbrace$, then for each such zero $\lambda_0$ there is a zero of the derivative $f^\prime$ in the closed unit disk $\lbrace z: \vert z-\lambda_0 \vert \leqslant 1 \rbrace$. This conjecture is known for $n \lt 9$, but only partial results
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Strominger–Yau–Zaslow conjecture for Calabi–Yau hypersurfaces in the Fermat family Acta Math. (IF 3.7) Pub Date : 2022-07-26 Yang Li
We produce special Lagrangian $T^n$-fibrations on the generic regions of some Calabi–Yau hypersurfaces in the Fermat family $X_s = \{Z_0\dots Z_{n+1}+ e^{-s} (Z_0^{n+2}+ \dots + Z_{n+1}^{n+2}) = 0\}\subset \mathbb{CP}^{n+1}$ near the large complex structure limit $s\to \infty$.
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Foliated corona decompositions Acta Math. (IF 3.7) Pub Date : 2022-07-26 Assaf Naor, Robert Young
We prove that the $L_4$ norm of the vertical perimeter of any measurable subset of the $3$-dimensional Heisenberg group $\mathbb{H}$ is at most a universal constant multiple of the (Heisenberg) perimeter of the subset. We show that this isoperimetric-type inequality is optimal in the sense that there are sets for which it fails to hold with the $L_4$ norm replaced by the $L_q$ norm for any $q \lt 4$
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Ancient low-entropy flows, mean-convex neighborhoods, and uniqueness Acta Math. (IF 3.7) Pub Date : 2022-07-01 Kyeongsu Choi, Robert Haslhofer, Or Hershkovits
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 $\varepsilon = \varepsilon (X) \gt 0$ such that the flow is mean convex in a space-time neighborhood of size $\varepsilon$ around $X$. The major difficulty
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An invariance principle for ergodic scale-free random environments Acta Math. (IF 3.7) Pub Date : 2022-07-01 Ewain Gwynne, Jason Miller, Scott Sheffield
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
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The Fuglede conjecture for convex domains is true in all dimensions Acta Math. (IF 3.7) Pub Date : 2022-07-01 Nir Lev, Máté Matolcsi
[no abstract]
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Generic family displaying robustly a fast growth of the number of periodic points Acta Math. (IF 3.7) Pub Date : 2022-01-10 Pierre Berger
For any $2 \leqslant r \leqslant \infty , n \geqslant 2$, we prove the existence of an open set $U$ of $C^r$‑self‑mappings of any $n$‑manifold so that a generic map $f$ in $U$ displays a fast growth of the number of periodic points: the number of its $n$‑periodic points grows as fast as asked. This complements the works of Martens–de Melo–van Strien, Kaloshin, Bonatti–Díaz–Fisher and Turaev, to give
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Gravitational instantons with faster than quadratic curvature decay. I Acta Math. (IF 3.7) Pub Date : 2022-01-10 Gao Chen, Xiuxiong Chen
In this paper, we study gravitational instantons (i.e., complete hyperkähler $4$‑manifolds with faster than quadratic curvature decay). We prove three main theorems: (1) Any gravitational instanton must have one of the following known ends: ALE, ALF, ALG, and ALH. (2) In the ALG and ALH non-splitting cases, it must be biholomorphic to a compact complex elliptic surface minus a divisor. Thus, we confirm
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Planar orthogonal polynomials and boundary universality in the random normal matrix model Acta Math. (IF 3.7) Pub Date : 2022-01-10 Håkan Hedenmalm, Aron Wennman
We obtain an asymptotic expansion of planar orthogonal polynomials with respect to exponentially varying weights, relevant for random matrix theory. As a consequence we show that the density of states in the random normal matrix model has universal error function transition across smooth interfaces for the limiting eigenvalue density. The key ingredient in the proof of the asymptotic expansion is the
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The Mirković–Vilonen basis and Duistermaat–Heckman measures Acta Math. (IF 3.7) Pub Date : 2021-11-24 Pierre Baumann, Joel Kamnitzer, Allen Knutson
Using the geometric Satake correspondence, the Mirković–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 $\mathfrak{g}$. We compute this action in terms of intersection multiplicities in the affine Grassmannian
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The number of closed ideals in $L(L_p)$ Acta Math. (IF 3.7) Pub Date : 2021-11-24 William B. Johnson, Gideon Schechtman
We show that there are $2^{2^{\aleph_0}}$ different closed ideals in the Banach algebra $L(L_p(0,1)), 1 \lt p \neq 2 \lt \infty$. This solves a problem in A. Pietsch’s 1978 book “Operator Ideals”. The proof is quite different from other methods of producing closed ideals in the space of bounded operators on a Banach space; in particular, the ideals are not contained in the strictly singular operators
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The KPZ fixed point Acta Math. (IF 3.7) Pub Date : 2021-11-24 Konstantin Matetski, Jeremy Quastel, Daniel Remenik
An explicit Fredholm determinant formula is derived for the multipoint distribution of the height function of the totally asymmetric simple exclusion process (TASEP) with arbitrary right-finite initial condition. The method is by solving the biorthogonal ensemble/non-intersecting path representation found by [54], [10]. The resulting kernel involves transition probabilities of a random walk forced
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Quotients of higher-dimensional Cremona groups Acta Math. (IF 3.7) Pub Date : 2021-06-01 Jérémy Blanc, Stéphane Lamy, Susanna Zimmermann
We study large groups of birational transformations $\operatorname{Bir}(X)$, where $X$ is a variety of dimension at least $3$, defined over $\mathbf{C}$ or a subfield of $\mathbf{C}$. Two prominent cases are when $X$ is the projective space $\mathbb{P}^n$, in which case $\operatorname{Bir}(X)$ is the Cremona group of rank $n$, or when $X \subset \mathbb{P}^{n+1}$ is a smooth cubic hypersurface. In
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The special fiber of the motivic deformation of the stable homotopy category is algebraic Acta Math. (IF 3.7) Pub Date : 2021-06-01 Bogdan Gheorghe, Guozhen Wang, Zhouli Xu
For each prime $p$, we define a $t$‑structure on the category $\,\widehat{\!S^{0,0}}/\tau\text{-}\mathbf{Mod}_{\mathrm{harm}}^b$ of harmonic $\mathbb{C}$-motivic left-module spectra over $\,\widehat{\!S^{0,0}}/\tau$, whose MGL-homology has bounded Chow–Novikov degree, such that its heart is equivalent to the abelian category of $p$‑completed $\mathrm{BP}_*\mathrm{BP}$-comodules that are concentrated
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The orbit method and analysis of automorphic forms Acta Math. (IF 3.7) Pub Date : 2021-03-30 Paul D. Nelson, Akshay Venkatesh
We develop the orbit method in a quantitative form, along the lines of microlocal analysis, and apply it to the analytic theory of automorphic forms. Our main global application is an asymptotic formula for averages of Gan–Gross–Prasad periods in arbitrary rank. The automorphic form on the larger group is held fixed, while that on the smaller group varies over a family of size roughly the fourth root
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A transcendental dynamical degree Acta Math. (IF 3.7) Pub Date : 2020-12-01 Jason P. Bell, Jeffrey Diller, Mattias Jonsson
We give an example of a dominant rational self-map of the projective plane whose dynamical degree is a transcendental number.
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Torsion points, Pell’s equation, and integration in elementary terms Acta Math. (IF 3.7) Pub Date : 2020-12-01 David Masser, Umberto Zannier
The main results of this paper involve general algebraic differentials $\omega$ on a general pencil of algebraic curves. We show how to determine whether $\omega$ is integrable in elementary terms for infinitely many members of the pencil. In particular, this corrects an assertion of James Davenport from 1981 and provides the first proof, even in rather strengthened form. We also indicate analogies
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On Thurston’s Euler class-one conjecture Acta Math. (IF 3.7) Pub Date : 2020-12-01 Mehdi Yazdi
In 1976, Thurston proved that taut foliations on closed hyperbolic $3$-manifolds have Euler class of norm at most $1$, and conjectured that conversely, any integral second cohomology class with norm equal to one is the Euler class of a taut foliation. This is the first from a series of two papers that together give a negative answer to Thurston’s conjecture. Here counter-examples have been constructed
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The fully marked surface theorem Acta Math. (IF 3.7) Pub Date : 2020-12-01 David Gabai, Mehdi Yazdi
In his seminal 1976 paper, Bill Thurston observed that a closed leaf $S$ of a codimension‑$1$ foliation on a compact $3$‑manifold has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on $[S]$, the homology class represented by $S$. The main result of this paper is a converse for taut foliations: if the Euler class of a taut foliation $\mathcal{F}$ evaluated on $[S]$
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Non-collision singularities in a planar 4-body problem Acta Math. (IF 3.7) Pub Date : 2020-06-23 Jinxin Xue
In this paper, we show that there is a Cantor set of initial conditions in the planar $4$‑body problem such that all four bodies escape to infinity in a finite time, avoiding collisions. This proves the Painlevé conjecture for the $4$‑body case, and thus settles the last open case of the conjecture.
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Rational homotopy theory of automorphisms of manifolds Acta Math. (IF 3.7) Pub Date : 2020-01-01 Alexander Berglund, Ib Madsen
We study the rational homotopy types of classifying spaces of automorphism groups of smooth simply connected manifolds of dimension at least five. We give dg Lie algebra models for the homotopy automorphisms and the block diffeomorphisms of such manifolds. Moreover, we use these models to calculate the rational cohomology of the classifying spaces of the homotopy automorphisms and block diffeomorphisms
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Purely unrectifiable metric spaces and perturbations of Lipschitz functions Acta Math. (IF 3.7) Pub Date : 2020-01-01 David Bate
We characterise purely $n$-unrectifiable subsets $S$ of a complete metric space $X$ with finite Hausdorff $n$-measure by studying arbitrarily small perturbations of elements of the set of all bounded 1-Lipschitz functions $f\colon X \to \mathbb R^m$ with respect to the supremum norm. In one such characterisation it is shown that, if $S$ has positive lower density almost everywhere, then the set of
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Ancient solutions to the Ricci flow in dimension $3$ Acta Math. (IF 3.7) Pub Date : 2020-01-01 Simon Brendle
It is known from work of Perelman that any finite-time singularity of the Ricci flow on a compact three-manifold is modeled on an ancient $\kappa$-solution. We prove that the every noncompact ancient $\kappa$-solution in dimension $3$ is isometric to either the shrinking cylinders (or a quotient thereof), or the Bryant soliton.
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Rigid connections and $F$-isocrystals Acta Math. (IF 3.7) Pub Date : 2020-01-01 Hélène Esnault, Michael Groechenig
An irreducible integrable connection $(E,\nabla)$ on a smooth projective complex variety $X$ is called rigid if it gives rise to an isolated point of the corresponding moduli space $\mathcal{M}_{dR}(X)$. According to Simpson's motivicity conjecture, irreducible rigid flat connections are of geometric origin, that is, arise as subquotients of a Gaus-Manin connection of a family of smooth projective
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Small cancellation labellings of some infinite graphs and applications Acta Math. (IF 3.7) Pub Date : 2020-01-01 Damian Osajda
We construct small cancellation labellings for some infinite sequences of finite graphs of bounded degree. We use them to define infinite graphical small cancellation presentations of groups. This technique allows us to provide examples of groups with exotic properties: - We construct the first examples of finitely generated coarsely non-amenable groups (that is, groups without Guoliang Yu's Property
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Stable solutions to semilinear elliptic equations are smooth up to dimension $9$ Acta Math. (IF 3.7) Pub Date : 2020-01-01 Xavier Cabré, Alessio Figalli, Xavier Ros-Oton, Joaquim Serra
In this paper we prove the following long-standing conjecture: stable solutions to semilinear elliptic equations are bounded (and thus smooth) in dimension $n \leq 9$. This result, that was only known to be true for $n\leq4$, is optimal: $\log(1/|x|^2)$ is a $W^{1,2}$ singular stable solution for $n\geq10$. The proof of this conjecture is a consequence of a new universal estimate: we prove that, in
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Correction to “On topological cyclic homology” Acta Math. (IF 3.7) Pub Date : 2019-01-01 Thomas Nikolaus,Peter Scholze
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Sharp estimates for oscillatory integral operators via polynomial partitioning Acta Math. (IF 3.7) Pub Date : 2019-01-01 Larry Guth, Jonathan Hickman, Marina Iliopoulou
The sharp range of $L^p$-estimates for the class of H\"ormander-type oscillatory integral operators is established in all dimensions under a positive-definite assumption on the phase. This is achieved by generalising a recent approach of the first author for studying the Fourier extension operator, which utilises polynomial partitioning arguments.
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Counterexamples to Strassen’s direct sum conjecture Acta Math. (IF 3.7) Pub Date : 2019-01-01 Yaroslav Shitov
The multiplicative complexity of systems of bilinear forms (and, in particular, the famous question of fast matrix multiplication) is an important area of research in modern theory of computation. One of the foundational papers on the topic is Strassen’s work [20], which contains an O(n 7/ ln ) algorithm for the multiplication of two n×n matrices. In his subsequent paper [21] published in 1973, Strassen
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$\hat{G}$-local systems on smooth projective curves are potentially automorphic Acta Math. (IF 3.7) Pub Date : 2019-01-01 Gebhard Böckle, Michael Harris, Chandrashekhar Khare, Jack A. Thorne
Let $X$ be a smooth, projective, geometrically connected curve over a finite field $\mathbb{F}_q$, and let $G$ be a split semisimple algebraic group over $\mathbb{F}_q$. Its dual group $\widehat{G}$ is a split reductive group over $\mathbb{Z}$. Conjecturally, any $l$-adic $\widehat{G}$-local system on $X$ (equivalently, any conjugacy class of continuous homomorphisms $\pi_1(X) \to \widehat{G}(\ove
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Irreducibility of random polynomials of large degree Acta Math. (IF 3.7) Pub Date : 2019-01-01 Emmanuel Breuillard, Péter P. Varjú
We consider random polynomials with independent identically distributed coefficients with a fixed law. Assuming the Riemann hypothesis for Dedekind zeta functions, we prove that such polynomials are irreducible and their Galois groups contain the alternating group with high probability as the degree goes to infinity. This settles a conjecture of Odlyzko and Poonen conditionally on RH for Dedekind zeta
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Strong property (T) for higher rank lattices Acta Math. (IF 3.7) Pub Date : 2019-01-01 Mikael de la Salle
We prove that every lattice in a product of higher rank simple Lie groups or higher rank simple algebraic groups over local fields has Vincent Lafforgue’s strong property (T). Over non-archimedean local fields, we also prove that they have strong Banach proerty (T) with respect to all Banach spaces with nontrivial type, whereas in general we obtain such a result with additional hypotheses on the Banach
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Bounds on the topology and index of minimal surfaces Acta Math. (IF 3.7) Pub Date : 2019-01-01 William H. Meeks, Joaquín Pérez, Antonio Ros
We prove that for every nonnegative integer $g$, there exists a bound on the number of ends of a complete, embedded minimal surface $M$ in $\mathbb{R}^3$ of genus $g$ and finite topology. This bound on the finite number of ends when $M$ has at least two ends implies that $M$ has finite stability index which is bounded by a constant that only depends on its genus.
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Maximization of the second non-trivial Neumann eigenvalue Acta Math. (IF 3.7) Pub Date : 2019-01-01 Dorin Bucur, Antoine Henrot
In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of R N with prescribed measure m attains its maximum on the union of two disjoint balls of measure m 2. As a consequence, the P{\'o}lya conjecture for the Neumann eigenvalues holds for the second eigenvalue and for arbitrary domains. We moreover prove that a relaxed form of the same inequality
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Bogoliubov theory in the Gross–Pitaevskii limit Acta Math. (IF 3.7) Pub Date : 2019-01-01 Chiara Boccato, Christian Brennecke, Serena Cenatiempo, Benjamin Schlein
We consider Bose gases consisting of $N$ particles trapped in a box with volume one and interacting through a repulsive potential with scattering length of the order $N^{-1}$(Gross-Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum, up to errors vanishing as $N \to \infty$. Our results confirm Bogoliubov's predictions.
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The linear stability of the Schwarzschild solution to gravitational perturbations Acta Math. (IF 3.7) Pub Date : 2019-01-01 Mihalis Dafermos, Gustav Holzegel, Igor Rodnianski
We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to a linearised Kerr metric. We express the equations in a suitable double null gauge
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Local homology and Serre categories Acta Math. (IF 3.7) Pub Date : 2019-01-01 Zahra Barqsouz, Seadat Ollah Faramarzi
We show some results about local homology modules when they are in a Serre subcategory of the category of R-modules. For an ideal a of R, we also define the concept of the condition C on a Serre category, which seems dual to the condition Ca in Melkersson [1]. As a main result we show that for an Artinian R-module M and any Serre subcategory S of the category of R-modules and a non-negative integer
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Measure Estimates, Harnack Inequalities and Ricci Lower Bound Acta Math. (IF 3.7) Pub Date : 2018-11-01 Yu Wang, Xiangwen Zhang
On a Riemannian metric-measure space, we establish an Alexandrov-Bakelman-Pucci type measure estimate connecting Bakry-\'Emery Ricci curvature lower bound, modified Laplacian and the measure of certain special sets. We apply this estimate to prove Harnack inequalities for the modified Laplacian operator and fully non-linear operators. These inequalities seem not available in the literature; And our
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Movable intersection and bigness criterion Acta Math. (IF 3.7) Pub Date : 2018-11-01 Jian Xiao
In this note, we give a Morse-type bigness criterion for the difference of two pseudo-effective $(1,1)$-classes by using movable intersections. And with this result we give a Morse-type bigness criterion for the difference of two movable $(n-1,n-1)$-classes.
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Systems of holomorphic multivalued projections on complex manifolds Acta Math. (IF 3.7) Pub Date : 2018-11-01 Kamil Drzyzga
Let M be a submanifold of a connected Stein manifold X. We construct a global system of holomorphic multivalued projections X −→ M . In particular, for every locally bounded family F ⊂ O(M) we get a continuous extension operator F −→ O(X).
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On the structure of band edges of $2$-dimensional periodic elliptic operators Acta Math. (IF 3.7) Pub Date : 2018-01-01 Nikolay Filonov,Ilya Kachkovskiy
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On topological cyclic homology Acta Math. (IF 3.7) Pub Date : 2018-01-01 Thomas Nikolaus, Peter Scholze
Topological cyclic homology is a refinement of Connes--Tsygan's cyclic homology which was introduced by Bokstedt--Hsiang--Madsen in 1993 as an approximation to algebraic $K$-theory. There is a trace map from algebraic $K$-theory to topological cyclic homology, and a theorem of Dundas--Goodwillie--McCarthy asserts that this induces an equivalence of relative theories for nilpotent immersions, which
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Examples of finite free complexes of small rank and small homology Acta Math. (IF 3.7) Pub Date : 2018-01-01 Srikanth B. Iyengar, Mark E. Walker
This work concerns finite free complexes over commutative noetherian rings, in particular over group algebras of elementary abelian groups. The main contribution is the construction of complexes such that the total rank of their underlying free modules, or the total length of their homology, is less than predicted by various conjectures in the theory of transformation groups and in local algebra.
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Restriction estimates using polynomial partitioning II Acta Math. (IF 3.7) Pub Date : 2018-01-01 Larry Guth
We improve the estimates in the restriction problem in dimension $n \ge 4$. To do so, we establish a weak version of a $k$-linear restriction estimate for any $k$. The exponents in this weak $k$-linear estimate are sharp for all $k$ and $n$.
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Character bounds for finite groups of Lie type Acta Math. (IF 3.7) Pub Date : 2018-01-01 Roman Bezrukavnikov, Martin W. Liebeck, Aner Shalev, Pham Huu Tiep
We establish new bounds on character values and character ratios for finite groups $G$ of Lie type, which are considerably stronger than previously known bounds, and which are best possible in many cases. These bounds have the form $|\chi(g)| \le \chi(1)^{\alpha_g}$, and give rise to a variety of applications, for example to covering numbers and mixing times of random walks on such groups. In particular
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Stable rationality of quadric surface bundles over surfaces Acta Math. (IF 3.7) Pub Date : 2018-01-01 Brendan Hassett, Alena Pirutka, Yuri Tschinkel
We study rationality properties of quadric surface bundles over the projective plane. We exhibit families of smooth projective complex fourfolds of this type over connected bases, containing both rational and non-rational fibers.
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Semiclassical measures on hyperbolic surfaces have full support Acta Math. (IF 3.7) Pub Date : 2018-01-01 Semyon Dyatlov, Long Jin
We show that each limiting semiclassical measure obtained from a sequence of eigenfunctions of the Laplacian on a compact hyperbolic surface is supported on the entire cosphere bundle. The key new ingredient for the proof is the fractal uncertainty principle, first formulated in [arXiv:1504.06589] and proved for porous sets in [arXiv:1612.09040].
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Algebraic actions of discrete groups: the $p$-adic method Acta Math. (IF 3.7) Pub Date : 2018-01-01 Serge Cantat, Junyi Xie
We study groups of automorphisms and birational transformations of quasi-projective varieties. Two methods are combined; the first one is based on p-adic analysis, the second makes use of isoperimetric inequalities and LangWeil estimates. For instance, we show that if SL n(Z) acts faithfully on a complex quasi-projective variety X by birational transformations, then dim(X) ≥ n−1 and X is rational if