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European Mathematical Society Publishing House
2024-03-29 12:56:30
86
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=21&update_since=2024-03-29
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
21
2019
1
Double dimers, conformal loop ensembles and isomonodromic deformations
Julien
Dubédat
Columbia University, New York, USA
Dimers, Schramm–Loewner Evolutions, monodromy
The double-dimer model consists in superimposing two independent, identically distributed perfect matchings on a planar graph, which produces an ensemble of non-intersecting loops. In [20], Kenyon established conformal invariance in the small mesh limit by considering topological observables of the model parameterized by SL$_2 (\mathbb C)$ representations of the fundamental group of the punctured domain. The scaling limit is conjectured to be CLE$_4$, the Conformal Loop Ensemble at $\kappa = 4$ [36]. In support of this conjecture, we prove that a large subclass of these topological correlators converge to their putative CLE$_4$ limit. Both the small mesh limit of the double-dimer correlators and the corresponding CLE$_4$ correlators are identified in terms of the -functions introduced by Jimbo, Miwa and Ueno [14] in the context of isomonodromic deformations.
Probability theory and stochastic processes
Ordinary differential equations
1
54
10.4171/JEMS/830
http://www.ems-ph.org/doi/10.4171/JEMS/830
9
21
2018
Tait colorings, and an instanton homology for webs and foams
Peter
Kronheimer
Harvard University, Cambridge, USA
Tomasz
Mrowka
Massachusetts Institute of Technology, Cambridge, USA
Tait coloring, four-color theorem, instanton, Floer homology, web, foam, 3-manifold
We use $SO(3)$ gauge theory to define a functor from a category of unoriented webs and foams to the category of finite-dimensional vector spaces over the field of two elements. We prove a non-vanishing theorem for this $SO(3)$ instanton homology of webs, using Gabai’s sutured manifold theory. It is hoped that the non-vanishing theorem may support a program to provide a new proof of the four-color theorem.
Manifolds and cell complexes
Combinatorics
55
119
10.4171/JEMS/831
http://www.ems-ph.org/doi/10.4171/JEMS/831
9
18
2018
Mean-field limit for collective behavior models with sharp sensitivity regions
José
Carrillo
Imperial College London, UK
Young-Pil
Choi
Inha University, Incheon, Korea
Maxime
Hauray
Université d’Aix-Marseille, Marseille, France
Samir
Salem
Université Paris-Dauphine, France
Mean-field limits, sharp vision regions, weak-strong stability
We rigorously show the mean-field limit for a large class of swarming individual based models with local sharp sensitivity regions. For instance, these models include nonlocal repulsive-attractive forces locally averaged over sharp vision cones and Cucker–Smale interactions with discontinuous communication weights.We define global-in-time solutions through a differential inclusion system corresponding to the particle descriptions. We estimate the error between the solutions to the differential inclusion system and weak solutions to the expected limiting kinetic equation by employing tools from optimal transport theory. Quantitative bounds on the expansion of the 1-Wasserstein distance along flows based on a weak-strong stability estimate are obtained. We also provide various examples of realistic sensitivity sets satisfying the assumptions of our main results.
Biology and other natural sciences
Mechanics of deformable solids
Fluid mechanics
121
161
10.4171/JEMS/832
http://www.ems-ph.org/doi/10.4171/JEMS/832
9
21
2018
Minkowski valuations on lattice polytopes
Károly
Böröczky
Hungarian Academy of Sciences, Budapest, Hungary
Monika
Ludwig
Technische Universität Wien, Austria
Minkowski valuation, lattice polytope, Betke–Kneser theorem
A complete classification is established of Minkowski valuations on lattice polytopes that intertwine the special linear group over the integers and are translation invariant. In the contravariant case, the only such valuations are multiples of projection bodies. In the equivariant case, the only such valuations are generalized difference bodies combined with multiples of the newly defined discrete Steiner point.
Convex and discrete geometry
163
197
10.4171/JEMS/833
http://www.ems-ph.org/doi/10.4171/JEMS/833
9
21
2018
Symbolic dynamics for three-dimensional flows with positive topological entropy
Yuri
Lima
Universidade Federal do Ceará, Fortaleza, Brazil
Omri
Sarig
Weizmann Institute of Science, Rehovot, Israel
Markov partitions, symbolic dynamics, geodesic flows, Pesin theory
We construct symbolic dynamics on sets of full measure (with respect to an ergodic measure of positive entropy) for $C^{1+\epsilon}$ flows on closed smooth three-dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a closed $C^\infty$ surface has at least const $\times e^{hT}/T$ simple closed orbits of period less than $T$ , whenever the topological entropy $h$ is positive—and without further assumptions on the curvature.
Dynamical systems and ergodic theory
Differential geometry
199
256
10.4171/JEMS/834
http://www.ems-ph.org/doi/10.4171/JEMS/834
9
25
2018
A note on positive-definite, symplectic four-manifolds
Jennifer
Hom
Georgia Institute of Technology, Atlanta, USA
Tye
Lidman
North Carolina State University, Raleigh, USA
Heegaard Floer homology, four-manifolds
We prove that a positive-definite smooth four-manifold with $b_2^+\geq 2$ and having either no 1-handles or no 3-handles cannot admit a symplectic structure.
Manifolds and cell complexes
257
270
10.4171/JEMS/835
http://www.ems-ph.org/doi/10.4171/JEMS/835
10
1
2018
Every Borel automorphism without finite invariant measures admits a two-set generator
Michael
Hochman
The Hebrew University of Jerusalem, Israel
Borel dynamics, entropy, generators, ergodic theory
We show that if an automorphism of a standard Borel space does not admit finite invariant measures, then it has a two-set generator. This implies that if the entropies of invariant probability measures of a Borel system are all less than log $k$, then the system admits a $k$-set generator, and that a wide class of hyperbolic-like systems are classified completely at the Borel level by entropy and periodic points counts.
Dynamical systems and ergodic theory
271
317
10.4171/JEMS/836
http://www.ems-ph.org/doi/10.4171/JEMS/836
10
1
2018
2
Gradient stability for the Sobolev inequality: the case $p\geq 2$
Alessio
Figalli
ETH Zürich, Switzerland
Robin
Neumayer
Northwestern University, Evanston, USA
Sobolev inequality, stability, quantitative inequalities
We show a strong form of the quantitative Sobolev inequality in $\mathbb{R}^n$ for $p\geq 2$, where the deficit of a function $u\in \dot W^{1,p} $ controls $\| \nabla u -\nabla v\|_{L^p}$ for an extremal function $v$ in the Sobolev inequality.
Functional analysis
Real functions
319
354
10.4171/JEMS/837
http://www.ems-ph.org/doi/10.4171/JEMS/837
10
3
2018
Counting generic measures for a subshift of linear growth
Van
Cyr
Bucknell University, Lewisburg, USA
Bryna
Kra
Northwestern University, Evanston, USA
Subshift, automorphism, block complexity, interval exchange transformation
In 1984 Boshernitzan proved an upper bound on the number of ergodic measures for a minimal subshift of linear block growth and asked if it could be lowered without further assumptions on the shift. We answer this question, showing that Boshernitzan’s bound is sharp. We further prove that the same bound holds for the, a priori, larger set of nonatomic generic measures, and that this bound remains valid even if one drops the assumption of minimality. Applying these results to interval exchange transformations, we give an upper bound on the number of nonatomic generic measures of a minimal IET, answering a question recently posed by Chaika and Masur.
Dynamical systems and ergodic theory
Computer science
355
380
10.4171/JEMS/838
http://www.ems-ph.org/doi/10.4171/JEMS/838
10
12
2018
Barycentric straightening and bounded cohomology
Jean-François
Lafont
Ohio State University, Columbus, USA
Shi
Wang
Indiana University, Bloomington, USA
Barycenter method, bounded cohomology, semisimple Lie group, Dupont’s problem.
We study the barycentric straightening of simplices in higher rank irreducible symmetric spaces of non-compact type.We show that, for an $n$-dimensional symmetric space of rank $r \geq 2$ (excluding SL(3,$ \mathbb R$)/SO(3) and SL(4, $\mathbb R$)/SO(4)), the $p$-Jacobian has uniformly bounded norm, provided $p \geq n–r+2$. As a consequence, for the corresponding non-compact, connected, semisimple real Lie group $G$, in degrees $p \geq n-r+2$, every degree $p$ cohomology class has a bounded representative. This answers Dupont’s problem in small codimension. We also give examples of symmetric spaces where the barycentrically straightened simplices of dimension $n–r$ have unbounded volume, showing that the range in which we obtain boundedness of the $p$-Jacobian is very close to optimal.
Manifolds and cell complexes
Differential geometry
381
403
10.4171/JEMS/839
http://www.ems-ph.org/doi/10.4171/JEMS/839
10
12
2018
An effective criterion for Eulerian multizeta values in positive characteristic
Chieh-Yu
Chang
National Tsing Hua University, Hsinchu, Taiwan
Matthew
Papanikolas
Texas A&M University, College Station, USA
Jing
Yu
National Taiwan University, Taipei, Taiwan
Multizeta values, Eulerian, Carlitz tensor powers, Carlitz polylogarithms, Anderson–Thakur polynomials
Characteristic $p$ multizeta values were initially studied by Thakur, who defined them as analogues of classical multiple zeta values of Euler. In the present paper we establish an effective criterion for Eulerian multizeta values, which characterizes when a multizeta value is a rational multiple of a power of the Carlitz period. The resulting "$t$-motivic" algorithm can tell us whether any given multizeta value is Eulerian or not. We also prove that if $\zeta_{A}(s_{1},\ldots,s_{r})$ is Eulerian, then $\zeta_{A}(s_{2},\ldots,s_{r})$ has to be Eulerian. This was conjectured by Lara Rodríguez and Thakur for the zeta-like case from numerical data. Our methods apply equally well to values of Carlitz multiple polylogarithms at algebraic points and can also be extended to determine zeta-like multizeta values.
Number theory
405
440
10.4171/JEMS/840
http://www.ems-ph.org/doi/10.4171/JEMS/840
10
12
2018
Additive triples of bijections, or the toroidal semiqueens problem
Sean
Eberhard
London, UK
Freddie
Manners
Stanford University, USA
Rudi
Mrazović
University of Zagreb, Croatia
Hardy–Littlewood circle method, transversals in Latin squares, permutations
We prove an asymptotic for the number of additive triples of bijections $\{1,\dots,n\}\to\mathbb{Z}/n\mathbb{Z}$, that is, the number of pairs of bijections $\pi_1,\pi_2\colon \{1,\dots,n\}\to\mathbb{Z}/n\mathbb{Z}$ such that the pointwise sum $\pi_1+\pi_2$ is also a bijection. This problem is equivalent to counting the number of orthomorphisms or complete mappings of $\mathbb{Z}/n\mathbb{Z}$, to counting the number of arrangements of $n$ mutually nonattacking semiqueens on an $n\times n$ toroidal chessboard, and to counting the number of transversals in a cyclic Latin square. The method of proof is a version of the Hardy–Littlewood circle method from analytic number theory, adapted to the group $(\mathbb{Z}/n\mathbb{Z})^n$.
Combinatorics
Number theory
441
463
10.4171/JEMS/841
http://www.ems-ph.org/doi/10.4171/JEMS/841
10
24
2018
Average decay of the Fourier transform of measures with applications
Renato
Lucà
Universität Basel, Switzerland
Keith
Rogers
Instituto de Ciencias Matemáticas, Madrid, Spain
Hausdorff dimension, Fourier transform, Schrödinger equation
We consider spherical averages of the Fourier transform of fractal measures and improve the lower bound on the rate of decay by taking advantage of multilinear estimates. Maximal estimates with respect to fractal measures are deduced for the Schrödinger and wave equations. This refines the almost everywhere convergence of the solution to its initial datum as time tends to zero. A consequence is that the solution to the wave equation cannot diverge on a $(d-1)$-dimensional manifold if the data belongs to the energy space $\dot{H}^1(\mathbb{R}^d)\times L^2(\mathbb{R}^d)$.
Fourier analysis
Measure and integration
465
506
10.4171/JEMS/842
http://www.ems-ph.org/doi/10.4171/JEMS/842
10
25
2018
Normality along squares
Michael
Drmota
Technische Universität Wien, Austria
Christian
Mauduit
Université d'Aix-Marseille, Marseille, France
Joël
Rivat
Université d'Aix-Marseille, France
Normal sequences, Thue–Morse sequence, exponential sums, correlation
The goal of this work is to show a first example of an almost periodic zero entropy sequence (in the sense of symbolic dynamical systems) whose subsequence along squares is a normal sequence. As an application, this provides a new method to produce normal numbers in a given base.
Number theory
Dynamical systems and ergodic theory
507
548
10.4171/JEMS/843
http://www.ems-ph.org/doi/10.4171/JEMS/843
10
25
2018
Reflectionless measures for Calderón–Zygmund operators II: Wolff potentials and rectifiability
Benjamin
Jaye
Clemson University, USA
Fedor
Nazarov
Kent State University, USA
Reflectionless measure, Riesz transform, singular integral operator, uniformly rectifiable set, Wolff potential
We continue our study of the reflectionless measures associated to an $s$-dimensional Calderón–Zygmund operator (CZO) acting in $\mathbb R^d$ with $s\in (0, d)$. Here, our focus will be the study of CZOs that are rigid, in the sense that they have few reflectionless measures associated to them. Our goal is to prove that the rigidity properties of a CZO $T$ impose strong geometric conditions upon the support of any measure $\mu$ for which $T$ is a bounded operator in $L^2 (\mu)$. In this way, we shall reduce certain well-known problems at the interface of harmonic analysis and geometric measure theory to a description of reflectionless measures of singular integral operators. What is more, we show that this approach yields promising new results.
Fourier analysis
Potential theory
549
583
10.4171/JEMS/844
http://www.ems-ph.org/doi/10.4171/JEMS/844
11
2
2018
Extremal metrics for the $Q^\prime$-curvature in three dimensions
Jeffrey
Case
Penn State University, University Park, USA
Chin-Yu
Hsiao
Academia Sinica, Taipei, Taiwan
Paul
Yang
Princeton University, USA
$Q^\prime$-curvature, pseudo-Einstein structures, CR manifolds, pseudodifferential operator, Green function asymptotics, Beckner–Onofri inequality
We construct contact forms with constant $Q^\prime$-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the $II$-functional from conformal geometry. Two crucial steps are to show that the $P^\prime$-operator can be regarded as an elliptic pseudodifferential operator and to compute the leading order terms of the asymptotic expansion of the Green function for $\sqrt{P^\prime}$.
Global analysis, analysis on manifolds
Several complex variables and analytic spaces
585
626
10.4171/JEMS/845
http://www.ems-ph.org/doi/10.4171/JEMS/845
11
9
2018
3
Integral points on conic log K3 surfaces
Yonatan
Harpaz
Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France
Integral points, log K3 surfaces, the fibration method, descent
Adapting a powerful method of Swinnerton–Dyer, we give explicit sufficient conditions for the existence of integral points on certain schemes which are fibered into affine conics. This includes, in particular, cases where the scheme is geometrically a smooth log K3 surface.
Number theory
Algebraic geometry
627
664
10.4171/JEMS/846
https://www.ems-ph.org/doi/10.4171/JEMS/846
11
20
2018
Estimates for solutions of Dirac equations and an application to a geometric elliptic-parabolic problem
Qun
Chen
Wuhan University, China
Jürgen
Jost
Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
Linlin
Sun
Wuhan University, China
Miaomiao
Zhu
Shanghai Jiao Tong University, China
Dirac equation, existence, uniqueness, chiral boundary condition, Dirac-harmonic map flow
We develop estimates for the solutions and derive existence and uniqueness results of various local boundary value problems for Dirac equations that improve all relevant results known in the literature. With these estimates at hand, we derive a general existence, uniqueness and regularity theorem for solutions of Dirac equations with such boundary conditions. We also apply these estimates to a new nonlinear elliptic-parabolic problem, the Dirac-harmonic heat flow on Riemannian spin manifolds. This problem is motivated by the supersymmetric nonlinear $\sigma$-model and combines a harmonic heat flow type equation with a Dirac equation that depends nonlinearly on the flow.
Partial differential equations
Differential geometry
665
707
10.4171/JEMS/847
https://www.ems-ph.org/doi/10.4171/JEMS/847
11
20
2018
Stability conditions on Fano threefolds of Picard number 1
Chunyi
Li
University of Edinburgh, UK
Stability condition, Fano threefolds, Bogomolov–Gieseker type inequality
We prove the conjectural Bogomolov–Gieseker type inequality for tilt-stable objects on each Fano threefold $X$ of Picard number 1. In view of the previous works [1], [2] and [3] on Bridgeland stability conditions, this induces an open subset of geometric stability conditions on D$^b(X)$. We also get a new stronger bound for Chern characters of slope semistable sheaves on $X$.
Algebraic geometry
709
726
10.4171/JEMS/848
https://www.ems-ph.org/doi/10.4171/JEMS/848
11
23
2018
Eigenvalues of minimal Cantor systems
Fabien
Durand
Université de Picardie Jules Verne, Amiens, France
Alexander
Frank
Universidad de Chile, Santiago, Chile
Alejandro
Maass
Universidad de Chile, Santiago, Chile
Minimal Cantor systems, Bratteli–Vershik representations, eigenvalues
In this article we give necessary and sufficient conditions for a complex number to be a continuous eigenvalue of a minimal Cantor system. Similarly, for minimal Cantor systems of finite rank, we provide necessary and sufficient conditions for having a measure-theoretical eigenvalue. These conditions are established from the combinatorial information on the Bratteli–Vershik representations of such systems. As an application, from any minimal Cantor system, we construct a strong orbit equivalent system without irrational continuous eigenvalues which shares all measure-theoretical eigenvalues with the original system. In a second application a minimal Cantor system is constructed satisfying the so-called maximal continuous eigenvalue group property.
Dynamical systems and ergodic theory
General topology
727
775
10.4171/JEMS/849
https://www.ems-ph.org/doi/10.4171/JEMS/849
11
23
2018
All couplings localization for quasiperiodic operators with monotone potentials
Svetlana
Jitomirskaya
University of California, Irvine, USA
Ilya
Kachkovskiy
Michigan State University, East Lansing, USA
Anderson localization, quasiperiodic Schrödinger operator, purely point spectrum
We establish Anderson localization for quasiperiodic operator families of the form $$(H(x)\psi)(m)=\psi(m+1)+\psi(m-1)+\lambda v(x+m\alpha)\psi(m)$$ for all coupling constants $\lambda > 0$ and all Diophantine frequencies $\alpha$, provided that $v$ is a 1-periodic function satisfying a Lipschitz monotonicity condition on [0,1). The localization is uniform on any energy interval on which the Lyapunov exponent is bounded from below.
Operator theory
Difference and functional equations
Quantum theory
777
795
10.4171/JEMS/850
https://www.ems-ph.org/doi/10.4171/JEMS/850
11
26
2018
Correspondance de Langlands locale $p$-adique et changement de poids
Pierre
Colmez
Université Pierre et Marie Curie, Paris, France
Local Langlands correspondence, weight 1 modular form, locally analytic representation, Kirillov model, Drinfeld upper half-plane, Breuil–Strauch conjecture
If $f$ is a modular form of weight $≥ 1$, the representation of $\mathbf {GL}_2(\mathbf Q_p)$ attached to it by the $p$-adic local Langlands correspondence encodes the representation attached by the classical correspondence. We introduce weight shifting techniques that allow us to extend the result to weight 1. We also give a full description of the Jordan–Hölder components of the locally analytic representations appearing in the correspondence.
Number theory
Topological groups, Lie groups
797
838
10.4171/JEMS/851
https://www.ems-ph.org/doi/10.4171/JEMS/851
11
27
2018
Arthur's multiplicity formula for certain inner forms of special orthogonal and symplectic groups
Olivier
Taïbi
École Normale Supérieure de Lyon, France
Automorphic forms, endoscopy, trace formula, rigid inner forms
Let $\mathbf G$ be a special orthogonal group or an inner form of a symplectic group over a number field $F$ such that there exists a non-empty set $S$ of real places of $F$ at which $\mathbf G$ has discrete series and outside of which $\mathbf G$ is quasi-split. We prove Arthur’s multiplicity formula for automorphic representations of $\mathbf G$ having algebraic regular infinitesimal character at all places in $S$.
Number theory
Topological groups, Lie groups
839
871
10.4171/JEMS/852
https://www.ems-ph.org/doi/10.4171/JEMS/852
12
3
2018
Tor as a module over an exterior algebra
David
Eisenbud
University of California, Berkeley, USA
Irena
Peeva
Cornell University, Ithaca, USA
Frank-Olaf
Schreyer
Universität des Saarlandes, Saarbrücken, Germany
Free resolutions, exterior algebras, Tor, Eisenbud operators
Let $S$ be a regular local ring with residue field $k$ and let $M$ be a finitely generated $S$-module. Suppose that $f_1,\dots ,f_c \in S$ is a regular sequence that annihilates $M$, and let $E$ be an exterior algebra over $k$ generated by $c$ elements. The homotopies for the $f_i$ on a free resolution of $M$ induce a natural structure of graded $E$-module on Tor$^{S}(M,k)$. In the case where $M$ is a high syzygy over the complete intersection $R:=S/(f_1,\dots,f_c)$ we describe this $E$-module structure in detail, including its minimal free resolution over $E$. Turning to Ext$_R(M, k)$ we show that, when $M$ is a high syzygy over $R$, the minimal free resolution of Ext$_R(M, k)$ as a module over the ring of CI operators is the Bernstein–Gel’fand–Gel’fand dual of the $E$-module Tor$^S(M, k)$. For the proof we introduce higher CI operators, and give a construction of a (generally non-minimal) resolution of $M$ over $S$ starting from a resolution of $M$ over $R$ and its higher CI operators.
Commutative rings and algebras
873
896
10.4171/JEMS/853
https://www.ems-ph.org/doi/10.4171/JEMS/853
11
29
2018
Azumaya algebras without involution
Asher
Auel
Yale University, New Haven, USA
Uriya
First
University of Haifa, Israel
Ben
Williams
University of British Columbia, Vancouver, Canada
Azumaya algebra, involution, classifying space, Brauer group, Clifford algebra, torsor, generic division algebra
Generalizing a theorem of Albert, Saltman showed that an Azumaya algebra $A$ over a ring represents a 2-torsion class in the Brauer group if and only if there is an algebra $A’$ in the Brauer class of $A$ admitting an involution of the first kind. Knus, Parimala, and Srinivas later showed that one can choose $A’$ such that deg $A’ = 2$ deg $A$. We show that 2 deg $A$ is the lowest degree one can expect in general. Specifically, we construct an Azumaya algebra A of degree 4 and period 2 such that the degree of any algebra $A’$ in the Brauer class of $A$ admitting an involution is divisible by 8. Separately, we provide examples of split and nonsplit Azumaya algebras of degree 2 admitting symplectic involutions, but no orthogonal involutions. These stand in contrast to the case of central simple algebras of even degree over fields, where the presence of a symplectic involution implies the existence of an orthogonal involution and vice versa.
Algebraic geometry
Associative rings and algebras
Algebraic topology
897
921
10.4171/JEMS/855
https://www.ems-ph.org/doi/10.4171/JEMS/855
12
10
2018
Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell–Weil rank
Michael
Stoll
Universität Bayreuth, Germany
Rational points on curves, uniform bounds, Chabauty’s method, $p$-adic integration, Mordell–Lang conjecture, Zilber–Pink conjectures
We show that there is a bound depending only on $g, r$ and [$K : \mathbb Q$] for the number of $K$-rational points on a hyperelliptic curve $C$ of genus $g$ over a number field $K$ such that the Mordell–Weil rank $r$ of its Jacobian is at most $g–3$. If $K = \mathbb Q$, an explicit bound is $8rg + 33(g–1) + 1$. The proof is based on Chabauty’s method; the new ingredient is an estimate for the number of zeros of an abelian logarithm on a $p$-adic ‘annulus’ on the curve, which generalizes the standard bound on disks. The key observation is that for a $p$-adic field $k$, the set of $k$-points on $C$ can be covered by a collection of disks and annuli whose number is bounded in terms of $g$ (and $k$). We also show, strengthening a recent result by Poonen and the author, that the lower density of hyperelliptic curves of odd degree over $\mathbb Q$ whose only rational point is the point at infinity tends to 1 uniformly over families defined by congruence conditions, as the genus $g$ tends to infinity.
Algebraic geometry
Number theory
923
956
10.4171/JEMS/857
https://www.ems-ph.org/doi/10.4171/JEMS/857
12
12
2018
4
Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves
Camille
Laurent
Université Pierre et Marie Curie Paris 6, France
Matthieu
Léautaud
École Polytechnique, Palaiseau, and Université Paris Diderot, France
Unique continuation, stability estimates, wave equation, control theory, Schrödinger equation
In this article, we first prove quantitative estimates associated to the unique continuation theorems for operators with partially analytic coefficients of Tataru [Tat95, Tat99b], Robbiano–Zuily [RZ98] and Hörmander [Hör97]. We provide local stability estimates that can be propagated, leading to global ones. Then, we specify the previous results to the wave operator on a Riemannian manifold $\mathcal M$ with boundary. For this operator, we also prove Carleman estimates and local quantitative unique continuation from and up to the boundary $\partial \mathcal M$. This allows us to obtain a global stability estimate from any open set $\Gamma$ of $\mathcal M$ or $\partial \mathcal M$, with the optimal time and dependence on the observation. As a first application, we compute a sharp lower estimate of the intensity of waves in the shadow of an obstacle. We also provide the cost of approximate controllability on the compact manifold $\mathcal M$: for any $T > 2\: \mathrm {sup}_{x \in \mathcal M} \mathrm {dist}(x,\Gamma)$, we can drive any data of $H^1_0 \times L^2$ in time $T$ to an $\varepsilon$-neighborhood of zero in $L^2 \times H^{-1}$, with a control located in $\Gamma$, at cost $e^{C/\varepsilon}$. We finally obtain related results for the Schrödinger equation.
Partial differential equations
Operator theory
Systems theory; control
957
1069
10.4171/JEMS/854
https://www.ems-ph.org/doi/10.4171/JEMS/854
11
29
2018
A stratified homotopy hypothesis
David
Ayala
Montana State University, Bozeman, USA
John
Francis
Northwestern University, Evanston, USA
Nick
Rozenblyum
University of Chicago, USA
Stratified spaces, $\infty$-categories, complete Segal spaces, quasi-categories, constructible bundles, constructible sheaves, exit-path category, striation sheaves, transversality sheaves, blowups, resolution of singularities
We show that conically smooth stratified spaces embed fully faithfully into $\infty$-categories. This articulates a stratified generalization of the homotopy hypothesis proposed by Grothendieck. Hence, each $\infty$-category defines a stack on conically smooth stratified spaces, and we identify the descent conditions it satisfies. These include $\mathbb R^1$-invariance and descent for open covers and blow-ups, analogous to sheaves for the h-topology in $\mathbb A^1$-homotopy theory. In this way, we identify $\infty$-categories as striation sheaves, which are those sheaves on conically smooth stratified spaces satisfying the indicated descent. We use this identification to construct by hand two remarkable examples of $\infty$-categories: $\mathcal B\mathsf{un}$, an $\infty$-category classifying constructible bundles; and $\mathcal E\mathsf {xit}$, the absolute exit-path $\infty$-category. These constructions are deeply premised on stratified geometry, the key geometric input being a characterization of conically smooth stratified maps between cones and the existence of pullbacks for constructible bundles.
Manifolds and cell complexes
Several complex variables and analytic spaces
Algebraic topology
1071
1178
10.4171/JEMS/856
https://www.ems-ph.org/doi/10.4171/JEMS/856
12
10
2018
The surreal numbers as a universal $H$-field
Matthias
Aschenbrenner
University of California Los Angeles, USA
Lou
van den Dries
University of Illinois at Urbana-Champaign, Urbana, USA
Joris
van der Hoeven
Ecole Polytechnique, Palaiseau, France
Surreal numbers, transseries, Hardy fields, differential fields
We show that the natural embedding of the differential field of transseries into Conway’s field of surreal numbers with the Berarducci–Mantova derivation is an elementary embedding. We also prove that any Hardy field embeds into the field of surreals with the Berarducci–Mantova derivation.
Field theory and polynomials
Mathematical logic and foundations
Ordinary differential equations
1179
1199
10.4171/JEMS/858
https://www.ems-ph.org/doi/10.4171/JEMS/858
1
7
2019
Convergence of the two-dimensional random walk loop-soup clusters to CLE
Titus
Lupu
Sorbonne Université, Paris, France
Conformal loop ensemble, Gaussian free field, loop-soup, metric graph, Poisson ensemble of Markov loops
We consider the random walk loop-soup of subcritical intensity parameter on the discrete half-plane $\mathtt{H}:=\mathbb{Z}\times\mathbb{N}$. We look at the clusters of discrete loops and show that the scaling limit of the outer boundaries of outermost clusters is a CLE$_{\kappa}$ conformal loop ensemble.
Probability theory and stochastic processes
Statistical mechanics, structure of matter
1201
1227
10.4171/JEMS/859
https://www.ems-ph.org/doi/10.4171/JEMS/859
12
27
2018
Primality testing with Gaussian periods
Hendrik
Lenstra, Jr.
Universiteit Leiden, Netherlands
Carl
Pomerance
Dartmouth College, Hanover, USA
Primality testing, constructing finite fields, Frobenius problem
We exhibit a deterministic algorithm that, for some effectively computable real number $c$, decides whether a given integer $n > 1$ is prime within time $\mathrm {log} n)^6\cdot(2+\mathrm {log\log} n)^c$. The same result, with 21/2 in place of $6$, was proved by Agrawal, Kayal, and Saxena. Our algorithm follows the same pattern as theirs, performing computations in an auxiliary ring extension of $\mathbb Z/n\mathbb Z$. We allow our rings to be generated by Gaussian periods rather than by roots of unity, which leaves us greater freedom in the selection of the auxiliary parameters and enables us to obtain a better run time estimate. The proof depends on results in analytic number theory and on the following theorem from additive number theory, which was provided by D. Bleichenbacher: if $t$ is a real number with $0 < t \le1$, and $S$ is an open subset of the interval $(0,t)$ with $\int_S\mathrm d x/x > t$, then each real number greater than or equal to 1 is in the additive semigroup generated by $S$. A byproduct of our main result is an improved algorithm for constructing finite fields of given characteristic and approximately given degree.
Number theory
Field theory and polynomials
1229
1269
10.4171/JEMS/861
https://www.ems-ph.org/doi/10.4171/JEMS/861
1
8
2019
5
Compactness results for triholomorphic maps
Costante
Bellettini
University of Cambridge, UK
Gang
Tian
Princeton University, USA and Peking University, Beijing, China
Almost stationary harmonic maps, hyperKähler manifolds, almost hyper-Hermitian manifolds, quantization of Dirichlet energy, bubbling set, regularity properties, Fueter sections
We consider triholomorphic maps from an almost hyper-Hermitian manifold $\mathcal M^{4m}$ into a (simply connected) hyperKähler manifold $\mathcal N^{4n}$. This notion entails that the map $u \in W^{1,2}$ satisfies a quaternionic del-bar equation. We work under the assumption that $u$ is locally strongly approximable in $W^{1,2}$ by smooth maps: then such maps are almost stationary harmonic, in a suitable sense (in the important special case that $\mathcal M$ is hyperKähler as well, then they are stationary harmonic). We show, by means of the bmo-$\mathscr{h}^1$-duality, that in this more general situation the classical $\varepsilon$-regularity result still holds and we establish the validity, for triholomorphic maps, of the $W^{2,1}$-conjecture (i.e. an a priori $W^{2,1}$-estimate in terms of the energy). We then address compactness issues for a weakly converging sequence $u_\ell \rightharpoonup u_\infty$ of strongly approximable triholomorphic maps $u_\ell:\mathcal M \to \mathcal N$ with uniformly bounded Dirichlet energies. The blow up analysis leads, as in the usual stationary setting, to the existence of a rectifiable blow-up set $\Sigma$ of codimension 2, away from which the sequence converges strongly. The defect measure $\Theta(x) {\mathcal H}^{4m-2} \lfloor \Sigma$ encodes the loss of energy in the limit and we prove that for a.e. point on $\Sigma$ the value of $\Theta$ is given by the sum of the energies of a (finite) number of smooth non-constant holomorphic bubbles (here the holomorphicity is to be understood with respect to a complex structure on $\mathcal N$ that depends on the chosen point on $\Sigma$). In the case that $\mathcal M$ is hyperKähler this quantization result was established by C. Y. Wang [41] with a different proof; our arguments rely on Lorentz spaces estimates. By means of a calibration argument and a homological argument we further prove that whenever the restriction of $\Sigma \cap (\mathcal M \setminus \mathrm{Sing}_{u_\infty})$ to an open set is covered by a Lipschitz connected graph, then actually this portion of $\Sigma$ is a smooth submanifold without boundary and it is pseudo-holomorphic for a (unique) almost complex structure on $\mathcal M$ (with $\Theta$ constant on this portion); moreover the bubbles originating at points of such a smooth piece are all holomorphic for a common complex structure on $\mathcal N$.
Global analysis, analysis on manifolds
Partial differential equations
Calculus of variations and optimal control; optimization
Differential geometry
1271
1317
10.4171/JEMS/860
https://www.ems-ph.org/doi/10.4171/JEMS/860
1
8
2019
Conformally Kähler, Einstein–Maxwell geometry
Vestislav
Apostolov
UQAM, Montréal, Canada
Gideon
Maschler
Clark University, Worcester, USA
Einstein–Maxwell, Kähler metrics, conformally Kähler, Einstein metrics K-stability, Futaki invariant, toric geometry, ambitoric structures, orbifolds, ambikähler metrics, extremal metrics
On a given compact complex manifold or orbifold $(M,J)$, we study the existence of Hermitian metrics $\tilde g$ in the conformal classes of Kähler metrics on $(M,J)$, such that the Ricci tensor of $\tilde g$ is of type (1, 1) with respect to the complex structure, and the scalar curvature of $\tilde g$ is constant. In real dimension 4, such Hermitian metrics provide a Riemannian counter-part of the Einstein–Maxwell equations in general relativity, and have been recently studied in [3, 34, 35, 33]. We show how the existence problem of such Hermitian metrics (which we call in any dimension conformally Kähler, Einstein–Maxwell metrics) fits into a formal momentum map interpretation, analogous to results by Donaldson and Fujiki [22, 25] in the constant scalar curvature Kähler case. This leads to a suitable notion of a Futaki invariant which provides an obstruction to the existence of conformally Kähler, Einstein–Maxwell metrics invariant under a certain group of automorphisms which are associated to a given Kähler class, a real holomorphic vector field on $(M, J)$, and a positive normalization constant. Specializing to the toric case, we further define a suitable notion of $K$-polystability and show it provides a (stronger) necessary condition for the existence of toric, conformally Kähler, Einstein–Maxwell metrics. We use the methods of [4] to show that on a compact symplectic toric 4-orbifold with second Betti number equal to 2, $K$-polystability is also a sufficient condition for the existence of (toric) conformally Kähler, Einstein–Maxwell metrics, and the latter are explicitly described as ambitoric in the sense of [3]. As an application, we exhibit many new examples of conformally Kähler, Einstein–Maxwell metrics defined on compact 4-orbifolds, and obtain a uniqueness result for the construction in [34].
Differential geometry
General
1319
1360
10.4171/JEMS/862
https://www.ems-ph.org/doi/10.4171/JEMS/862
1
8
2019
Optimal sweepouts of a Riemannian 2-sphere
Gregory
Chambers
Rice University, Houston, USA
Yevgeny
Liokumovich
Institute for Advanced Study, Princeton, USA
Homotopies of curves, sweepouts of Riemannian spheres, embedded geodesics, minmax constructions
Given a sweepout of a Riemannian 2-sphere which is composed of curves of length less than $L$, we construct a second sweepout composed of curves of length less than $L$ which are either constant curves or simple curves. This result, and the methods used to prove it, have several consequences; we answer a question of M. Freedman concerning the existence of min-max embedded geodesics, we partially answer a question due to N. Hingston and H.-B. Rademacher, and we also extend the results of [CL] concerning converting homotopies to isotopies in an effective way.
Differential geometry
1361
1377
10.4171/JEMS/863
https://www.ems-ph.org/doi/10.4171/JEMS/863
1
11
2019
Periods of modular forms on $\Gamma_0(N)$ and products of Jacobi theta functions
YoungJu
Choie
Pohang University of Science and Technology, Pohang City, Republic of Korea
Yoon Kyung
Park
Gongju National University of Education, Gongju, Republic of Korea
Don
Zagier
Max Planck Institute for Mathematics, Bonn, Germany
Period, Hecke eigenform, Jacobi theta series, parabolic cohomology, Rankin–Cohen brackets
Generalizing a result of [15] for modular forms of level one, we give a closed formula for the sum of all Hecke eigenforms on $\Gamma_0(N)$, multiplied by their odd period polynomials in two variables, as a single product of Jacobi theta series for any squarefree level $N$ . We also show that for $N = 2, 3$ and $5$ this formula completely determines the Fourier expansions all Hecke eigenforms of all weights on $\Gamma_0(N)$.
Number theory
1379
1410
10.4171/JEMS/864
https://www.ems-ph.org/doi/10.4171/JEMS/864
2
1
2019
Bounding cubic-triple product Selmer groups of elliptic curves
Yifeng
Liu
Yale University, New Haven, USA
Selmer group, Bloch–Kato conjecture, elliptic curve
Let $E$ be a modular elliptic curve over a totally real cubic field. We have a cubic-triple product motive over $\mathbb{Q}$ constructed from $E$ through multiplicative induction; it is of rank 8. We show that, under certain assumptions on $E$, the nonvanishing of the central critical value of the $L$-function attached to the motive implies that the dimension of the associated Bloch-Kato Selmer group is 0.
Number theory
Algebraic geometry
1411
1508
10.4171/JEMS/865
https://www.ems-ph.org/doi/10.4171/JEMS/865
2
1
2019
Long-range order in the 3-state antiferromagnetic Potts model in high dimensions
Ohad
Feldheim
The Hebrew University of Jerusalem, Israel
Yinon
Spinka
The University of British Columbia, Vancouver, Canada
Potts model, long-range order, phase transition, rigidity
We prove the existence of long-range order for the 3-state Potts antiferromagnet at low temperature on $\mathbb Z^d$ for sufficiently large $d$. In particular, we show the existence of six extremal and ergodic infinite-volume Gibbs measures, which exhibit spontaneous magnetization in the sense that vertices in one bipartition class have a much higher probability to be in one state than in either of the other two states. This settles the high-dimensional case of the Kotecký conjecture.
Probability theory and stochastic processes
Statistical mechanics, structure of matter
1509
1570
10.4171/JEMS/866
https://www.ems-ph.org/doi/10.4171/JEMS/866
2
1
2019
The Dynamical Manin–Mumford Conjecture and the Dynamical Bogomolov Conjecture for split rational maps
Dragos
Ghioca
University of British Columbia, Vancouver, Canada
Khoa
Nguyen
University of British Columbia and Pacific Institute for the Mathematical Sciences, Vancouver, Canada
Hexi
Ye
Zhejiang University, Hangzhou, China
Dynamical Manin–Mumford Conjecture, equidistribution of points of small height, symmetries of the Julia set of a rational function
We prove the Dynamical Bogomolov Conjecture for endomorphisms $\Phi:\mathbb P^1\times \mathbb P^1\lra \mathbb P^1\times \mathbb P^1$, where $\Phi(x,y):=(f(x), g(y))$ for any rational functions $f$ and $g$ defined over $\bar {\mathbb Q}$. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with a theorem of Levin regarding symmetries of the Julia set. Using a specialization theorem of Yuan and Zhang, we can prove the Dynamical Manin–Mumford Conjecture for endomorhisms $\Phi=(f,g)$ of $\mathbb P^1\times \mathbb P^1$, where $f$ and $g$ are rational functions defined over an arbitrary field of characteristic 0.
Dynamical systems and ergodic theory
1571
1594
10.4171/JEMS/869
https://www.ems-ph.org/doi/10.4171/JEMS/869
2
1
2019
6
Improved fractal Weyl bounds for hyperbolic manifolds (with an appendix by David Borthwick, Semyon Dyatlov and Tobias Weich)
Semyon
Dyatlov
Massachusetts Institute of Technology, Cambridge, USA
Resonances, hyperbolic quotients, fractal Weyl law
We give a new fractal Weyl upper bound for resonances of convex co-compact hyperbolic manifolds in terms of the dimension $n$ of the manifold and the dimension $\delta$ of its limit set. More precisely, we show that as $R\to\infty$, the number of resonances in the box $[R,R+1]+i[-\beta,0]$ is $\mathcal O(R^{m(\beta,\delta)+})$, where the exponent $m(\beta,\delta)=\mathrm {min}(2\delta+2\beta+1-n,\delta)$ changes its behavior at $\beta={n-1\over 2}-{\delta\over 2}$. In the case $\delta \delta-{n-1\over 2}\}$. Both results use the fractal uncertainty principle point of view recently introduced in [DyZa]. The appendix presents numerical evidence for the Weyl upper bound.
Partial differential equations
Number theory
1595
1639
10.4171/JEMS/867
https://www.ems-ph.org/doi/10.4171/JEMS/867
2
1
2019
Twists and braids for general 3-fold flops
Will
Donovan
University of Tokyo, Japan
Michael
Wemyss
University of Glasgow, UK
Birational geometry, complex 3-fold, flops, rational curve, contraction algebra, Dynkin type, homological algebra, derived category, autoequivalence, braid-type group, hyperplane arrangement, Deligne groupoid
Given a quasi-projective 3-fold $X$ with only Gorenstein terminal singularities, we prove that the flop functors beginning at $X$ satisfy higher degree braid relations, with the combinatorics controlled by a real hyperplane arrangement $\mathcal H$. This leads to a general theory, incorporating known special cases with degree 3 braid relations, in which we show that higher degree relations can occur even for two smooth rational curves meeting at a point. This theory yields an action of the fundamental group of the complexified complement $\pi_1(\mathbb{C}^n\backslash \mathcal H_\mathbb{C})$ on the derived category of $X$, for any such 3-fold that admits individually floppable curves. We also construct such an action in the more general case where individual curves may flop analytically, but not algebraically, and furthermore we lift the action to a form of affine pure braid group under the additional assumption that $X$ is $\mathbb Q$-factorial. Along the way, we produce two new types of derived autoequivalences. One uses commutative deformations of the scheme-theoretic fibre of a flopping contraction, and the other uses noncommutative deformations of the fibre with reduced scheme structure, generalising constructions of Toda and the authors [T07, DW1] which considered only the case when the flopping locus is irreducible. For type A flops of irreducible curves, we show that the two autoequivalences are related, but in other cases they are very different, with the noncommutative twist being linked to birational geometry via the Bridgeland–Chen [B02, C02] flop-flop functor.
Algebraic geometry
Associative rings and algebras
Category theory; homological algebra
Group theory and generalizations
1641
1701
10.4171/JEMS/868
https://www.ems-ph.org/doi/10.4171/JEMS/868
2
1
2019
Bessel functions and local converse conjecture of Jacquet
Jingsong
Chai
Hunan University, Changsha, China
Bessel functions, Howe vectors, local converse conjecture of Jacquet
In this paper, we prove the local converse conjecture of Jacquet over $p$-adic fields for GL$_n$ using Bessel functions.
Number theory
Topological groups, Lie groups
1703
1728
10.4171/JEMS/870
https://www.ems-ph.org/doi/10.4171/JEMS/870
2
1
2019
Cacti and cells
Ivan
Losev
University of Toronto, Canada
Cells, cactus group, wall-crossing functors, perverse equivalences
The goal of this paper is to construct an action of the cactus group of a Weyl group $W$ on $W$ that is nicely compatible with Kazhdan–Lusztig cells. The action is realized by the wall-crossing bijections that are combinatorial shadows of wall-crossing functors on the category $\mathcal O$.
Nonassociative rings and algebras
Combinatorics
Associative rings and algebras
Category theory; homological algebra
1729
1750
10.4171/JEMS/871
https://www.ems-ph.org/doi/10.4171/JEMS/871
2
5
2019
Spectral summation formula for GSp(4) and moments of spinor $L$-functions
Valentin
Blomer
Georg-August-Universität Göttingen, Germany
Siegel modular forms, spinor $L$-function, Petersson formula, moments of $L$-functions, non-vanishing, Böcherer conjecture
We compute the first and second moment of the spinor $L$-function at the central point of Siegel modular forms of large weight $k$ with power saving error term and give applications to non-vanishing.
Number theory
1751
1774
10.4171/JEMS/872
https://www.ems-ph.org/doi/10.4171/JEMS/872
2
12
2019
Constructions of $k$-regular maps using finite local schemes
Jarosław
Buczyński
University of Warsaw and Polish Academy of Sciences, Warsaw, Poland
Tadeusz
Januszkiewicz
Polish Academy of Sciences, Warsaw, Poland
Joachim
Jelisiejew
University of Warsaw, Poland
Mateusz
Michałek
Freie Universität Berlin, Germany, and Polish Academy of Sciences, Warsaw, Poland
$k$-regular embeddings, secants, punctual Hilbert scheme, finite Gorenstein schemes
A continuous map $\mathbb R^m \to \mathbb R^N$ or $\mathbb C^m \to \mathbb C^N$ is called $k$-regular if the images of any $k$ points are linearly independent. Given integers $m$ and $k$ a problem going back to Chebyshev and Borsuk is to determine the minimal value of $N$ for which such maps exist. The methods of algebraic topology provide lower bounds for $N$, but there are very few results on the existence of such maps for particular values $m$ and $k$. Using methods of algebraic geometry we construct $k$-regular maps. We relate the upper bounds on $N$ with the dimension of the locus of certain Gorenstein schemes in the punctual Hilbert scheme. The computations of the dimension of this family is explicit for $k \leq 9$, and we provide explicit examples for $k \leq 5$. We also provide upper bounds for arbitrary $m$ and $k$.
Differential geometry
Commutative rings and algebras
Algebraic geometry
Manifolds and cell complexes
1775
1808
10.4171/JEMS/873
https://www.ems-ph.org/doi/10.4171/JEMS/873
2
25
2019
Structure theory of metric measure spaces with lower Ricci curvature bounds
Andrea
Mondino
University of Warwick, Coventry, UK
Aaron
Naber
Northwestern University, Evanston, USA
Ricci curvature, optimal transport, unique tangent space, rectifiable space
We prove that a metric measure space $(X, \sf d, \mathfrak m)$ satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space $W^{1,2}$ is Hilbert is rectifiable. That is, a $\sf {RCD}^*(K,N)$-space is rectifiable, and in particular for $\mathfrak m$-a.e. point the tangent cone is unique and euclidean of dimension at most $N$. The proof is based on a maximal function argument combined with an original Almost Splitting Theorem via estimates on the gradient of the excess. We also show a sharp integral Abresh–Gromoll type inequality on the excess function and an Abresh–Gromoll-type inequality on the gradient of the excess. The argument is new even in the smooth setting.
Differential geometry
Global analysis, analysis on manifolds
1809
1854
10.4171/JEMS/874
https://www.ems-ph.org/doi/10.4171/JEMS/874
3
5
2019
Maximal surfaces in Anti-de Sitter space, width of convex hulls and quasiconformal extensions of quasisymmetric homeomorphisms
Andrea
Seppi
Université Grenoble Alpes, Gières, France
Anti-de Sitter geometry, maximal surfaces, universal Teichmüller space
We give upper bounds on the principal curvatures of a maximal surface of nonpositive curvature in three-dimensional Anti-de Sitter space, which only depend on the width of the convex hull of the surface. Moreover, given a quasisymmetric homeomorphism $\phi$, we study the relation between the width of the convex hull of the graph of $\phi$, as a curve in the boundary of infinity of Anti-de Sitter space, and the cross-ratio norm of $\phi$. As an application, we prove that if $\phi$ is a quasisymmetric homeomorphism of $\mathbb{R}\mathrm P^1$ with cross-ratio norm $||\phi||$, then ln $K\leq C||\phi||$, where $K$ is the maximal dilatation of the minimal Lagrangian extension of $\phi$ to the hyperbolic plane.
Manifolds and cell complexes
Functions of a complex variable
Differential geometry
1855
1913
10.4171/JEMS/875
https://www.ems-ph.org/doi/10.4171/JEMS/875
3
8
2019
7
The energy of a deterministic Loewner chain: Reversibility and interpretation via SLE$_{0+}$
Yilin
Wang
ETH Zürich, Switzerland
Loewner differential equation, Loewner energy, reversibility, quasiconformal mapping, Schramm–Loewner Evolution
We study some features of the energy of a deterministic chordal Loewner chain, which is defined as the Dirichlet energy of its driving function. In particular, using an interpretation of this energy as a large deviation rate function for SLE$_{\kappa}$ as $\kappa \to 0$ and the known reversibility of the SLE$_{\kappa}$ curves for small $\kappa$, we show that the energy of a deterministic curve from one boundary point $a$ of a simply connected domain $D$ to another boundary point $b$ is equal to the energy of its time-reversal, ie. of the same curve but viewed as going from $b$ to $a$ in $D$.
Functions of a complex variable
Probability theory and stochastic processes
1915
1941
10.4171/JEMS/876
https://www.ems-ph.org/doi/10.4171/JEMS/876
3
8
2019
Elliptic curves over $\mathbb{Q}_\infty$ are modular
Jack
Thorne
University of Cambridge, UK
Elliptic curves, modular forms, Iwasawa theory
We show that if $p$ is a prime, then all elliptic curves defined over the cyclotomic $\mathbb Z_p$-extension of $\mathbb Q$ are modular.
Number theory
1943
1948
10.4171/JEMS/877
https://www.ems-ph.org/doi/10.4171/JEMS/877
3
13
2019
The Monge–Ampère equation for non-integrable almost complex structures
Jianchun
Chu
Peking University, Beijing, China
Valentino
Tosatti
Northwestern University, Evanston, USA
Ben
Weinkove
Northwestern University, Evanston, USA
Monge–Ampère equation, almost complex manifolds, Calabi–Yau theorem
We show existence and uniqueness of solutions to the Monge–Ampère equation on compact almost complex manifolds with non-integrable almost complex structure.
Several complex variables and analytic spaces
Partial differential equations
Differential geometry
1949
1984
10.4171/JEMS/878
https://www.ems-ph.org/doi/10.4171/JEMS/878
3
13
2019
Regular cross sections of Borel flows
Konstantin
Slutsky
University of Illinois at Chicago, USA
Borel flow, flow under a function, suspension flow, cross section
Any free Borel flow is shown to admit a cross section with only two possible distances between adjacent points, answering an old question of Nadkarni. As an application of this result, we derive a short proof of the classification of Borel flows up to Lebesgue orbit equivalence.
Mathematical logic and foundations
Dynamical systems and ergodic theory
1985
2050
10.4171/JEMS/879
https://www.ems-ph.org/doi/10.4171/JEMS/879
3
14
2019
Continuity, positivity and simplicity of the Lyapunov exponents for quasi-periodic cocycles
Pedro
Duarte
Universidade de Lisboa, Portugal
Silvius
Klein
Pontifícia Universidade Católica do Rio de Janeiro, Brazil
Lyapunov exponents, quasi-periodic cocycles, large deviations, subharmonic functions, lattice Schrödinger operators
An analytic quasi-periodic cocycle is a linear cocycle over a fixed ergodic torus translation of one or several variables, where the fiber action depends analytically on the base point. Consider the space of all such cocycles of any given dimension and endow it with the uniform norm. Assume that the translation vector satisfies a generic Diophantine condition. We establish large deviation type estimates for the iterates of such cocycles, which, moreover, are stable under small perturbations of the cocycle. As a consequence of these uniform estimates, we obtain continuity properties of the Lyapunov exponents regarded as functions on this space of cocycles. This result builds upon our previous work on this topic and its proof uses an abstract continuity theorem for the Lyapunov exponents which we derived in a recent monograph. The new feature of this paper is extending the availability of such results to cocycles that are identically singular (i.e. non-invertible everywhere), in the several variables torus translation setting. This feature is exactly what allows us, through a simple limiting argument, to obtain criteria for the positivity and simplicity of the Lyapunov exponents of such cocycles. Specializing to the family of cocycles corresponding to a block Jacobi operator, we derive consequences on the continuity, positivity and simplicity of its Lyapunov exponents, and on the continuity of its integrated density of states.
Dynamical systems and ergodic theory
Potential theory
Several complex variables and analytic spaces
Quantum theory
2051
2106
10.4171/JEMS/880
https://www.ems-ph.org/doi/10.4171/JEMS/880
3
14
2019
Barycenters of polytope skeleta and counterexamples to the Topological Tverberg Conjecture, via constraints
Pavle
Blagojević
Institut SANU, Belgrad, Serbia and Freie Universität Berlin, Germany
Florian
Frick
Carnegie Mellon University, Pittsburgh, USA
Günter
Ziegler
Freie Universität Berlin, Germany
Topological Tverberg Theorem, polytope skeleta, constraint method, Generalized Van Kampen–Flores Theorem, Mabillard–Wagner generalized Whitney trick
Using the authors’ 2014 “constraint method,” we give a short proof for a 2015 result of Dobbins on representations of a point in a polytope as the barycenter of points in a skeleton, and show that the “$r$-fold Whitney trick” of Mabillard and Wagner (2014/2015) implies that the Topological Tverberg Conjecture for $r$-fold intersections fails dramatically for all $r$ that are not prime powers.
Convex and discrete geometry
Algebraic topology
2107
2116
10.4171/JEMS/881
https://www.ems-ph.org/doi/10.4171/JEMS/881
3
14
2019
Scattering profile for global solutions of the energy-critical wave equation
Thomas
Duyckaerts
Université Paris 13, Sorbonne Paris Cité, Villetaneuse, France
Carlos
Kenig
University of Chicago, USA
Frank
Merle
Université de Cergy-Pontoise, France
Wave equation, critical nonlinearity, global solution, radiation term
Consider the focusing energy-critical wave equation in space dimension 3, 4 or 5. We prove that any global solution which is bounded in the energy space converges in the exterior of wave cones to a radiation term which is a solution of the linear wave equation.
Partial differential equations
General
2117
2162
10.4171/JEMS/882
https://www.ems-ph.org/doi/10.4171/JEMS/882
3
20
2019
Super-approximation, II: the $p$-adic case and the case of bounded powers of square-free integers
Alireza
Salehi Golsefidy
University of California at San Diego, La Jolla, USA
Super-approximation, spectral gap, expanders, linear groups
Let $\Omega$ be a finite symmetric subset of GL$_n(\mathbb Z[1/q_0])$, and $\Gamma:=\langle \Omega \rangle$, and let $\pi_m$ be the group homomorphism induced by the quotient map $\mathbb Z[1/q_0] \to \mathbb Z[1/q_0] / m \mathbb Z[1/q_0]$. Then the family {Cay $(\pi_m (\Gamma),\pi_m(\Omega))\}_m$ of Cayley graphs is a family of expanders as $m$ ranges over fixed powers of square-free integers and powers of primes that are coprime to $q_0$ if and only if the connected component of the Zariski-closure of $\Gamma$ is perfect. Some of the immediate applications, e.g. orbit equivalence rigidity and largeness of certain $\ell$-adic Galois representations, are also discussed.
Topological groups, Lie groups
Combinatorics
2163
2232
10.4171/JEMS/883
https://www.ems-ph.org/doi/10.4171/JEMS/883
3
20
2019
8
Properties of the maximal entropy measure and geometry of Hénon attractors
Pierre
Berger
Université Paris 13, Villetaneuse, France
Hénon map, Lyapunov exponent, maximal entropy measure
We consider an abundant class of non-uniformly hyperbolic $C^2$-Hénon like diffeomorphisms called strongly regular and which corresponds to Benedicks–Carleson parameters. We prove the existence of $m > 0$ such that for any such diffeomorphism $f$, every invariant probability measure of $f$ has a Lyapunov exponent greater than $m$, answering a question of L. Carleson. Moreover, we show the existence and uniqueness of a measure of maximal entropy, which answers a question of M. Lyubich and Y. Pesin. We also prove that the maximal entropy measure is equidistributed on the periodic points and is finitarily Bernoulli, which gives an answer to a question of J.-P. Thouvenot. Finally, we show that the maximal entropy measure is exponentially mixing and satisfies the central limit theorem. The proof is based on a new construction of a Young tower for which the first return time coincides with the symbolic return time, and whose orbit is conjugate to a strongly positive recurrent Markov shift.
Dynamical systems and ergodic theory
2233
2299
10.4171/JEMS/884
https://www.ems-ph.org/doi/10.4171/JEMS/884
4
3
2019
Proof of the Log-Convex Density Conjecture
Gregory
Chambers
Rice University, Houston, USA
Manifold with density, isoperimetric problem
We completely characterize isoperimetric regions in $\mathbb R^n$ with density $e^h$, where $h$ is convex, smooth, and radially symmetric. In particular, balls around the origin constitute isoperimetric regions of any given volume, proving the Log-Convex Density Conjecture due to Kenneth Brakke.
Calculus of variations and optimal control; optimization
Differential geometry
2301
2332
10.4171/JEMS/885
https://www.ems-ph.org/doi/10.4171/JEMS/885
4
2
2019
Unsmoothable group actions on compact one-manifolds
Hyungryul
Baik
Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea
Sang-hyun
Kim
Korea Institute for Advanced Study (KIAS), Seoul, Republic of Korea
Thomas
Koberda
University of Virginia, Charlottesville, USA
Mapping class groups, right-angled Artin groups, circle actions
We show that no finite index subgroup of a sufficiently complicated mapping class group or braid group can act faithfully by $C^{1+\mathrm {bv}}$ diffeomorphisms on the circle, which generalizes a result of Farb–Franks, and which parallels a result of Ghys and Burger–Monod concerning differentiable actions of higher rank lattices on the circle. This answers a question of Farb, which has its roots in the work of Nielsen. We prove this result by showing that if a right-angled Artin group acts faithfully by $C^{1+\mathrm {bv}}$ diffeomorphisms on a compact one-manifold, then its defining graph has no subpath of length 3. As a corollary, we also show that no finite index subgroup of Aut$(F_n)$ or Out$(F_n)$ for $n \geq 3$, of the Torelli group for genus at least 3, and of each term of the Johnson filtration for genus at least 5, can act faithfully by $C^{1+\mathrm {bv}}$ diffeomorphisms on a compact one-manifold.
Manifolds and cell complexes
Group theory and generalizations
Dynamical systems and ergodic theory
2333
2353
10.4171/JEMS/886
https://www.ems-ph.org/doi/10.4171/JEMS/886
4
15
2019
Eta invariants and the hypoelliptic Laplacian
Jean-Michel
Bismut
Université Paris-Sud, Orsay, France
Spectral theory, Selberg trace formula, hypoelliptic equations, index theory and related fixed point theorems, eta invariants, Chern–Simons invariants, diffusion processes and stochastic analysis on manifolds
The purpose of this paper is to give a new proof of the results of Moscovici and Stanton on orbital integrals associated with eta invariants on compact locally symmetric spaces. Moscovici and Stanton used methods of harmonic analysis on reductive groups. Here, we combine our approach to orbital integrals that uses the hypoelliptic Laplacian with the introduction of a rotation on certain Clifford algebras. Probabilistic methods play an important role in establishing key estimates. In particular, we construct a suitable Itô calculus associated with certain hypoelliptic diffusions.
Global analysis, analysis on manifolds
Number theory
Partial differential equations
2355
2515
10.4171/JEMS/887
https://www.ems-ph.org/doi/10.4171/JEMS/887
4
15
2019
Long wave limit for Schrödinger maps
Pierre
Germain
New York University, New York, USA
Frédéric
Rousset
Université Paris-Sud, Orsay, France and Institut Universitaire de France
Schrödinger map, long wave limit, KdV equation
We study long wave limits for general Schrödinger map systems into Kähler manifolds with a constraining potential vanishing on a Lagrangian submanifold. We obtain KdV-type systems set on the tangent space of the submanifold. Our general theory is applied to study the long wave limits of the Gross–Pitaevskii equation and of the Landau–Lifshitz systems for ferromagnetic and anti-ferromagnetic chains.
Partial differential equations
Differential geometry
2517
2602
10.4171/JEMS/888
https://www.ems-ph.org/doi/10.4171/JEMS/888
4
24
2019
9
Convergence of a Newton algorithm for semi-discrete optimal transport
Jun
Kitagawa
Michigan State University, East Lansing, USA
Quentin
Mérigot
Université Paris-Sud, Orsay, France
Boris
Thibert
Université Grenoble Alpes, Grenoble, France
Optimal transport, Ma–Trudinger–Wang condition, Laguerre tessellation
A popular way to solve optimal transport problems numerically is to assume that the source probability measure is absolutely continuous while the target measure is finitely supported. We introduce a damped Newton algorithm in this setting, which is experimentally efficient, and we establish its global linear convergence for cost functions satisfying an assumption that appears in the regularity theory for optimal transport.
Calculus of variations and optimal control; optimization
Numerical analysis
2603
2651
10.4171/JEMS/889
https://www.ems-ph.org/doi/10.4171/JEMS/889
4
24
2019
Semiorthogonal decompositions of the categories of equivariant coherent sheaves for some reflection groups
Alexander
Polishchuk
University of Oregon, Eugene, USA and National Research University Higher School of Economics, Moscow, Russia
Michel
Van den Bergh
Hasselt University, Diepenbeek, Belgium
Derived category, semiorthogonal decomposition, equivariant sheaf, Springer correspondence, reflection group, Hochschild homology, equivariant cohomology
We consider the derived category $D^b_G(V)$ of coherent sheaves on a complex vector space $V$ equivariant with respect to an action of a finite reflection group $G$. In some cases, including Weyl groups of type $A$, $B$, $G_2$, $F_4$, as well as the groups $G(m,1,n)=(\mu_m)^n\rtimes S_n$, we construct a semiorthogonal decomposition of this category, indexed by the conjugacy classes of $G$. The pieces of this decompositions are equivalent to the derived categories of coherent sheaves on the quotient-spaces $V^g/C(g)$, where $C(g)$ is the centralizer subgroup of $g\in G$. In the case of the Weyl groups the construction uses some key results about the Springer correspondence, due to Lusztig, along with some formality statement generalizing a result of Deligne [23]. We also construct global analogs of some of these semiorthogonal decompositions involving derived categories of equivariant coherent sheaves on $C^n$, where $C$ is a smooth curve.
Algebraic geometry
Category theory; homological algebra
Algebraic topology
2653
2749
10.4171/JEMS/890
https://www.ems-ph.org/doi/10.4171/JEMS/890
5
7
2019
Consistent systems of linear differential and difference equations
Reinhard
Schäfke
Université de Strasbourg, France
Michael
Singer
North Carolina State University, Raleigh, USA
linear differential equations, linear difference equations, consistent systems, shift operator, q-difference equation, Mahler operator
We consider systems of linear differential and difference equations $$\delta Y(x)/dx = A(x) Y(x), \: \sigma s(Y(x)) = B(x)Y(x)$$ with $\delta = \frac{d}{dx}$, $\sigma$ a shift operator $\sigma(x) = x+a$, $q$-dilation operator $\sigma(x) = qx$ or Mahler operator $\sigma(x) = x^p$ and systems of two linear difference equations $$\sigma_1 Y(x) =A(x)Y(x), \: \sigma_2 Y(x) =B(x)Y(x)$$ with $(\sigma_1,\sigma_2)$ a sufficiently independent pair of shift operators, pair of $q$-dilation operators or pair of Mahler operators. Here $A(x)$ and $B(x)$ are $n\times n$ matrices with rational function entries. Assuming a consistency hypothesis, we show that such systems can be reduced to a system of a very simple form. Using this we characterize functions satisfying two linear scalar differential or difference equations with respect to these operators. We also indicate how these results have consequences both in the theory of automatic sets, leading to a new proof of Cobham's Theorem, and in the Galois theories of linear difference and differential equations, leading to hypertranscendence results.
Difference and functional equations
Ordinary differential equations
2751
2792
10.4171/JEMS/891
https://www.ems-ph.org/doi/10.4171/JEMS/891
5
13
2019
Parabolic dynamics and anisotropic Banach spaces
Paolo
Giulietti
Scuola Normale Superiore, Pisa, Italy
Carlangelo
Liverani
University of Tor Vergata, Rome, Italy
Horocycle flows, quantitative equidistribution, quantitative mixing, spectral theory, transfer operator, anisotropic spaces, cohomological equations
We investigate the relation between the distributions appearing in the study of ergodic averages of parabolic flows (e.g. in the work of Flaminio–Forni) and the ones appearing in the study of the statistical properties of hyperbolic dynamical systems (i.e. the eigendistributions of the transfer operator). In order to avoid, as much as possible, technical issues that would cloud the basic idea, we limit ourselves to a simple flow on the torus. Our main result is that, roughly, the growth of ergodic averages (and the characterization of coboundary regularity) of a parabolic flow is controlled by the eigenvalues of a suitable transfer operator associated to the renormalizing dynamics. The conceptual connection that we illustrate is expected to hold in considerable generality.
Dynamical systems and ergodic theory
2793
2858
10.4171/JEMS/892
https://www.ems-ph.org/doi/10.4171/JEMS/892
5
20
2019
A heuristic for boundedness of ranks of elliptic curves
Jennifer
Park
University of Michigan, Ann Arbor, USA
Bjorn
Poonen
Massachusetts Institute of Technology, Cambridge, USA
John
Voight
Dartmouth College, Hanover, USA
Melanie Matchett
Wood
University of Wisconsin, Madison, USA
Elliptic curve, rank, Shafarevich–Tate group
We present a heuristic that suggests that ranks of elliptic curves $E$ over $\mathbb Q$ are bounded. In fact, it suggests that there are only finitely many $E$ of rank greater than 21. Our heuristic is based on modeling the ranks and Shafarevich–Tate groups of elliptic curves simultaneously, and relies on a theorem counting alternating integer matrices of specified rank. We also discuss analogues for elliptic curves over other global fields.
Number theory
Algebraic geometry
2859
2903
10.4171/JEMS/893
https://www.ems-ph.org/doi/10.4171/JEMS/893
5
21
2019
Uniform K-stability and asymptotics of energy functionals in Kähler geometry
Sébastien
Boucksom
École Polytechnique, Palaiseau, France
Tomoyuki
Hisamoto
Nagoya University, Japan
Mattias
Jonsson
University of Michigan, Ann Arbor, USA, Chalmers University of Technology and University of Gothenburg, Sweden
K-stability, Kähler geometry, canonical metrics, non-Archimedean geometry
Consider a polarized complex manifold $(X, L)$ and a ray of positive metrics on $L$ defined by a positive metric on a test configuration for $(X, L)$. For many common functionals in Kähler geometry, we prove that the slope at infinity along the ray is given by evaluating the non-Archimedean version of the functional (as defined in our earlier paper [BHJ17]) at the non-Archimedean metric on $L$ defined by the test configuration. Using this asymptotic result, we show that coercivity of the Mabuchi functional implies uniform K-stability, as defined in [Der15, BHJ17]. As a partial converse, we show that uniform K-stability implies coercivity of the Mabuchi functional when restricted to Bergman metrics.
Differential geometry
Algebraic geometry
Several complex variables and analytic spaces
2905
2944
10.4171/JEMS/894
https://www.ems-ph.org/doi/10.4171/JEMS/894
5
24
2019
10
Mosco convergence for $H$ (curl) spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems
Hongyu
Liu
Hong Kong Baptist University, Kowloon, Hong Kong
Luca
Rondi
Università degli Studi di Trieste and Università degli Studi di Milano, Italy
Jingni
Xiao
Hong Kong Baptist University, Hong Kong, and Rutgers University, Piscataway, USA
Maxwell equations, Mosco convergence, higher integrability, scattering, inverse scattering, polyhedral scatterers, stability
This paper is concerned with the scattering problem for time-harmonic electromagnetic waves, due to the presence of scatterers and of inhomogeneities in the medium. We prove a sharp stability result for the solutions to the direct electromagnetic scattering problem, with respect to variations of the scatterer and of the inhomogeneity, under minimal regularity assumptions for both of them. The stability result leads to bounds on solutions to the scattering problems which are uniform for an extremely general class of admissible scatterers and inhomogeneities. These uniform bounds are a key step in tackling the challenging stability issue for the corresponding inverse electromagnetic scattering problem. In this paper we establish two optimal stability results of logarithmic type for the determination of polyhedral scatterers by a minimal number of electromagnetic scattering measurements. In order to prove the stability result for the direct electromagnetic scattering problem, we study two fundamental issues in the theory of Maxwell equations: Mosco convergence for $H$ (curl) spaces and higher integrability properties of solutions to Maxwell equations in nonsmooth domains.
Optics, electromagnetic theory
Partial differential equations
Calculus of variations and optimal control; optimization
2945
2993
10.4171/JEMS/895
https://www.ems-ph.org/doi/10.4171/JEMS/895
5
23
2019
Arithmetic and representation theory of wild character varieties
Tamás
Hausel
Institute of Science and Technology Austria, Klosterneuburg, Austria
Martin
Mereb
Universidad de Buenos Aires, Argentina
Michael Lennox
Wong
Universität Duisburg-Essen, Germany
Irregular connection, character variety, Hitchin system, Yokonuma–Hecke algebra, Macdonald polynomials
We count points over a finite field on wild character varieties of Riemann surfaces for singularities with regular semisimple leading term. The new feature in our counting formulas is the appearance of characters of Yokonuma–Hecke algebras. Our result leads to the conjecture that the mixed Hodge polynomials of these character varieties agree with the previously conjectured perverse Hodge polynomials of certain twisted parabolic Higgs moduli spaces, indicating the possibility of a $P=W$ conjecture for a suitable wild Hitchin system.
Algebraic geometry
Associative rings and algebras
Group theory and generalizations
2995
3052
10.4171/JEMS/896
https://www.ems-ph.org/doi/10.4171/JEMS/896
6
6
2019
Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups
Guitang
Lan
Universität Mainz, Germany
Mao
Sheng
University of Science and Technology of China, Hefei, China
Kang
Zuo
Universität Mainz, Germany
Higgs bundles, $p$-adic Hodge theory
Let $k$ be the algebraic closure of a finite field of odd characteristic $p$ and $X$ a smooth projective scheme over the Witt ring $W(k)$ which is geometrically connected in characteristic zero. We introduce the notion of Higgs–de Rham flow and prove that the category of periodic Higgs–de Rham flows over $X/W(k)$ is equivalent to the category of Fontaine modules, hence further equivalent to the category of crystalline representations of the étale fundamental group $\pi_1(X_K)$ of the generic fiber of $X$, after Fontaine–Laffaille and Faltings. Moreover, we prove that every semistable Higgs bundle over the special fiber $X_k$ of $X$ of rank $\leq p$ initiates a semistable Higgs–de Rham flow and thus those of rank $\leq p-1$ with trivial Chern classes induce $k$-representations of $\pi_1(X_K)$. A fundamental construction in this paper is the inverse Cartier transform over a truncated Witt ring. In characteristic $p$, it was constructed by Ogus–Vologodsky in the nonabelian Hodge theory in positive characteristic; in the affine local case, our construction is related to the local Ogus–Vologodsky correspondence of Shiho.
Algebraic geometry
Number theory
3053
3112
10.4171/JEMS/897
https://www.ems-ph.org/doi/10.4171/JEMS/897
6
12
2019
Classification of a family of non-almost-periodic free Araki–Woods factors
Cyril
Houdayer
Université Paris-Sud, Orsay, France
Dimitri
Shlyakhtenko
University of California Los Angeles, USA
Stefaan
Vaes
Katholieke Universiteit Leuven, Belgium
Free Araki–Woods factors, free product von Neumann algebras, Popa’s deformation/ rigidity theory, type III factors
We obtain a complete classification of a large class of non-almost-periodic free Araki–Woods factors $\Gamma(\mu, m)''$ up to isomorphism. We do this by showing that free Araki–Woods factors $\Gamma(\mu, m)''$ arising from finite symmetric Borel measures $\mu$ on $\mathbb{R}$ whose atomic part $\mu_a$ is nonzero and not concentrated on $\{0\}$ have the joint measure class $\mathcal C(\bigvee_{k \geq 1} \mu^{\ast k})$ as an invariant. Our key technical result is a deformation/rigidity criterion for the unitary conjugacy of two faithful normal states. We use this to also deduce rigidity and classification theorems for free product von Neumann algebras.
Functional analysis
3113
3142
10.4171/JEMS/898
https://www.ems-ph.org/doi/10.4171/JEMS/898
6
13
2019
Density of positive Lyapunov exponents for symplectic cocycles
Disheng
Xu
Royal Institute of Technology, Stockholm, Sweden
Lyapunov exponents, Schrödinger operator on strips, symplectic cocycles, Kotani theory, monotonic cocycles, Hermitian symmetric spaces, Siegel upper half-plane, Shilov boundary, Lagrangian Grassmannian
We prove that Sp$(2d,\mathbb R)$-cocycles, HSp($2d$)-cocycles and pseudo-unitary cocycles with at least one non-zero Lyapunov exponent are dense in all usual regularity classes for non periodic dynamical systems. For Schrödinger operator on the strip, we prove a similar result for density of positive Lyapunov exponents. This generalizes a result of A. Avila [2] to higher dimensions.
Dynamical systems and ergodic theory
Ordinary differential equations
Operator theory
Statistical mechanics, structure of matter
3143
3190
10.4171/JEMS/899
https://www.ems-ph.org/doi/10.4171/JEMS/899
6
17
2019
Amenability of groups is characterized by Myhill's Theorem. With an appendix by Dawid Kielak
Laurent
Bartholdi
Georg-August-Universität Göttingen, Germany
Cellular automata, Moore–Myhill theorem, amenability of groups, Ore domains, localization
We prove a converse to Myhill's "Garden-of-Eden" theorem and obtain in this manner a characterization of amenability in terms of cellular automata: A group $G$ is amenable if and only if every cellular automaton with carrier $G$ that has gardens of Eden also has mutually erasable patterns. This answers a question by Schupp, and solves a conjecture by Ceccherini-Silberstein, Machì and Scarabotti. Furthermore, for non-amenable $G$ the cellular automaton with carrier $G$ that has gardens of Eden but no mutually erasable patterns may also be assumed to be linear. An appendix by Dawid Kielak shows that group rings without zero divisors are Ore domains precisely when the group is amenable, answering a conjecture attributed to Guba.
Dynamical systems and ergodic theory
Associative rings and algebras
Abstract harmonic analysis
3191
3197
10.4171/JEMS/900
https://www.ems-ph.org/doi/10.4171/JEMS/900
6
19
2019
Compactness and finite forcibility of graphons
Roman
Glebov
Ben Gurion University of the Negev, Beer-Sheva, Israel
Daniel
Král'
Masaryk University, Brno, Czech Republic, and University of Warwick, Coventry, UK
Jan
Volec
Emory University, Atlanta, USA
Graph limits, extremal combinatorics
Graphons are analytic objects associated with convergent sequences of graphs. Problems from extremal combinatorics and theoretical computer science led to a study of graphons determined by finitely many subgraph densities, which are referred to as finitely forcible. Following the intuition that such graphons should have finitary structure, Lovász and Szegedy conjectured that the topological space of typical vertices of a finitely forcible graphon is always compact. We disprove the conjecture by constructing a finitely forcible graphon such that the associated space is not compact. The construction method gives a general framework for constructing finitely forcible graphons with non-trivial properties.
Combinatorics
3199
3223
10.4171/JEMS/901
https://www.ems-ph.org/doi/10.4171/JEMS/901
6
24
2019
Decomposition of Brownian loop-soup clusters
Wei
Qian
University of Cambridge, UK and ETH Zürich, Switzerland
Wendelin
Werner
ETH Zürich, Switzerland
Brownian loop-soups, Schramm–Loewner evolutions
We study the structure of Brownian loop-soup clusters in two dimensions. Among other things, we obtain the following decomposition of the clusters with critical intensity: If one conditions a loop-soup cluster on its outer boundary $\partial$ (which is known to be an SLE$_4$-type loop), then the union of all excursions away from $\partial$ by all the Brownian loops in the loop-soup that touch $\partial$ is distributed exactly like the union of all excursions of a Poisson point process of Brownian excursions in the domain enclosed by $\partial$. A related result that we derive and use is that the couplings of the Gaussian Free Field (GFF) with CLE$_4$ via level lines (by Miller–Sheffield), of the square of the GFF with loop-soups via occupation times (by Le Jan), and of the CLE$_4$ with loop-soups via loop-soup clusters (by Sheffield and Werner) can be made to coincide. An instrumental role in our proof of this fact is played by Lupu’s description of CLE$_4$ as limits of discrete loop-soup clusters.
Probability theory and stochastic processes
General
3225
3253
10.4171/JEMS/902
https://www.ems-ph.org/doi/10.4171/JEMS/902
6
24
2019
An elementary proof of the rank-one theorem for BV functions
Annalisa
Massaccesi
Università di Verona, Italy
Davide
Vittone
Università di Padova, Italy
Functions of bounded variation, rank-one theorem
We provide a simple proof of a result, due to G. Alberti, concerning a rank-one property for the singular part of the derivative of vector-valued functions of bounded variation.
Calculus of variations and optimal control; optimization
Measure and integration
3255
3258
10.4171/JEMS/903
https://www.ems-ph.org/doi/10.4171/JEMS/903
6
26
2019
11
On melting and freezing for the 2D radial Stefan problem
Mahir
Hadžić
King's College London, UK
Pierre
Raphaël
Université de Nice Sophia Antipolis, Nice, France
Stefan problem, finite time melting, singularity formation
We consider the two dimensional free boundary Stefan problem describing the evolution of a spherically symmetric ice ball $\{r\leq \lambda (t)\}$. We revisit the pioneering analysis of [31] and prove the existence in the radial class of finite time melting regimes $$\lambda(t)=\left\{\begin{array}{ll} (T-t)^{1/2}e^{-\frac{\sqrt{2}}{2}\sqrt{|\ln(T-t)|}+O(1)}\\ (c+o(1))\frac{(T-t)^{\frac{k+1}{2}}}{|\ln (T-t)|^{\frac{k+1}{2k}}}, \ \ k\in \mathbb N^*\end{array}\right. \quad\text{ as } t\to T$$ which respectively correspond to the fundamental stable melting rate, and a sequence of codimension $k$ excited regimes. Our analysis fully revisits a related construction for the harmonic heat flow in [60] by introducing a new and canonical functional framework for the study of type II (i.e. non-self-similar) blow up. We also show a deep duality between the construction of the melting regimes and the derivation of a discrete sequence of global-in-time freezing regimes $$\lambda_\infty - \lambda(t)\sim\left\{\begin{array}{ll} \frac{1}{\log t}\\ \frac{1}{t^{k}(\log t)^{2}}, \ \ k\in \mathbb N^*\end{array}\right.\quad\text{ as } t\to +\infty$$ which correspond respectively to the fundamental stable freezing rate, and excited regimes which are codimension $k$ stable.
Partial differential equations
Fluid mechanics
3259
3341
10.4171/JEMS/904
https://www.ems-ph.org/doi/10.4171/JEMS/904
7
1
2019
Anosov representations and dominated splittings
Jairo
Bochi
Pontificia Universidad Católica de Chile, Santiago, Chile
Rafael
Potrie
Universidad de la República, Montevideo, Uruguay
Andrés
Sambarino
Université Pierre et Marie Curie, Paris, France
Discrete subgroups of Lie groups, linear cocycles, dominated splitting, coarse geometry, hyperbolic groups
We provide a link between Anosov representations introduced by Labourie and dominated splitting of linear cocycles. This allows us to obtain equivalent characterizations for Anosov representations and to recover recent results due to Guéritaud–Guichard–Kassel–Wienhard [GGKW] and Kapovich–Leeb–Porti [KLP2] by different methods. We also give characterizations in terms of multicones and cone types inspired by the work of Avila–Bochi–Yoccoz [ABY] and Bochi–Gourmelon [BG]. Finally, we provide a new proof of the higher rank Morse Lemma of Kapovich–Leeb–Porti [KLP2].
Topological groups, Lie groups
Group theory and generalizations
Dynamical systems and ergodic theory
Differential geometry
3343
3414
10.4171/JEMS/905
https://www.ems-ph.org/doi/10.4171/JEMS/905
7
19
2019
Heat flow and quantitative differentiation
Tuomas
Hytönen
University of Helsinki, Finland
Assaf
Naor
Princeton University, USA
Quantitative differentiation, uniform convexity, Littlewood–Paley theory, heat semigroup, metric embeddings
For every Banach space $(Y,\|\cdot\|_Y)$ that admits an equivalent uniformly convex norm we prove that there exists $c=c(Y)\in (0,\infty)$ with the following property. Suppose that $n\in \mathbb N$ and that $X$ is an $n$-dimensional normed space with unit ball $B_X$. Then for every $1$-Lipschitz function $f:B_X\to Y$ and for every $\e\in (0,1/2]$ there exists a radius $r\ge \exp(-1/\epsilon^{cn})$, a point $x\in B_X$ with $x+rB_X\subset B_X$, and an affine mapping $\Lambda:X\to Y$ such that $\|f(y)-\Lambda(y)\|_Y\le \epsilon r$ for every $y\in x+rB_X$. This is an improved bound for a fundamental quantitative differentiation problem that was formulated by Bates, Johnson, Lindenstrauss, Preiss and Schechtman (1999), and consequently it yields a new proof of Bourgain's discretization theorem (1987) for uniformly convex targets. The strategy of our proof is inspired by Bourgain's original approach to the discretization problem, which takes the affine mapping $\Lambda$ to be the first order Taylor polynomial of a time-$t$ Poisson evolute of an extension of $f$ to all of $X$ and argues that, under appropriate assumptions on $f$, there must exist a time $t\in (0,\infty)$ at which $\Lambda$ is (quantitatively) invertible. However, in the present context we desire a more stringent conclusion, namely that $\Lambda$ well-approximates $f$ on a macroscopically large ball, in which case we show that for our argument to work one cannot use the Poisson semigroup. Nevertheless, our strategy does succeed with the Poisson semigroup replaced by the heat semigroup. As a crucial step of our proof, we establish a new uniformly convex-valued Littlewood–Paley–Stein $\mathcal{G}$-function inequality for the heat semigroup; influential work of Martínez, Torrea and Xu (2006) obtained such an inequality for subordinated Poisson semigroups but left the important case of the heat semigroup open. As a byproduct, our proof also yields a new and simple approach to the classical Dorronsoro theorem (1985) even for real-valued functions.
Fourier analysis
Approximations and expansions
Functional analysis
Probability theory and stochastic processes
3415
3466
10.4171/JEMS/906
https://www.ems-ph.org/doi/10.4171/JEMS/906
7
19
2019
A Variational Tate Conjecture in crystalline cohomology
Matthew
Morrow
Sorbonne Université, Paris, France
Variational Hodge, Tate conjecture, topological cyclic homology
Given a smooth, proper family of varieties in characteristic $p > 0$, and a cycle $z$ on a fibre of the family, we consider a Variational Tate Conjecture characterising, in terms of the crystalline cycle class of $z$, whether $z$ extends cohomologically to the entire family. This is a characteristic $p$ analogue of Grothendieck’s Variational Hodge Conjecture. We prove the conjecture for divisors, and an infinitesimal variant of the conjecture for cycles of higher codimension; the former result can be used to reduce the $\ell$-adic Tate conjecture for divisors over finite fields to the case of surfaces.
Algebraic geometry
$K$-theory
3467
3511
10.4171/JEMS/907
https://www.ems-ph.org/doi/10.4171/JEMS/907
7
19
2019
Observability and unique continuation inequalities for the Schrödinger equation
Gengsheng
Wang
Tianjin University, China
Ming
Wang
China University of Geosciences, Wuhan, China
Yubiao
Zhang
Tianjin University, China
Observability, unique continuation, controllability, free Schrödinger equation
We present several observability and unique continuation inequalities for the free Schrödinger equation in the whole space. The observations in these inequalities are made either at two points in time or one point in time. These inequalities correspond to different kinds of controllability for the free Schrödinger equation. We also show that the observability inequality at two points in time is equivalent to the uncertainty principle given in [21].
Systems theory; control
Partial differential equations
3513
3572
10.4171/JEMS/908
https://www.ems-ph.org/doi/10.4171/JEMS/908
7
16
2019
12
Optimal packings of bounded degree trees
Felix
Joos
University of Birmingham, UK
Jaehoon
Kim
Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea
Daniela
Kühn
University of Birmingham, UK
Deryk
Osthus
University of Birmingham, UK
Trees, graph decompositions, packings, quasirandomness
We prove that if $T_1,\dots, T_n$ is a sequence of bounded degree trees such that $T_i$ has $i$ vertices, then $K_n$ has a decomposition into $T_1,\dots, T_n$. This shows that the tree packing conjecture of Gyárfás and Lehel from 1976 holds for all bounded degree trees (in fact, we can allow the first $o(n)$ trees to have arbitrary degrees). Similarly, we show that Ringel's conjecture from 1963 holds for all bounded degree trees. We deduce these results from a more general theorem, which yields decompositions of dense quasi-random graphs into suitable families of bounded degree graphs. Our proofs involve Szemerédi's regularity lemma, results on Hamilton decompositions of robust expanders, random walks, iterative absorption as well as a recent blow-up lemma for approximate decompositions.
Combinatorics
General
3573
3647
10.4171/JEMS/909
https://www.ems-ph.org/doi/10.4171/JEMS/909
8
5
2019
Finite field restriction estimates based on Kakeya maximal operator estimates
Mark
Lewko
New York, USA
Fourier transform, restriction conjecture, Kakeya conjecture, finite fields
In the finite field setting, we show that the restriction conjecture associated to any one of a large family of $d=2n+1$ dimensional quadratic surfaces implies the $n+1$-dimensional Kakeya conjecture (Dvir's theorem). This includes the case of the paraboloid over finite fields in which $−1$ is a square. We are able to partially reverse this implication using the sharp Kakeya maximal operator estimates of Ellenberg, Oberlin and Tao to establish the first finite field restriction estimates beyond the Stein–Tomas exponent in this setting.
Fourier analysis
Number theory
Convex and discrete geometry
3649
3707
10.4171/JEMS/910
https://www.ems-ph.org/doi/10.4171/JEMS/910
8
5
2019
Parabolic implosion for endomorphisms of $\mathbb C^2$
Fabrizio
Bianchi
Imperial College London, UK, and Université de Lille, Villeneuve-d'Ascq, France
Holomorphic dynamics, maps tangent to the identity, parabolic implosion
We give an estimate of the discontinuity of the large Julia set for a perturbation of a class of maps tangent to the identity, by means of a two-dimensional Lavaurs theorem. We adapt to our situation a strategy due to Bedford, Smillie and Ueda in the semiattracting setting. We also prove the discontinuity of the filled Julia set for such perturbations of regular polynomials.
Partial differential equations
Dynamical systems and ergodic theory
3709
3737
10.4171/JEMS/911
https://www.ems-ph.org/doi/10.4171/JEMS/911
8
8
2019
Building-like geometries of finite Morley rank
Isabel
Müller
Universität Münster, Germany
Katrin
Tent
Universität Münster, Germany
Almost strongly minimal structure, 2-ample, Hrushovski construction, building, geometry
For any $n \geq 6$ we construct almost strongly minimal geometries of type $\bullet \overset{n}{-} \bullet \overset{n}{-}\bullet$ which are 2-ample but not 3-ample.
Mathematical logic and foundations
Geometry
3739
3757
10.4171/JEMS/912
https://www.ems-ph.org/doi/10.4171/JEMS/912
8
9
2019
Rational points on log Fano threefolds over a finite field
Yoshinori
Gongyo
University of Tokyo, Japan
Yusuke
Nakamura
University of Tokyo, Japan
Hiromu
Tanaka
University of Tokyo, Japan
Rational points, Fano varieties, Witt rationality, rational chain connectedness
We prove the $W\mathcal{O}$-rationality of klt threefolds and the rational chain connectedness of klt Fano threefolds over a perfect field of characteristic $p > 5$. As a consequence, any klt Fano threefold over a finite field has a rational point.
Algebraic geometry
3759
3795
10.4171/JEMS/913
https://www.ems-ph.org/doi/10.4171/JEMS/913
8
13
2019
Multiple mixing and parabolic divergence in smooth area-preserving flows on higher genus surfaces
Adam
Kanigowski
Penn State University, State College, USA
Joanna
Kułaga-Przymus
Polish Acadamy of Science, Warsaw, and Nicolaus Copernicus University, Torun, Poland
Corinna
Ulcigrai
University of Bristol, UK
Mixing, multiple mixing, area-preserving flows, parabolic divergence, Ratner’s property, special flows, interval exchange transformations, logarithmic singularities
We consider typical area-preserving flows on higher genus surfaces and prove that the flow restricted to mixing minimal components is mixing of all orders, thus answering affirmatively Rokhlin’s multiple mixing question in this context. The main tool is a variation of the Ratner property originally proved by Ratner for the horocycle flow, i.e. the switchable Ratner property introduced by Fayad and Kanigowski for special flows over rotations. This property, which is of independent interest, provides a quantitative description of the parabolic behavior of these flows and has implications for joining classification. The main result is formulated in the language of special flows over interval exchange transformations with asymmetric logarithmic singularities. We also prove a strengthening of one of Fayad and Kanigowski’s main results, by showing that Arnold’s flows are mixing of all orders for almost every location of the singularities.
Dynamical systems and ergodic theory
3797
3855
10.4171/JEMS/914
https://www.ems-ph.org/doi/10.4171/JEMS/914
8
30
2019
Stability versions of Erdős–Ko–Rado type theorems via isoperimetry
David
Ellis
Queen Mary, University of London, UK
Nathan
Keller
Bar-Ilan University, Ramat Gan, Israel
Noam
Lifshitz
The Hebrew University of Jerusalem, Israel
Erdős–Ko–Rado theorem, stability version, Ahlswede–Khachatrian theorem, isoperimetry, Erdős matching conjecture, cross-intersecting families, discrete Fourier analysis
Erdős–Ko–Rado (EKR) type theorems yield upper bounds on the sizes of families of sets, subject to various intersection requirements on the sets in the family. Stability versions of such theorems assert that if the size of a family is close to the maximum possible size, then the family itself must be close (in some appropriate sense) to a maximum-sized family. In this paper, we present an approach to obtaining stability versions of EKR-type theorems, via isoperimetric inequalities for subsets of the hypercube. Our approach is rather general, and allows the leveraging of a wide variety of exact EKR-type results into strong stability versions of these results, without going into the proofs of the original results. We use this approach to obtain tight stability versions of the EKR theorem itself and of the Ahlswede–Khachatrian theorem on $t$-intersecting families of $k$-element subsets of $\{1,\dots,n\}$ (for $k < n/(t+1)$), and to show that, somewhat surprisingly, all these results hold when the ‘intersection’ requirement is replaced by a much weaker requirement. Other examples include stability versions of Frankl’s recent result on the Erdős matching conjecture, the Ellis–Filmus–Friedgut proof of the Simonovits–Sós conjecture, and various EKR-type results on $r$-wise (cross-)$t$-intersecting families.
Combinatorics
General
3857
3902
10.4171/JEMS/915
https://www.ems-ph.org/doi/10.4171/JEMS/915
8
30
2019