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European Mathematical Society Publishing House
2024-03-28 23:34:24
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=17&update_since=2024-03-28
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
17
2015
1
Partial hyperbolicity and homoclinic tangencies
Sylvain
Crovisier
Université Paris-Sud, ORSAY CEDEX, FRANCE
Martin
Sambarino
Universidad de la República, MONTEVIDEO, URUGUAY
Dawei
Yang
Soochow University, SUZHOU, CHINA
Homoclinic tangency, heterodimensional cycle, hyperbolic diffeomorphism, generic dynamics, homoclinic class, partial hyperbolicity
We show that any diffeomorphism of a compact manifold can be $C^1$ approximated by diffeomorphisms exhibiting a homoclinic tangency or by diffeomorphisms having a partial hyperbolic structure.
Dynamical systems and ergodic theory
General
1
49
10.4171/JEMS/497
http://www.ems-ph.org/doi/10.4171/JEMS/497
Kloosterman sums in residue classes
Valentin
Blomer
Georg-August-Universität Göttingen, GÖTTINGEN, GERMANY
Djordje
Milićević
Bryn Mawr College, BRYN MAWR, UNITED STATES
Kloosterman sums, Kuznetsov formula, arithmetic progressions, Linnik's conjecture
We prove upper bounds for sums of Kloosterman sums against general arithmetic weight functions. In particular, we obtain power cancellation in sums of Kloosterman sums over arithmetic progressions, which is of square-root strength in any xed primitive congruence class up to bounds towards the Ramanujan conjecture.
Number theory
General
51
69
10.4171/JEMS/498
http://www.ems-ph.org/doi/10.4171/JEMS/498
Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation
Marcel
Guàrdia
Université Paris 7 Denis Diderot, PARIS CEDEX 13, FRANCE
Vadim
Kaloshin
University of Maryland, COLLEGE PARK, UNITED STATES
Hamiltonian partial differential equations, nonlinear Schrödinger equation, transfer of energy, growth of Sobolev norms, normal forms of Hamiltonian fixed points
We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix $s>1$. Recently Colliander, Keel, Staffilani, Tao and Takaoka proved the existence of solutions with $s$-Sobolev norm growing in time. We establish the existence of solutions with polynomial time estimates. More exactly, there is $c>0$ such that for any $\mathcal K\gg 1$ we find a solution $u$ and a time $T$ such that $\| u(T)\|_{H^s}\geq\mathcal K \| u(0)\|_{H^s}$. Moreover, the time $T$ satisfies the polynomial bound $0
Partial differential equations
Dynamical systems and ergodic theory
71
149
10.4171/JEMS/499
http://www.ems-ph.org/doi/10.4171/JEMS/499
Quantitative spectral gap for thin groups of hyperbolic isometries
Michael
Magee
University of California at Santa Cruz, SANTA CRUZ, UNITED STATES
Spectral gap, thin groups
Let $\Lambda$ be a subgroup of an arithmetic lattice in $\mathrm{SO}(n+1 , 1)$. The quotient $\mathbb{H}^{n+1} / \Lambda$ has a natural family of congruence covers corresponding to ideals in a ring of integers. We establish a super-strong approximation result for Zariski-dense $\Lambda$ with some additional regularity and thickness properties. Concretely, this asserts a quantitative spectral gap for the Laplacian operators on the congruence covers. This generalizes results of Sarnak and Xue (1991) and Gamburd (2002).
Number theory
General
Topological groups, Lie groups
151
187
10.4171/JEMS/500
http://www.ems-ph.org/doi/10.4171/JEMS/500
Stacks of group representations
Paul
Balmer
UCLA, LOS ANGELES, UNITED STATES
Restriction, extension, monad, stack, modular representations, finite group, ring object, descent, endotrivial representation
We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group $G$, the derived and the stable categories of representations of a subgroup $H$ can be constructed out of the corresponding category for $G$ by a purely triangulated-categorical construction, analogous to étale extension in algebraic geometry. In the case of finite groups, we then use descent methods to investigate when modular representations of the subgroup $H$ can be extended to $G$. We show that the presheaves of plain, derived and stable representations all form stacks on the category of finite $G$-sets (or the orbit category of $G$), with respect to a suitable Grothendieck topology that we call the sipp topology. When $H$ contains a Sylow subgroup of~$G$, we use sipp Čech cohomology to describe the kernel and the image of the homomorphism $T(G)\to T(H)$, where $T(-)$ denotes the group of endotrivial representations.
Group theory and generalizations
General
Algebraic geometry
Category theory; homological algebra
189
228
10.4171/JEMS/501
http://www.ems-ph.org/doi/10.4171/JEMS/501
2
Géométries relatives
Thomas
Blossier
Université Lyon 1, VILLEURBANNE CEDEX, FRANCE
Amador
Martin-Pizarro
Université Lyon 1, VILLEURBANNE CEDEX, FRANCE
Frank Olaf
Wagner
Université Lyon 1, VILLEURBANNE CEDEX, FRANCE
Model Theory, amalgamation methods, CM-triviality, geometry, group, definability
In this paper, we shall study type-definable groups in a simple theory with respect to one or several stable reducts. While the original motivation came from the analysis of definable groups in structures obtained by Hrushovski's amalgamation method, the notions introduced are in fact more general, and in particular can be applied to certain expansions of algebraically closed fields by operators.
Mathematical logic and foundations
General
229
258
10.4171/JEMS/502
http://www.ems-ph.org/doi/10.4171/JEMS/502
Cheeger inequalities for unbounded graph Laplacians
Frank
Bauer
Harvard University, CAMBRIDGE, UNITED STATES
Matthias
Keller
The Hebrew University, JERUSALEM, ISRAEL
Radosław
Wojciechowski
York College of The City University of New York, JAMAICA, UNITED STATES
Isoperimetric inequality, intrinsic metric, Schrödinger operators, weighted graphs, curvature, volume growth
We use the concept of intrinsic metrics to give a new definition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if the vertex degrees are unbounded.
Operator theory
Combinatorics
Partial differential equations
Difference and functional equations
259
271
10.4171/JEMS/503
http://www.ems-ph.org/doi/10.4171/JEMS/503
Geometry of non-holonomic diffusion
Simon
Hochgerner
Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND
Tudor
Ratiu
Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND
Non-holonomic system, symmetry, measure, reduction, diffusion, Brownian motion, generator, Chaplygin system, snakeboard, two-wheeled carriage
We study stochastically perturbed non-holonomic systems from a geometric point of view. In this setting, it turns out that the probabilistic properties of the perturbed system are intimately linked to the geometry of the constraint distribution. For $G$-Chaplygin systems, this yields a stochastic criterion for the existence of a smooth preserved measure. As an application of our results we consider the motion planning problem for the noisy two-wheeled robot and the noisy snakeboard.
Dynamical systems and ergodic theory
General
Global analysis, analysis on manifolds
Systems theory; control
273
319
10.4171/JEMS/504
http://www.ems-ph.org/doi/10.4171/JEMS/504
First steps in stable Hamiltonian topology
Kai
Cieliebak
Universität Augsburg, AUGSBURG, GERMANY
Evgeny
Volkov
Universität Augsburg, AUGSBURG, GERMANY
Hamiltonian structure, contact structure, integrable system
In this paper we study topological properties of stable Hamiltonian structures. In particular, we prove the following results in dimension three: The space of stable Hamiltonian structures modulo homotopy is discrete; stable Hamiltonian structures are generically Morse-Bott (i.e. all closed orbits are Bott nondegenerate) but not Morse; the standard contact structure on $S^3$ is homotopic to a stable Hamiltonian structure which cannot be embedded in $\mathbb R^4$. Moreover, we derive a structure theorem for stable Hamiltonian structures in dimension three, study sympectic cobordisms between stable Hamiltonian structures, and discuss implications for the foundations of symplectic field theory.
Differential geometry
General
Dynamical systems and ergodic theory
321
404
10.4171/JEMS/505
http://www.ems-ph.org/doi/10.4171/JEMS/505
Category $\mathcal O$ for quantum groups
Henning Haahr
Andersen
University of Aarhus, AARHUS C, DENMARK
Volodymyr
Mazorchuk
Uppsala Universitet, UPPSALA, SWEDEN
Quantized highest weights modules, specialization at roots of unity, tensor decompositions, tilting modules
In this paper we study the BGG-categories $\mathcal O_q$ associated to quantum groups. We prove that many properties of the ordinary BGG-category $\mathcal O$ for a semisimple complex Lie algebra carry over to the quantum case. Of particular interest is the case when $q$ is a complex root of unity. Here we prove a tensor decomposition for both simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for $\mathcal O$ and for finite dimensional $U_q$-modules we are able to determine all irreducible characters as well as the characters of all indecomposable tilting modules in $\mathcal O_q$. As a consequence of these results we are able to recover also a known result, namely that the generic quantum case behaves like the classical category $\mathcal O$.
Nonassociative rings and algebras
General
Group theory and generalizations
405
431
10.4171/JEMS/506
http://www.ems-ph.org/doi/10.4171/JEMS/506
Robust optimality of Gaussian noise stability
Elchanan
Mossel
University of California, BERKELEY, UNITED STATES
Joe
Neeman
University of Texas at Austin, AUSTIN, UNITED STATES
Gaussian noise sensitivity, isoperimetry, influence, Max-Cut
We prove that under the Gaussian measure, half-spaces are uniquely the most noise stable sets. We also prove a quantitative version of uniqueness, showing that a set which is almost optimally noise stable must be close to a half-space. This extends a theorem of Borell, who proved the same result but without uniqueness, and it also answers a question of Ledoux, who asked whether it was possible to prove Borell’s theorem using a direct semigroup argument. Our quantitative uniqueness result has various applications in diverse fields.
Probability theory and stochastic processes
Real functions
Computer science
433
482
10.4171/JEMS/507
http://www.ems-ph.org/doi/10.4171/JEMS/507
3
Minimality of toric arrangements
Giacomo
d'Antonio
Universität Bremen, BREMEN, GERMANY
Emanuele
Delucchi
Université de Fribourg, FRIBOURG, SWITZERLAND
Toric arrangements, discrete Morse theory, minimal CW-complexes, torsion in cohomology
We prove that the complement of a toric arrangement has the homotopy type of a minimal CW-complex. As a corollary we deduce that the integer cohomology of these spaces is torsionfree. We apply discrete Morse theory to the toric Salvetti complex, providing a sequence of cellular collapses that leads to a minimal complex.
Convex and discrete geometry
Manifolds and cell complexes
483
521
10.4171/JEMS/508
http://www.ems-ph.org/doi/10.4171/JEMS/508
Skolem–Mahler–Lech type theorems and Picard–Vessiot theory
Michael
Wibmer
RWTH Aachen, AACHEN, GERMANY
Linear difference equations, Picard–Vessiot theory, Skolem–Mahler–Lech theorem, dynamical Mordell–Lang conjecture
We show that three problems involving linear difference equations with rational function coefficients are essentially equivalent. The first problem is the generalization of the classical Skolem–Mahler–Lech theorem to rational function coefficients. The second problem is whether or not for a given linear difference equation there exists a Picard–Vessiot extension inside the ring of sequences. The third problem is a certain special case of the dynamical Mordell–Lang conjecture. This allows us to deduce solutions to the first two problems in a particular but fairly general special case.
Field theory and polynomials
General
Number theory
Difference and functional equations
523
533
10.4171/JEMS/509
http://www.ems-ph.org/doi/10.4171/JEMS/509
De Rham cohomology and homotopy Frobenius manifolds
Vladimir
Dotsenko
Trinity College, DUBLIN, IRELAND
Sergey
Shadrin
University of Amsterdam, AMSTERDAM, NETHERLANDS
Bruno
Vallette
Université de Nice Sophia Antipolis, NICE CEDEX 02, FRANCE
De Rham cohomology, homotopy Frobenius manifold, Poisson/Jacobi/contact manifold, multicomplex, Batalin–Vilkovisky algebra
We endow the de Rham cohomology of any Poisson or Jacobi manifold with a natural homotopy Frobenius manifold structure. This result relies on a minimal model theorem for multicomplexes and a new kind of a Hodge degeneration condition.
Global analysis, analysis on manifolds
Algebraic geometry
Category theory; homological algebra
Differential geometry
535
547
10.4171/JEMS/510
http://www.ems-ph.org/doi/10.4171/JEMS/510
Definable orthogonality classes in accessible categories are small
Joan
Bagaria
Universitat de Barcelona, BARCELONA, SPAIN
Carles
Casacuberta
Universitat de Barcelona, BARCELONA, SPAIN
A.R.D.
Mathias
Université de la Réunion, SAINTE CLOTILDE, REUNION (FRENCH)
Jiří
Rosický
Masaryk University, BRNO, CZECH REPUBLIC
Supercompact cardinal, extendible cardinal, Lévy hierarchy, accessible category, reflective subcategory, cohomological localization
We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopěnka's principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class $\mathcal S$ of morphisms in a locally presentable category $\mathcal C$ of structures, the orthogonal class of objects is a small-orthogonality class (hence reflective) can be proved in ZFC if $\mathcal S$ is $\mathbf \Sigma_1$, while it follows from the existence of a proper class of supercompact cardinals if $\mathcal S$ is $\mathbf \Sigma_2$, and from the existence of a proper class of what we call $C(n)$-extendible cardinals if $\mathcal S$ is $\mathbf \Sigma_{n+2}$ for $n \ge 1$. These cardinals form a new hierarchy, and we show that Vopěnka's principle is equivalent to the existence of $C(n)$-extendible cardinals for all $n$. As a consequence of our approach, we prove that the existence of cohomological localizations of simplicial sets, a long-standing open problem in algebraic topology, is implied by the existence of arbitrarily large supercompact cardinals. This follows from the fact that $E^*$-equivalence classes are $\mathbf \Sigma_2$, where $E$ denotes a spectrum treated as a parameter. In contrast with this fact, $E_*$-equivalence classes are $\mathbf \Sigma_1$, from which it follows (as is well known) that the existence of homological localizations is provable in ZFC.
Mathematical logic and foundations
Nonassociative rings and algebras
General topology
549
589
10.4171/JEMS/511
http://www.ems-ph.org/doi/10.4171/JEMS/511
Vector bundles on plane cubic curves and the classical Yang–Baxter equation
Igor
Burban
Universität zu Köln, KÖLN, GERMANY
Thilo
Henrich
Universität Bonn, BONN, GERMANY
Yang–Baxter equations, elliptic fibrations, vector bundles on curves of genus one, derived categories, Massey products
In this article, we develop a geometric method to construct solutions of the classical Yang–Baxter equation, attaching a family of classical r{matrices to the Weierstrass family of plane cubic curves and a pair of coprime positive integers. It turns out that all elliptic $r$-matrices arise in this way from smooth cubic curves. For the cuspidal cubic curve, we prove that the obtained solutions are rational and compute them explicitly. We also describe them in terms of Stolin's classi cation and prove that they are degenerations of the corresponding elliptic solutions.
Associative rings and algebras
Algebraic geometry
Category theory; homological algebra
591
644
10.4171/JEMS/512
http://www.ems-ph.org/doi/10.4171/JEMS/512
On the K-theory of the C*-algebra generated by the left regular representation of an Ore semigroup
Joachim
Cuntz
Universität Münster, MÜNSTER, GERMANY
Siegfried
Echterhoff
Universität Münster, MÜNSTER, GERMANY
Xin
Li
Queen Mary University of London, LONDON, UNITED KINGDOM
K-theory, semigroup C*-algebra, $ax + b$-semigroup, purely infinite
We compute the K-theory of C*-algebras generated by the left regular representation of left Ore semigroups satisfying certain regularity conditions. Our result describes the K-theory of these semigroup C*-algebras in terms of the K-theory for the reduced group C*-algebras of certain groups which are typically easier to handle. Then we apply our result to specific semigroups from algebraic number theory.
Functional analysis
General
Number theory
Group theory and generalizations
645
687
10.4171/JEMS/513
http://www.ems-ph.org/doi/10.4171/JEMS/513
Poincaré inequalities and rigidity for actions on Banach spaces
Piotr
Nowak
, WARSZAWA, POLAND
Poincaré inequality, Kazhdan’s property (T); affine isometric action
The aim of this paper is to extend the framework of the spectral method for proving property (T) to the class of reflexive Banach spaces and present a condition implying that every affine isometric action of a given group $G$ on a reflexive Banach space $X$ has a fixed point. This last property is a strong version of Kazhdan's property (T) and is equivalent to the fact that $H^1(G,\pi)=0$ for every isometric representation $\pi$ of $G$ on $X$. The condition is expressed in terms of $p$-Poincar\'{e} constants and we provide examples of groups, which satisfy such conditions and for which $H^1(G,\pi)$ vanishes for every isometric representation $\pi$ on an $L_p$ space for some $p>2$. Our methods allow to estimate such a $p$ explicitly and yield several interesting applications. In particular, we obtain quantitative estimates for vanishing of 1-cohomology with coefficients in uniformly bounded representations on a Hilbert space. We also give lower bounds on the conformal dimension of the boundary of a hyperbolic group in the Gromov density model.
Topological groups, Lie groups
Functional analysis
689
709
10.4171/JEMS/514
http://www.ems-ph.org/doi/10.4171/JEMS/514
4
Finiteness results for Abelian tree models
Jan
Draisma
Eindhoven University of Technology, EINDHOVEN, NETHERLANDS
Rob
Eggermont
Eindhoven University of Technology, EINDHOVEN, NETHERLANDS
Phylogenetic tree models, tensor rank, noetherianity up to symmetry, applied algebraic geometry
Equivariant tree models are statistical models used in the reconstruction of phylogenetic trees from genetic data. Here equivariant§ refers to a symmetry group imposed on the root distribution and on the transition matrices in the model. We prove that if that symmetry group is Abelian, then the Zariski closures of these models are defined by polynomial equations of bounded degree, independent of the tree. Moreover, we show that there exists a polynomial-time membership test for that Zariski closure. This generalises earlier results on tensors of bounded rank, which correspond to the case where the group is trivial and the tree is a star, and implies a qualitative variant of a quantitative conjecture by Sturmfels and Sullivant in the case where the group and the alphabet coincide. Our proofs exploit the symmetries of an infinite-dimensional projective limit of Abelian star models.
Commutative rings and algebras
Algebraic geometry
Linear and multilinear algebra; matrix theory
Statistics
711
738
10.4171/JEMS/515
http://www.ems-ph.org/doi/10.4171/JEMS/515
Optimal bounds for the colored Tverberg problem
Pavle
Blagojević
Freie Universität Berlin, BERLIN, GERMANY
Benjamin
Matschke
, BONN, GERMANY
Günter
Ziegler
Freie Universität Berlin, BERLIN, GERMANY
Optimal colored Tverberg theorem, Bárány–Larman conjecture, equivariant obstruction theory, chessboard complexes
We prove a “Tverberg type” multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Bárány et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of B´ar´any & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory.
Convex and discrete geometry
Algebraic topology
739
754
10.4171/JEMS/516
http://www.ems-ph.org/doi/10.4171/JEMS/516
1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs
Sebastian
Hensel
University of Chicago, CHICAGO, UNITED STATES
Piotr
Przytycki
McGill University, MONTREAL, CANADA
Richard
Webb
University College London, LONDON, UNITED KINGDOM
Gromov hyperbolic, slim triangle, curve graph, arc graph, unicorn
We describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.
Group theory and generalizations
General
755
762
10.4171/JEMS/517
http://www.ems-ph.org/doi/10.4171/JEMS/517
Strong density for higher order Sobolev spaces into compact manifolds
Pierre
Bousquet
Université de Toulouse, TOULOUSE CEDEX 9, FRANCE
Augusto
Ponce
Université Catholique de Louvain, LOUVAIN-LA-NEUVE, FRANCE
Jean
Van Schaftingen
Université Catholique de Louvain, LOUVAIN-LA-NEUVE, BELGIUM
Strong density, Sobolev maps, higher order Sobolev spaces, homotopy, topological singularity
Given a compact manifold $N^n$, an integer $k \in \mathbb{N}_*$ and an exponent $1 \le p < \infty$, we prove that the class $C^\infty(\overline{Q}^m; N^n)$ of smooth maps on the cube with values into $N^n$ is dense with respect to the strong topology in the Sobolev space $W^{k, p}(Q^m; N^n)$ when the homotopy group $\pi_{\lfloor kp \rfloor}(N^n)$ of order $\lfloor kp \rfloor$ is trivial. We also prove density of maps that are smooth except for a set of dimension $m - \lfloor kp \rfloor - 1$, without any restriction on the homotopy group of $N^n$.
Global analysis, analysis on manifolds
Functional analysis
763
817
10.4171/JEMS/518
http://www.ems-ph.org/doi/10.4171/JEMS/518
Trudinger–Moser inequality on the whole plane with the exact growth condition
Slim
Ibrahim
University of Victoria, VICTORIA, B.C., CANADA
Nader
Masmoudi
New York University, NEW YORK, UNITED STATES
Kenji
Nakanishi
Kyoto University, KYOTO, JAPAN
Sobolev critical exponent, Trudinger-Moser inequality, concentration compactness, nonlinear Schrödinger equation, ground state
Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to $L^{\infty}$. It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modi ed versions are available. We prove a precised version of the latter, giving necessary and sufficient conditions for the boundedness, as well as for the compactness, in terms of the growth and decay of the nonlinear function. It is tightly related to the ground state of the nonlinear Schrödinger equation (or the nonlinear Klein-Gordon equation), for which the range of the time phase (or the mass constant) as well as the energy is given by the best constant of the inequality.
Partial differential equations
General
Functional analysis
819
835
10.4171/JEMS/519
http://www.ems-ph.org/doi/10.4171/JEMS/519
Matrix coefficients, counting and primes for orbits of geometrically finite groups
Amir
Mohammadi
The University of Texas at Austin, AUSTIN, UNITED STATES
Hee
Oh
Yale University, NEW HAVEN, UNITED STATES
Geometrically finite group, matrix coefficients, mixing, sieve, spectral gap, equidistribution
Let $G:=\mathrm {SO}(n,1)^\circ$ and $\Gamma(n-1)/2$ for $n=2,3$ and when $\delta>n-2$ for $n\geq 4$, we obtain an effective archimedean counting result for a discrete orbit of $\Gamma$ in a homogeneous space $H \backslash G$ where $H$ is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family $\{\mathcal B_T\subset H \backslash G \}$ of compact subsets, there exists $\eta>0$ such that $$\#[e]\Gamma\cap \mathcal B_T=\mathcal M(\mathcal B_T) +O(\mathcal M(\mathcal B_T)^{1-\eta})$$ for an explicit measure $\mathcal M$ on $H\backslash G$ which depends on $\Gamma$. We also apply the affine sieve and describe the distribution of almost primes on orbits of $\Gamma$ in arithmetic settings. One of key ingredients in our approach is an effective asymptotic formula for the matrix coefficients of $L^2(\Gamma \backslash G)$ that we prove by combining methods from spectral analysis, harmonic analysis and ergodic theory. We also prove exponential mixing of the frame flows with respect to the Bowen-Margulis-Sullivan measure.
Number theory
Category theory; homological algebra
$K$-theory
Ordinary differential equations
837
897
10.4171/JEMS/520
http://www.ems-ph.org/doi/10.4171/JEMS/520
Witten's Conjecture for many four-manifolds of simple type
Paul
Feehan
Rutgers University, PISCATAWAY, UNITED STATES
Thomas
Leness
Florida International University, MIAMI, UNITED STATES
Cobordisms, Donaldson invariants, Seiberg-Witten invariants, smooth four-dimensional manifolds, $\SO(3)$ monopoles, Yang-Mills gauge theory
We prove that Witten's Conjecture [40] on the relationship between the Donaldson and Seiberg-Witten series for a four-manifold of Seiberg-Witten simple type with $b_1=0$ and odd $b_2^+\ge 3$ follows from our $\SO(3)$-monopole cobordism formula [6] when the four-manifold has $c_1^2\ge \chi_h-3$ or is abundant.
Manifolds and cell complexes
Global analysis, analysis on manifolds
899
923
10.4171/JEMS/521
http://www.ems-ph.org/doi/10.4171/JEMS/521
Counting arithmetic subgroups and subgroup growth of virtually free groups
Amichai
Eisenmann
The Hebrew University of Jerusalem, JERUSALEM, ISRAEL
Arithmetic subgroups, counting lattices, subgroup growth, virtually free groups
Let $K$ be a $p$-adic field, and let $H=PSL_2(K)$ endowed with the Haar measure determined by giving a maximal compact subgroup measure $1$. Let $AL_H(x)$ denote the number of conjugacy classes of arithmetic lattices in $H$ with co-volume bounded by $x$. We show that under the assumption that $K$ does not contain the element $\zeta+\zeta^{-1}$, where $\zeta$ denotes the $p$-th root of unity over $\mathbb{Q}_p$, we have $$\lim_{x\rightarrow\infty}\frac{\log AL_H(x)}{x\log x}=q-1$$ where $q$ denotes the order of the residue field of $K$. The main innovation of this paper is the proof of a sharp bound on subgroup growth of lattices in $H$ as above.
Topological groups, Lie groups
Group theory and generalizations
925
953
10.4171/JEMS/522
http://www.ems-ph.org/doi/10.4171/JEMS/522
Quantization commutes with reduction in the non-compact setting: the case of holomorphic discrete series
Paul-Emile
Paradan
Université Montpellier II, MONTPELLIER CEDEX, FRANCE
Holomorphic discrete series, moment map, reduction, geometric quantization, transversally elliptic symbol
In this paper we show that the multiplicities of holomorphic discrete series representations relative to reductive subgroups satisfy the credo “quantization commutes with reduction”.
Global analysis, analysis on manifolds
General
$K$-theory
Manifolds and cell complexes
955
990
10.4171/JEMS/523
http://www.ems-ph.org/doi/10.4171/JEMS/523
Towards the Jacquet conjecture on the Local Converse Problem for $p$-adic $\mathrm {GL}_n$
Dihua
Jiang
University of Minnesota, MINNEAPOLIS, UNITED STATES
Chufeng
Nien
National Cheng Kung University, TAINAN, TAIWAN
Shaun
Stevens
University of East Anglia, NORWICH, UNITED KINGDOM
Irreducible admissible representation, Whittaker model, local gamma factor, local converse theorem
The Local Converse Problem is to determine how the family of the local gamma factors $\gamma(s,\pi\times\tau,\psi)$ characterizes the isomorphism class of an irreducible admissible generic representation $\pi$ of $\mathrm {GL}_n(F)$, with $F$ a non-archimedean local field, where $\tau$ runs through all irreducible supercuspidal representations of $\mathrm {GL}_r(F)$ and $r$ runs through positive integers. The Jacquet conjecture asserts that it is enough to take $r=1,2,\ldots,\left[\frac{n}{2}\right]$. Based on arguments in the work of Henniart and of Chen giving preliminary steps towards the Jacquet conjecture, we formulate a general approach to prove the Jacquet conjecture. With this approach, the Jacquet conjecture is proved under an assumption which is then verified in several cases, including the case of level zero representations.
Number theory
General
Topological groups, Lie groups
991
1007
10.4171/JEMS/524
http://www.ems-ph.org/doi/10.4171/JEMS/524
Calculus of variations with differential forms
Saugata
Bandyopadhyay
IISER Kolkata, MOHANPUR, INDIA
Bernard
Dacorogna
Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND
Swarnendu
Sil
Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND
Calculus of variations, differential forms, quasiconvexity, polyconvexity and ext. one convexity
We study integrals of the form $\int_{\Omega}f\left( d\omega\right)$, where $1\leq k\leq n$, $f:\Lambda^{k}\rightarrow\mathbb{R}$ is continuous and $\omega$ is a $\left(k-1\right)$-form. We introduce the appropriate notions of convexity, namely ext. one convexity, ext. quasiconvexity and ext. polyconvexity. We study their relations, give several examples and counterexamples. We finally conclude with an application to a minimization problem.
Calculus of variations and optimal control; optimization
1009
1039
10.4171/JEMS/525
http://www.ems-ph.org/doi/10.4171/JEMS/525
5
Sharp isoperimetric inequalities and model spaces for the Curvature-Dimension-Diameter condition
Emanuel
Milman
Technion - Israel Institute of Technology, HAIFA, ISRAEL
Isoperimetric inequality, generalized Ricci tensor, manifold with density, geodesically convex, model space
We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex support are bounded from above (possibly infinitely). Our inequalities are sharp for sets of any given measure and with respect to all parameters (curvature, dimension and diameter). Moreover, for each choice of parameters, we identify the model spaces which are extremal for the isoperimetric problem. In particular, we recover the Gromov–Lévy and Bakry–Ledoux isoperimetric inequalities, which state that whenever the curvature is strictly positively bounded from below, these model spaces are the $n$-sphere and Gauss space, corresponding to generalized dimension being $n$ and $\infty$, respectively. In all other cases, which seem new even for the classical Riemannian-volume measure, it turns out that there is no single model space to compare to, and that a simultaneous comparison to a natural one-parameter family of model spaces is required, nevertheless yielding a sharp result.
Several complex variables and analytic spaces
Differential geometry
1041
1078
10.4171/JEMS/526
http://www.ems-ph.org/doi/10.4171/JEMS/526
Quantitative stability for sumsets in $\mathbb R^n$
Alessio
Figalli
University of Texas at Austin, AUSTIN, UNITED STATES
David
Jerison
Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES
Quantitative stability, sumsets, Freiman’s theorem
Given a measurable set $A\subset \mathbb R^n$ of positive measure, it is not difficult to show that $|A+A|=|2A|$ if and only if $A$ is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If $(|A+A|-|2A|)/|A|$ is small, is $A$ close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between $A$ and its convex hull in terms of $(|A+A|-|2A|)/|A|$.
Calculus of variations and optimal control; optimization
Ordinary differential equations
1079
1106
10.4171/JEMS/527
http://www.ems-ph.org/doi/10.4171/JEMS/527
Invariance of the Gibbs measure for the Benjamin–Ono equation
Yu
Deng
Princeton University, PRINCETON, UNITED STATES
Benjamin–Ono equation, Gibbs measure, measure invariance, global well-posedness
In this paper we consider the periodic Benjemin-Ono equation.We establish the invariance of the Gibbs measure associated to this equation, thus answering a question raised in Tzvetkov [28]. As an intermediate step, we also obtain a local well-posedness result in Besov-type spaces rougher than $L^2$, extending the $L^2$ well-posedness result of Molinet [20].
Dynamical systems and ergodic theory
Partial differential equations
1107
1198
10.4171/JEMS/528
http://www.ems-ph.org/doi/10.4171/JEMS/528
Front propagation for nonlinear diffusion equations on the hyperbolic space
Hiroshi
Matano
University of Tokyo, TOKYO, JAPAN
Fabio
Punzo
Università degli Studi di Milano, MILANO, ITALY
Alberto
Tesei
Università di Roma La Sapienza, ROMA, ITALY
Semilinear parabolic equations, hyperbolic space, extinction and propagation, asymptotical symmetry of solutions, horospheric waves
We study the Cauchy problem in the hyperbolic space $\mathbb{H}^n (n\ge2)$ for the semilinear heat equation with forcing term, which is either of KPP type or of Allen-Cahn type. Propagation and extinction of solutions, asymptotical speed of propagation and asymptotical symmetry of solutions are addressed. With respect to the corresponding problem in the Euclidean space $\mathbb R^n$ new phenomena arise, which depend on the properties of the diffusion process in $ \mathbb{H}^n$. We also investigate a family of travelling wave solutions, named horospheric waves, which have properties similar to those of plane waves in $\mathbb R^n$.
Partial differential equations
1199
1227
10.4171/JEMS/529
http://www.ems-ph.org/doi/10.4171/JEMS/529
Weakly regular $T^2$-symmetric spacetimes. The global geometry of future Cauchy developments
Philippe
LeFloch
Université Pierre et Marie Curie, PARIS, FRANCE
Jacques
Smulevici
Université Paris-Sud, ORSAY CEDEX, FRANCE
Einstein equations, $T^2$ symmetry, vacuum spacetime, weakly regular, energy space, global geometry
We provide a geometric well-posedness theory for the Einstein equations within the class of weakly regular vacuum spacetimes with $T^2$-symmetry, as defined in the present paper, and we investigate their global causal structure. Our assumptions allow us to give a meaning to the Einstein equations under weak regularity as well as to solve the initial value problem under the assumed symmetry. First, introducing a frame adapted to the symmetry and identifying certain cancellation properties taking place in the standard expressions of the connection and the curvature, we formulate the initial value problem for the Einstein field equations under the proposed weak regularity assumptions. Second, considering the Cauchy development of any weakly regular initial data set and denoting by $R$ the area of the orbits of symmetry, we establish the existence of a global foliation by the level sets of $R$ such that $R$ grows to infinity in the future direction. Our weak regularity assumptions only require that $R$ is Lipschitz continuous while the metric coefficients describing the initial geometry of the symmetry orbits are in the Sobolev space $H^1$ and the remaining coefficients have even weaker regularity.
Relativity and gravitational theory
Partial differential equations
1229
1292
10.4171/JEMS/530
http://www.ems-ph.org/doi/10.4171/JEMS/530
6
Regularity of Lipschitz free boundaries for the thin one-phase problem
Daniela
De Silva
Columbia University, NEW YORK, UNITED STATES
Ovidiu
Savin
Columbia University, NEW YORK, UNITED STATES
Energy minimizers, one-phase free boundary problem, monotonicity formula
We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional $$E(u,\Omega) = \int_\Omega |\nabla u|^2 dX + \mathcal{H}^n(\{u>0\} \cap \{x_{n+1} = 0\}), \quad \Omega \subset \mathbb R^{n+1},$$ among all functions $u\ge 0$ which are fixed on $\partial \Omega$.
Partial differential equations
1293
1326
10.4171/JEMS/531
http://www.ems-ph.org/doi/10.4171/JEMS/531
Cloaking via anomalous localized resonance for doubly complementary media in the quasistatic regime
Hoai-Minh
Nguyen
EPFL SB CAMA, LAUSANNE, SWITZERLAND
Cloaking, anomalous localized resonance, negative index materials, complementary media
This paper is devoted to the study of cloaking via anomalous localized resonance (CALR) in the two- and three-dimensional quasistatic regimes. CALR associated with negative index materials was discovered by Milton and Nicorovici [21] for constant plasmonic structures in the two-dimensional quasistatic regime. Two key features of this phenomenon are the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others, and the connection between the localized resonance and the blow up of the power of the fields as the loss of the material goes to 0. An important class of negative index materials for which the localized resonance might appear is the class of reflecting complementary media introduced in [24]. It was shown in [29] that the complementarity property is not enough to ensure a connection between the blow up of the power and the localized resonance. In this paper, we study CALR for a subclass of complementary media called doubly complementary media. This class is rich enough to allow us to cloak an arbitrary source concentrating on an arbitrary smooth bounded manifold of codimension 1 placed in an arbitrary medium via anomalous localized resonance; the cloak is independent of the source. The following three properties are established for doubly complementary media: P1. CALR appears if and only if the power blows up; P2. The power blows up if the source is located "near” the plasmonic structure; P3. The power remains bounded if the source is far away from the plasmonic structure. Property P2, the blow up of the power, is in fact established for reflecting complementary media. The proofs are based on several new observations and ideas. One of the difficulties is to handle the localized resonance. To this end, we extend the reflecting and removing localized singularity techniques introduced in [24–26], and implement the separation of variables for Cauchy problems for a general shell. The results in this paper are inspired by and imply recent ones of Ammari et al. [3] and Kohn et al. [16] in two dimensions and extend theirs to general non-radial core-shell structures in both two and three dimensions.
Partial differential equations
Optics, electromagnetic theory
1327
1365
10.4171/JEMS/532
http://www.ems-ph.org/doi/10.4171/JEMS/532
Expansion in finite simple groups of Lie type
Emmanuel
Breuillard
Université Paris-Sud 11, ORSAY CEDEX, FRANCE
Ben
Green
University of Oxford, OXFORD, UNITED KINGDOM
Robert
Guralnick
University of Southern California, LOS ANGELES, UNITED STATES
Terence
Tao
University of California Los Angeles, LOS ANGELES, UNITED STATES
Finite simple groups, expander graphs, product theorems
We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the Bourgain-Gamburd method and on the main result of our companion paper [BGGT].
Group theory and generalizations
1367
1434
10.4171/JEMS/533
http://www.ems-ph.org/doi/10.4171/JEMS/533
Absolute continuity, Lyapunov exponents and rigidity I: geodesic flows
Artur
Avila
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
Marcelo
Viana
, RIO DE JANEIRO, BRAZIL
Amie
Wilkinson
University of Chicago, CHICAGO, UNITED STATES
Lyapunov exponent, geodesic flow, partial hyperbolicity, disintegration, absolute continuity, rigidity
We consider volume-preserving perturbations of the time-one map of the geodesic flow of a compact surface with negative curvature. We show that if the Liouville measure has Lebesgue disintegration along the center foliation then the perturbation is itself the time-one map of a smooth volume-preserving flow, and that otherwise the disintegration is necessarily atomic.
Dynamical systems and ergodic theory
1435
1462
10.4171/JEMS/534
http://www.ems-ph.org/doi/10.4171/JEMS/534
A variational analysis of a gauged nonlinear Schrödinger equation
Alessio
Pomponio
Politecnico di Bari, BARI, ITALY
David
Ruiz
Universidad de Granada, GRANADA, SPAIN
Gauged Schrödinger equations, Chern-Simons theory, variational methods, concentration compactness
This paper is motivated by a gauged Schrödinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: $$ - \Delta u(x) + \left( \omega + \frac{h^2(|x|)}{|x|^2} + \int_{|x|}^{+\infty} \frac{h(s)}{s} u^2(s)\, ds \right) u(x) = |u(x)|^{p-1}u(x),$$ where $$ h(r)= \frac{1}{2}\int_0^{r} s u^2(s) \, ds.$$ This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for $p\in(1,3)$, the functional may be bounded from below or not, depending on $\omega $. Quite surprisingly, the threshold value for $\omega $ is explicit. From this study we prove existence and non-existence of positive solutions.
Partial differential equations
1463
1486
10.4171/JEMS/535
http://www.ems-ph.org/doi/10.4171/JEMS/535
On the motion of a curve by its binormal curvature
Robert
Jerrard
University of Toronto, TORONTO, ONTARIO, CANADA
Didier
Smets
UPMC, Université Paris 06,, PARIS CEDEX 05, FRANCE
Binormal curvature flow, integral current, oriented varifold
We propose a weak formulation for the binormal curvature flow of curves in $\mathbb R^3$. This formulation is sufficiently broad to consider integral currents as initial data, and sufficiently strong for the weak-strong uniqueness property to hold, as long as self-intersections do not occur. We also prove a global existence theorem in that framework.
Differential geometry
Fluid mechanics
1487
1515
10.4171/JEMS/536
http://www.ems-ph.org/doi/10.4171/JEMS/536
GCD sums from Poisson integrals and systems of dilated functions
Christoph
Aistleitner
Technische Universität Graz, GRAZ, AUSTRIA
István
Berkes
Technische Universität Graz, GRAZ, AUSTRIA
Kristian
Seip
University of Trondheim, TRONDHEIM, NORWAY
GCD sums and matrices, Carleson–Hunt inequality, Poisson integral, polydisc, spectral norm, convergence of series of dilated functions
Upper bounds for GCD sums of the form $$\sum_{k,{\ell}=1}^N\frac{(\mathrm {gcd}(n_k,n_{\ell}))^{2\alpha}}{(n_k n_{\ell})^\alpha}$$ are established, where $(n_k)_{1 \leq k \leq N}$ is any sequence of distinct positive integers and $0
Number theory
General
1517
1546
10.4171/JEMS/537
http://www.ems-ph.org/doi/10.4171/JEMS/537
7
A maximum principle for systems with variational structure and an application to standing waves
Nicholas
Alikakos
University of Athens, ATHENS, GREECE
Giorgio
Fusco
Università degli Studi dell'Aquila, L'AQUILA, ITALY
Vector Allen–Cahn equation, standing waves, periodic domains, maximum principle for (vector) minimizers
We establish via variational methods the existence of a standing wave together with an estimate on the convergence to its asymptotic states for a bistable system of partial differential equations on a periodic domain. The main tool is a replacement lemma which has as a corollary a maximum principle for minimizers.
Partial differential equations
1547
1567
10.4171/JEMS/538
http://www.ems-ph.org/doi/10.4171/JEMS/538
Determinantal Barlow surfaces and phantom categories
Christian
Böhning
Universität Hamburg, HAMBURG, GERMANY
Hans-Christian
Graf von Bothmer
Universität Hamburg, HAMBURG, GERMANY
Ludmil
Katzarkov
University of Miami, CORAL GABLES, UNITED STATES
Pawel
Sosna
Universität Hamburg, HAMBURG, GERMANY
Derived categories, exceptional collections, semiorthogonal decompositions, Hochschild homology, Barlow surfaces
We prove that the bounded derived category of the surface $S$ constructed by Barlow admits a length 11 exceptional sequence consisting of (explicit) line bundles. Moreover, we show that in a small neighbourhood of $S$ in the moduli space of determinantal Barlow surfaces, the generic surface has a semiorthogonal decomposition of its derived category into a length 11 exceptional sequence of line bundles and a category with trivial Grothendieck group and Hochschild homology, called a phantom category. This is done using a deformation argument and the fact that the derived endomorphism algebra of the sequence is constant. Applying Kuznetsov’s results on heights of exceptional sequences, we also show that the sequence on $S$ itself is not full and its (left or right) orthogonal complement is also a phantom category.
Algebraic geometry
Category theory; homological algebra
1569
1592
10.4171/JEMS/539
http://www.ems-ph.org/doi/10.4171/JEMS/539
Crystal bases for the quantum queer superalgebra
Dimitar
Grantcharov
University of Texas at Arlington, ARLINGTON, UNITED STATES
Ji Hye
Jung
Seoul National University, SEOUL, SOUTH KOREA
Seok-Jin
Kang
Seoul National University, SEOUL, SOUTH KOREA
Masaki
Kashiwara
Kyoto University, KYOTO, JAPAN
Myungho
Kim
Seoul National University, SEOUL, SOUTH KOREA
Quantum queer superalgebras, crystal bases, odd Kashiwara operators
In this paper, we develop the crystal basis theory for the quantum queer superalgebra $U_q(\mathfrak{q}(n))$. We define the notion of crystal bases and prove the tensor product rule for $U_q(\mathfrak{q}(n))$-modules in the category $\mathcal{O}^{\ge0}_{\mathrm {int}}$. Our main theorem shows that every $U_q(\mathfrak{q}(n))$-module in the category $\mathcal{O}^{\ge0}_{\mathrm {int}}$ has a unique crystal basis.
Nonassociative rings and algebras
General
Quantum theory
1593
1627
10.4171/JEMS/540
http://www.ems-ph.org/doi/10.4171/JEMS/540
Isometries of quadratic spaces
Eva
Bayer-Fluckiger
MA C3 635 (Batiment MA), LAUSANNE, SWITZERLAND
Quadratic space, isometry, orthogonal group, minimal polynomial
Let $k$ be a global field of characteristic not 2, and let $f \in k[X]$ be an irreducible polynomial. We show that a non-degenerate quadratic space has an isometry with minimal polynomial $f$ if and only if such an isometry exists over all the completions of $k$. This gives a partial answer to a question of Milnor.
Number theory
1629
1656
10.4171/JEMS/541
http://www.ems-ph.org/doi/10.4171/JEMS/541
The discriminant and oscillation lengths for contact and Legendrian isotopies
Vincent
Colin
Université de Nantes, NANTES, FRANCE
Sheila
Sandon
Université de Strasbourg, STRASBOURG, FRANCE
Bi-invariant metrics, contactomorphism group, discriminant and translated points of contactomorphisms, Legendrian isotopies, orderability of contact manifolds, generating functions
We define an integer-valued non-degenerate bi-invariant metric (the \emph{discriminant metric}) on the universal cover of the identity component of the contactomorphism group of any contact manifold. This metric has a very simple geometric definition, based on the notion of discriminant points of contactomorphisms. Using generating functions we prove that the discriminant metric is unbounded for the standard contact structures on $\mathbb{R}^{2n}\times S^1$ and $\mathbb{R}P^{2n+1}$. On the other hand we also show by elementary arguments that the discriminant metric is bounded for the standard contact structures on $\mathbb{R}^{2n+1}$ and $S^{2n+1}$. As an application of these results we get that the contact fragmentation norm is unbounded for $\mathbb{R}^{2n}\times S^1$ and $\mathbb{R}P^{2n+1}$. By elaborating on the construction of the discriminant metric we then define an integer-valued bi-invariant pseudo-metric, that we call the \emph{oscillation pseudo-metric}, which is non-degenerate if and only if the contact manifold is orderable in the sense of Eliashberg and Polterovich and, in this case, it is compatible with the partial order. Finally we define the discriminant and oscillation lengths of a Legendrian isotopy, and prove that they are unbounded for $T^{\ast}B\times S^1$ for any closed manifold $B$, for $\mathbb{R}P^{2n+1}$ and for some $3$-dimensional circle bundles.
Differential geometry
General
1657
1685
10.4171/JEMS/542
http://www.ems-ph.org/doi/10.4171/JEMS/542
Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space
Andrea
Nahmod
University of Massachusetts, AMHERST, UNITED STATES
Gigliola
Staffilani
Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES
Supercritical nonlinear Schrödinger equation, almost sure well-posedness, random data
In this paper we prove an almost sure local well-posedness result for the periodic 3D quintic nonlinear Schrödinger equation in the supercritical regime, that is below the critical space $H^1(\mathbb T^3)$. We also prove a long time existence result; more precisely we prove that for fixed $T>0$ there exists a set $\Sigma_T$, $\mathbb P(\Sigma_T) > 0$ such that any data $\phi^{\omega}(x) \in H^{\gamma}(\mathbb T^3), \gamma1$. In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space $H^1(\mathbb T^3)$, that is in the supercritical scaling regime.
Partial differential equations
Dynamical systems and ergodic theory
1687
1759
10.4171/JEMS/543
http://www.ems-ph.org/doi/10.4171/JEMS/543
A descriptive view of unitary group representations
Simon
Thomas
Rutgers University, PISCATAWAY, UNITED STATES
Unitary dual, Borel equivalence relation, amenable group
In this paper, we will study the relative complexity of the unitary duals of countable groups. In particular, we will explain that if $G$ and $H$ are countable amenable non-type I groups, then the unitary duals of $G$ and $H$ are Borel isomorphic.
Topological groups, Lie groups
Mathematical logic and foundations
Dynamical systems and ergodic theory
Abstract harmonic analysis
1761
1787
10.4171/JEMS/544
http://www.ems-ph.org/doi/10.4171/JEMS/544
An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise
Viorel
Barbu
Romanian Academy, IASI, ROMANIA
Michael
Röckner
Universität Bielefeld, BIELEFELD, GERMANY
Maximal monotone operator, stochastic integral, operatorial equations
In this paper, we develop a new general approach to the existence and uniqueness theory of infinite dimensional stochastic equations of the form $$dX+A(t)X dt=X dW \mbox \mathrm {in} (0,T)\times H,$$ where $A(t)$ is a nonlinear monotone and demicontinuous operator from $V$ to $V'$, coercive and with polynomial growth. Here, $V$ is a reflexive Banach space continuously and densely embedded in a Hilbert space $H$ of (generalized) functions on a domain $\mathcal O\subset\mathbb R^d$ and $V'$ is the dual of $V$ in the duality induced by $H$ as pivot space. Furthermore, $W$ is a Wiener process in $H$. The new approach is based on an operatorial reformulation of the stochastic equation which is quite robust under perturbation of $A(t)$. This leads to new existence and uniqueness results of a larger class of equations with linear multiplicative noise than the one treatable by the known approaches. In addition, we obtain regularity results for the solutions with respect to both the time and spatial variable which are sharper than the classical ones. New applications include stochastic partial differential equations, as e.g. stochastic transport equations.
Probability theory and stochastic processes
Operator theory
1789
1815
10.4171/JEMS/545
http://www.ems-ph.org/doi/10.4171/JEMS/545
8
On the duality between $p$-modulus and probability measures
Luigi
Ambrosio
Scuola Normale Superiore, PISA, ITALY
Simone
Di Marino
Scuola Normale Superiore, PISA, ITALY
Giuseppe
Savaré
Università di Pavia, PAVIA, ITALY
$p$-Modulus, capacity, duality, Sobolev functions
Motivated by recent developments on calculus in metric measure spaces $(X,\mathrm d,\mathrm m)$, we prove a general duality principle between Fuglede's notion [15] of $p$-modulus for families of finite Borel measures in $(X,\mathrm d)$ and probability measures with barycenter in $L^q(X,\mathrm m)$, with $q$ dual exponent of $p\in (1,\infty)$. We apply this general duality principle to study null sets for families of parametric and non-parametric curves in $X$. In the final part of the paper we provide a new proof, independent of optimal transportation, of the equivalence of notions of weak upper gradient based on $p$-podulus [21, 23] and suitable probability measures in the space of curves ([6, 7]).
Measure and integration
Functional analysis
Calculus of variations and optimal control; optimization
1817
1853
10.4171/JEMS/546
http://www.ems-ph.org/doi/10.4171/JEMS/546
Blow up for the critical gKdV equation. II: Minimal mass dynamics
Yvan
Martel
École Polytechnique, PALAISEAU CEDEX, FRANCE
Frank
Merle
Université de Cergy-Pontoise, CERGY-PONTOISE CEDEX, FRANCE
Pierre
Raphaël
Université de Nice Sophia Antipolis, NICE CEDEX 02, FRANCE
Generalized Korteweg–de Vries equation, blow up, minimal mass solution, uniqueness of threshold solution
We consider the mass critical (gKdV) equation $u_t + (u_{xx} + u^5)_x =0$ for initial data in $H^1$. We first prove the existence and uniqueness in the energy space of a minimal mass blow up solution and give a sharp description of the corresponding blow up soliton-like bubble. We then show that this solution is the universal attractor of all solutions near the ground state which have a defocusing behavior. This allows us to sharpen the description of near soliton dynamics obtained in [29].
Partial differential equations
1855
1925
10.4171/JEMS/547
http://www.ems-ph.org/doi/10.4171/JEMS/547
Gap universality of generalized Wigner and $\beta$-ensembles
László
Erdős
Institute of Scienceand Technology Austria, KLOSTERNEUBURG, AUSTRIA
Horng-Tzer
Yau
Harvard University, CAMBRIDGE, UNITED STATES
$\beta$-ensembles, Wigner-Dyson-Gaudin-Mehta universality, gap distribution, log-gas
We consider generalized Wigner ensembles and general $\beta$-ensembles with analytic potentials for any $\beta \ge 1 $. The recent universality results in particular assert that the local averages of consecutive eigenvalue gaps in the bulk of the spectrum are universal in the sense that they coincide with those of the corresponding Gaussian $\beta$-ensembles. In this article, we show that local averaging is not necessary for this result, i.e. we prove that the single gap distributions in the bulk are universal. In fact, with an additional step, our result can be extended to any potential $C^4(\mathbb R)$.
Linear and multilinear algebra; matrix theory
Statistical mechanics, structure of matter
1927
2036
10.4171/JEMS/548
http://www.ems-ph.org/doi/10.4171/JEMS/548
Asymptotic analysis and sign-changing bubble towers for Lane–Emden problems
Francesca
De Marchis
Università di Roma Tor Vergata, ROMA, ITALY
Isabella
Ianni
Seconda Università di Napoli, CASERTA, ITALY
Filomena
Pacella
Università di Roma La Sapienza, ROMA, ITALY
Superlinear elliptic boundary value problems, sign-changing solutions, asymptotic analysis, bubble towers
We consider the semilinear Lane–Emden problem \begin{equation}\label{problemAbstract}\left\{\begin{array}{lr}-\Delta u= |u|^{p-1}u\qquad \mbox{ in }\Omega\\ u=0\qquad\qquad\qquad\mbox{ on }\partial \Omega \end{array}\right.\tag{$\mathcal E_p$} \end{equation} where $p>1$ and $\Omega$ is a smooth bounded domain of $\mathbb R^2$. The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of \eqref{problemAbstract}, as $p\to+\infty$. Among other results we show, under some symmetry assumptions on $\Omega$, that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as $p\to+\infty$, and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of the Liouville problem in $\mathbb R^2$.
Partial differential equations
2037
2068
10.4171/JEMS/549
http://www.ems-ph.org/doi/10.4171/JEMS/549
Hybrid sup-norm bounds for Hecke–Maass cusp forms
Nicolas
Templier
Princeton University, PRINCETON, UNITED STATES
Automorphic forms, trace formula, amplification, diophantine approximation
Let $f$ be a Hecke–Maass cusp form of eigenvalue $\lambda$ and square-free level $N$. Normalize the hyperbolic measure such that $\mathrm {vol}(Y_0(N))=1$ and the form $f$ such that $\|{f}\|_2=1$. It is shown that $\|{f}\|_\infty \ll_\epsilon \lambda^{\frac{5}{24}+\epsilon} N^{\frac{1}{3}+\epsilon}$ for all $\epsilon>0$. This generalizes simultaneously the current best bounds in the eigenvalue and level aspects.
Number theory
General
Algebraic geometry
2069
2082
10.4171/JEMS/550
http://www.ems-ph.org/doi/10.4171/JEMS/550
9
A new function space and applications
Jean
Bourgain
Institute for Advanced Study, PRINCETON, UNITED STATES
Haim
Brezis
Rutgers University, Hill Center, Busch Campus, PISCATAWAY, UNITED STATES
Petru
Mironescu
Université Lyon 1, VILLEURBANNE CEDEX, FRANCE
BMO, VMO, BV, Sobolev spaces, integer-valued functions, constant function, isoperimetric inequality
We define a new function space $B$, which contains in particular BMO, BV, and $W^{1/p,p}$, $1 < p < \infty$. We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several inequalities involving $L^p$ norms of integer-valued functions in $B$. We introduce a significant closed subspace, $B_0$, of $B$, containing in particular VMO and $W^{1/p,p}$, $1 \le p < \infty$. The above mentioned estimates imply in particular that integer-valued functions belonging to $B_0$ are necessarily constant. This framework provides a "common roof" to various, seemingly unrelated, statements asserting that integer-valued functions satisfying some kind of regularity condition must be constant.
Functional analysis
2083
2101
10.4171/JEMS/551
http://www.ems-ph.org/doi/10.4171/JEMS/551
Stability properties for quasilinear parabolic equations with measure data
Marie-Françoise
Bidaut-Véron
Université de Tours, TOURS, FRANCE
Quoc-Hung
Nguyen
Université de Tours, TOURS, FRANCE
Quasilinear parabolic equations, measure data, renormalized solutions, stability, Landes-time approximations, Steklov time-averages
Let $\Omega$ be a bounded domain of $\mathbb{R}^{N}$, and $Q=\Omega \times(0,T).$ We study problems of the model type \[ \left\{ \begin{array} [c]{l}% {u_{t}}-{\Delta_{p}}u=\mu\qquad\text{in }Q,\\ {u}=0\qquad\text{on }\partial\Omega\times(0,T),\\ u(0)=u_{0}\qquad\text{in }\Omega, \end{array} \right. \] where $p>1$, $\mu\in\mathcal{M}_{b}(Q)$ and $u_{0}\in L^{1}(\Omega).$ Our main result is a \textit{stability theorem }extending the results of Dal Maso, Murat, Orsina, Prignet, for the elliptic case, valid for quasilinear operators $u\longmapsto\mathcal{A}(u)=$div$(A(x,t,\nabla u))$.
Partial differential equations
2103
2135
10.4171/JEMS/552
http://www.ems-ph.org/doi/10.4171/JEMS/552
A strong maximum principle for the Paneitz operator and a non-local flow for the $Q$-curvature
Matthew
Gursky
University of Notre Dame, NOTRE DAME, UNITED STATES
Andrea
Malchiodi
Scuola Normale Superiore, PISA, ITALY
$Q$-curvature, Paneitz operator, conformal geometry, non-local flow
In this paper we consider Riemannian manifolds $(M^n,g)$ of dimension $n \geq 5$, with semi-positive $Q$-curvature and non-negative scalar curvature. Under these assumptions we prove (i) the Paneitz operator satisfies a strong maximum principle; (ii) the Paneitz operator is a positive operator; and (iii) its Green's function is strictly positive. We then introduce a non-local flow whose stationary points are metrics of constant positive $Q$-curvature. Modifying the test function construction of Esposito-Robert, we show that it is possible to choose an initial conformal metric so that the flow has a sequential limit which is smooth and positive, and defines a conformal metric of constant positive $Q$-curvature.
Partial differential equations
2137
2173
10.4171/JEMS/553
http://www.ems-ph.org/doi/10.4171/JEMS/553
Topological classification of multiaxial $U(n)$-actions (with an appendix by Jared Bass)
Sylvain
Cappell
New York University, NEW YORK, UNITED STATES
Shmuel
Weinberger
University of Chicago, CHICAGO, UNITED STATES
Min
Yan
Hong Kong University of Science and Technology, HONG KONG, CHINA
Transformation group, topological manifold, stratified space, multiaxial, surgery, assembly map
This paper begins the classification of topological actions on manifolds by compact, connected, Lie groups beyond the circle group. It treats multiaxial topological actions of unitary and symplectic groups without the dimension restrictions used in earlier works by M. Davis and W. C. Hsiang on differentiable actions. The general results are applied to give detailed calculations for topological actions homotopically modeled on standard multiaxial representation spheres. In the present topological setting, Schubert calculus of complex Grassmannians surprisingly enters in the calculations, yielding a profusion of “fake” representation spheres compared with the paucity in the previously studied smooth setting.
Manifolds and cell complexes
2175
2208
10.4171/JEMS/554
http://www.ems-ph.org/doi/10.4171/JEMS/554
Flexibility of surface groups in classical simple Lie groups
Inkang
Kim
KIAS, SEOUL, SOUTH KOREA
Pierre
Pansu
Université Paris-Sud 11, ORSAY CEDEX, FRANCE
Algebraic group, symmetric space, rigidity, group cohomology, moduli space
We show that a surface group of high genus contained in a classical simple Lie group can be deformed to become Zariski dense, unless the Lie group is $SU(p,q)$ (resp. $SO^* (2n)$, $n$ odd) and the surface group is maximal in some $S(U(p,p) \times U(q-p)) \subset SU(p,q)$ (resp. $SO^* (2n-2) \times SO(2) \subset SO^* (2n)$). This is a converse, for classical groups, to a rigidity result of S. Bradlow, O. García-Prada and P. Gothen.
Geometry
Manifolds and cell complexes
2209
2242
10.4171/JEMS/555
http://www.ems-ph.org/doi/10.4171/JEMS/555
Bistable traveling waves for monotone semiflows with applications
Jian
Fang
Harbin Institute of Technology, HARBIN, CHINA
Xiao-Qiang
Zhao
Memorial University of Newfoundland, ST. JOHN'S, CANADA
Monotone semiflows, traveling waves, bistable dynamics, periodic habitat
This paper is devoted to the study of traveling waves for monotone evolution systems of bistable type. In an abstract setting, we establish the existence of traveling waves for discrete and continuous-time monotone semiflows in homogeneous and periodic habitats. The results are then extended to monotone semiflows with weak compactness. We also apply the theory to four classes of evolution systems.
Dynamical systems and ergodic theory
General
Partial differential equations
2243
2288
10.4171/JEMS/556
http://www.ems-ph.org/doi/10.4171/JEMS/556
Extremal metrics and lower bound of the modified K-energy
Yuji
Sano
Kumamoto University, KUMAMOTO, JAPAN
Carl
Tipler
Université du Québec à Montréal, MONTRÉAL, QC, CANADA
Kähler manifolds, extremal metrics, K-energy
We provide a new proof of a result of X.X. Chen and G.Tian [5]: for a polarized extremal Kähler manifold, the minimum of the modified K-energy is attained at an extremal metric. The proof uses an idea of C. Li [16] adapted to the extremal metrics using some weighted balanced metrics.
Differential geometry
Several complex variables and analytic spaces
2289
2310
10.4171/JEMS/557
http://www.ems-ph.org/doi/10.4171/JEMS/557
The Brauer category and invariant theory
Gustav
Lehrer
University of Sydney, SYDNEY, AUSTRALIA
R.B.
Zhang
University of Sydney, SYDNEY, AUSTRALIA
Brauer category, invariant theory, second fundamental theorem, quantum group
A category of Brauer diagrams, analogous to Turaev's tangle category, is introduced, a presentation of the category is given, and full tensor functors are constructed from this category to the category of tensor representations of the orthogonal group O$(V)$ or the symplectic group Sp$(V)$ over any field of characteristic zero. The first and second fundamental theorems of invariant theory for these classical groups are generalised to the category theoretic setting. The major outcome is that we obtain presentations for the endomorphism algebras of the module $V^{\otimes r}$, which are new in the classical symplectic case and in the orthogonal and symplectic quantum case, while in the orthogonal classical case, the proof we give here is more natural than in our earlier work. These presentations are obtained by appending to the standard presentation of the Brauer algebra of degree $r$ one additional relation. This relation stipulates the vanishing of a single element of the Brauer algebra which is quasi-idempotent, and which we describe explicitly both in terms of diagrams and algebraically. In the symplectic case, if dim $V=2n$, the element is precisely the central idempotent in the Brauer subalgebra of degree $n+1$, which corresponds to its trivial representation. Since this is the Brauer algebra of highest degree which is semisimple, our generator is an exact analogue for the Brauer algebra of the Jones idempotent of the Temperley-Lieb algebra. In the orthogonal case the additional relation is also a quasi-idempotent in the integral Brauer algebra. Both integral and quantum analogues of these results are given, the latter of which involve the BMW algebras.
Algebraic geometry
Linear and multilinear algebra; matrix theory
Nonassociative rings and algebras
Group theory and generalizations
2311
2351
10.4171/JEMS/558
http://www.ems-ph.org/doi/10.4171/JEMS/558
Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model
Christophe
Sabot
Université Lyon 1, VILLEURBANNE CEDEX, FRANCE
Pierre
Tarrès
CNRS and Université Paris-Dauphine, PARIS CEDEX 16, FRANCE
Self-interacting random walk, reinforcement, random walk in random environment, sigma models, supersymmetry, de Finetti theorem
Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986 [8], is a random process which takes values in the vertex set of a graph $G$ and is more likely to cross edges it has visited before. We show that it can be represented in terms of a vertex-reinforced jump process (VRJP) with independent gamma conductances; the VRJP was conceived by Werner and first studied by Davis and Volkov [10, 11], and is a continuous-time process favouring sites with more local time. We calculate, for any finite graph $G$, the limiting measure of the centred occupation time measure of VRJP, and interpret it as a supersymmetric hyperbolic sigma model in quantum field theory, introduced by Zirnbauer in 1991 [35]. This enables us to deduce that VRJP and ERRWare positive recurrent on any graph of bounded degree for large reinforcement, and that the VRJP is transient on $\mathbb Z^d , d \geq 3$, for small reinforcement, using results of Disertori and Spencer [15] and Disertori, Spencer and Zirnbauer [16].
Probability theory and stochastic processes
Quantum theory
2353
2378
10.4171/JEMS/559
http://www.ems-ph.org/doi/10.4171/JEMS/559
Torsion points on families of simple abelian surfaces and Pell's equation over polynomial rings (with an appendix by E. V. Flynn)
David
Masser
Universität Basel, BASEL, SWITZERLAND
Umberto
Zannier
Scuola Normale Superiore, PISA, ITALY
Torsion point, abelian surface scheme, Pell equation, Jacobian variety, Chabauty’s theorem
In recent papers we proved a special case of a variant of Pink's Conjecture for a variety inside a semiabelian scheme: namely for any curve inside anything isogenous to a product of two elliptic schemes. Here we go beyond the elliptic situation by settling the crucial case of any simple abelian surface scheme defined over the field of algebraic numbers, thus confirming an earlier conjecture of Shou-Wu Zhang. This is of particular relevance in the topic, also in view of very recent counterexamples by Bertrand. Furthermore there are applications to the study of Pell equations over polynomial rings; for example we deduce that there are at most finitely many complex $t$ for which there exist $A,B \neq 0$ in ${\mathbb C}[X]$ with $A^2 – DB^2 = 1$ for $D = X^6 + X + t$. We also consider equations $A^2 – DB^2 = c'X + c$, where the situation is quite different.
Number theory
Algebraic geometry
2379
2416
10.4171/JEMS/560
http://www.ems-ph.org/doi/10.4171/JEMS/560
10
Generating series and asymptotics of classical spin networks
Francesco
Costantino
Université Paul Sabatier, TOULOUSE, FRANCE
Julien
Marché
École Polytechnique, PALAISEAU, FRANCE
Spin networks, generating series, asymptotical behavior, saddle point method, coherent states
We study classical spin networks with group SU$_2$. In the first part, using Gaussian integrals, we compute their generating series in the case where the edges are equipped with holonomies; this generalizes Westbury’s formula. In the second part, we use an integral formula for the square of the spin network and perform stationary phase approximation under some non-degeneracy hypothesis. This gives a precise asymptotic behavior when the labels are rescaled by a constant going to infinity.
Manifolds and cell complexes
Quantum theory
Relativity and gravitational theory
2417
2452
10.4171/JEMS/561
http://www.ems-ph.org/doi/10.4171/JEMS/561
Factorization of point configurations, cyclic covers, and conformal blocks
Michele
Bolognesi
Université de Montpellier, MONTPELLIER, FRANCE
Noah
Giansiracusa
University of Georgia, ATHENS, UNITED STATES
GIT, factorization, ramified cover, conformal blocks
We describe a relation between the invariants of $n$ ordered points in projective $d$-space and of points contained in a union of two linear subspaces. This yields an attaching map for GIT quotients parameterizing point configurations in these spaces, and we show that it respects the Segre product of the natural GIT polarizations. Associated to a configuration supported on a rational normal curve is a cyclic cover, and we show that if the branch points are weighted by the GIT linearization and the rational normal curve degenerates, then the admissible covers limit is a cyclic cover with weights as in this attaching map. We find that both GIT polarizations and the Hodge class for families of cyclic covers yield line bundles on $\overline M_{0,n}$ with functorial restriction to the boundary. We introduce a notion of divisorial factorization, abstracting an axiom from rational conformal field theory, to encode this property and show that it determines the isomorphism class of these line bundles. Consequently, we obtain a unified, geometric proof of two recent results on conformal block bundles, one by Fedorchuk and one by Gibney and the second author.
Algebraic geometry
General
2453
2471
10.4171/JEMS/562
http://www.ems-ph.org/doi/10.4171/JEMS/562
Brauer relations in finite groups
Alex
Bartel
University of Warwick, COVENTRY, UNITED KINGDOM
Tim
Dokchitser
University of Bristol, BRISTOL, UNITED KINGDOM
Brauer relations, finite groups, permutation groups, Burnside ring, rational representations
If $G$ is a non-cyclic finite group, non-isomorphic $G$-sets $X, Y$ may give rise to isomorphic permutation representations $\mathbb C[X ]\cong \mathbb C[Y]$. Equivalently, the map from the Burnside ring to the rational representation ring of $G$ has a kernel. Its elements are called Brauer relations, and the purpose of this paper is to classify them in all finite groups, extending the Tornehave–Bouc classification in the case of $p$-groups.
$K$-theory
Group theory and generalizations
2473
2512
10.4171/JEMS/563
http://www.ems-ph.org/doi/10.4171/JEMS/563
On Kakeya–Nikodym averages, $L^p$-norms and lower bounds for nodal sets of eigenfunctions in higher dimensions
Matthew
Blair
University of New Mexico, ALBUQUERQUE, UNITED STATES
Christopher
Sogge
The Johns Hopkins University, BALTIMORE, UNITED STATES
Eigenfunctions, $L^p$-norms, Kakeya averages
We extend a result of the second author [27, Theorem 1.1] to dimensions $d \geq 3$ which relates the size of $L^p$-norms of eigenfunctions for $2 < p < 2(d+1) / d-1$ to the amount of $L^2$-mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee [22] and a variable coefficient variant of an "$\epsilon$ removal lemma" of Tao and Vargas [35]. We also use Hörmander's [20] $L^2$ oscillatory integral theorem and the Cartan–Hadamard theorem to show that, under the assumption of nonpositive curvature, the $L^2$-norm of eigenfunctions $e_{\lambda}$ over unit-length tubes of width $\lambda^{-1/2}$ goes to zero. Using our main estimate, we deduce that, in this case, the $L^p$-norms of eigenfunctions for the above range of exponents is relatively small. As a result, we can slightly improve the known lower bounds for nodal sets in dimensions $d \ge 3$ of Colding and Minicozzi [10] in the special case of (variable) nonpositive curvature.
Global analysis, analysis on manifolds
Partial differential equations
Fourier analysis
2513
2543
10.4171/JEMS/564
http://www.ems-ph.org/doi/10.4171/JEMS/564
Soft local times and decoupling of random interlacements
Serguei
Popov
University of Campinas - UNICAMP, CAMPINAS, BRAZIL
Augusto
Teixeira
IMPA, RIO DE JANEIRO, BRAZIL
Random interlacements, stochastic domination, soft local time, connectivity decay, smoothening of discrete sets
In this paper we establish a decoupling feature of the random interlacement process $\mathcal{I}^u \subset \mathbb Z^d$ at level $u$, $d \geq 3$. Roughly speaking, we show that observations of $\mathcal{I}^u$ restricted to two disjoint subsets $A_1$ and $A_2$ of $\mathbb Z^d$ are approximately independent, once we add a sprinkling to the process $\mathcal{I}^u$ by slightly increasing the parameter $u$. Our results differ from previous ones in that we allow the mutual distance between the sets $A_1$ and $A_2$ to be much smaller than their diameters. We then provide an important application of this decoupling for which such flexibility is crucial. More precisely, we prove that, above a certain critical threshold $u_{**}$, the probability of having long paths that avoid $\mathcal{I}^u$ is exponentially small, with logarithmic corrections for $d=3$. To obtain the above decoupling, we first develop a general method for comparing the trace left by two Markov chains on the same state space. This method is based in what we call the soft local time of a chain. In another crucial step towards our main result, we also prove that any discrete set can be "smoothened" into a slightly enlarged discrete set, for which its equilibrium measure behaves in a regular way. Both these auxiliary results are interesting in themselves and are presented independently from the rest of the paper.
Probability theory and stochastic processes
Statistical mechanics, structure of matter
2545
2593
10.4171/JEMS/565
http://www.ems-ph.org/doi/10.4171/JEMS/565
Prime numbers along Rudin–Shapiro sequences
Christian
Mauduit
Université d'Aix-Marseille, MARSEILLE CEDEX 9, FRANCE
Joël
Rivat
Université d'Aix-Marseille, MARSEILLE CEDEX 9, FRANCE
Rudin–Shapiro sequence, prime numbers, Möbius function, exponential sums
For a large class of digital functions $f$, we estimate the sums $\sum_{n \leq x} \Lambda(n) f(n)$ (and $\sum_{n \leq x} \mu(n) f(n)$, where $\Lambda$ denotes the von Mangoldt function (and $\mu$ the Möbius function). We deduce from these estimates a Prime Number Theorem (and a Möbius randomness principle) for sequences of integers with digit properties including the Rudin-Shapiro sequence and some of its generalizations.
Number theory
2595
2642
10.4171/JEMS/566
http://www.ems-ph.org/doi/10.4171/JEMS/566
A presentation by generators and relations of Nichols algebras of diagonal type and convex orders on root systems
Iván Ezequiel
Angiono
Universidad Nacional de Cordoba, CORDOBA, ARGENTINA
Nichols algebras, quantized enveloping algebras, pointed Hopf algebras
We obtain a presentation by generators and relations of any Nichols algebra of diagonal type with finite root system. We prove that the defining ideal is finitely generated. The proof is based on Kharchenko’s theory of PBW bases of Lyndon words. We prove that the lexicographic order on Lyndon words is convex for PBW generators and so the PBW basis is orthogonal with respect to the canonical non-degenerate form associated to the Nichols algebra.
Associative rings and algebras
General
Nonassociative rings and algebras
2643
2671
10.4171/JEMS/567
http://www.ems-ph.org/doi/10.4171/JEMS/567
Spreading and vanishing in nonlinear diffusion problems with free boundaries
Yihong
Du
School of Science and Technology, ARMIDALE, AUSTRALIA
Bendong
Lou
Tongji University, SHANGHAI, CHINA
Nonlinear diffusion equation, free boundary problem, asymptotic behavior, monostable, bistable, combustion, sharp threshold, spreading speed
We study nonlinear diffusion problems of the form $u_t=u_{xx}+f(u)$ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special $f(u)$ of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any $f(u)$ which is $C^1$ and satisfies $f(0)=0$, we show that the omega limit set $\omega(u)$ of every bounded positive solution is determined by a stationary solution. For monostable, bistable and combustion types of nonlinearities, we obtain a rather complete description of the long-time dynamical behavior of the problem; moreover, by introducing a parameter $\sigma$ in the initial data, we reveal a threshold value $\sigma^*$ such that spreading ($\lim_{t \to \infty}u= 1$) happens when $\sigma > \sigma^*$, vanishing ($\lim_{t \to \infty}u=0$) happens when $\sigma < \sigma^*$, and at the threshold value $\sigma^*$, $\omega(u)$ is different for the three different types of nonlinearities. When spreading happens, we make use of "semi-waves" to determine the asymptotic spreading speed of the front.
Partial differential equations
2673
2724
10.4171/JEMS/568
http://www.ems-ph.org/doi/10.4171/JEMS/568
11
High-order phase transitions in the quadratic family
Daniel
Coronel
Universidad Andres Bello, SANTIAGO, CHILE
Juan
Rivera-Letelier
Pontifica Universidad Católica de Chile, SANTIAGO, CHILE
Quadratic family, thermodynamic formalism, phase transition
We give the first example of a transitive quadratic map whose real and complex geometric pressure functions have a high-order phase transition. In fact, we show that this phase transition resembles a Kosterlitz-Thouless singularity: Near the critical parameter the geometric pressure function behaves as $x \mapsto \mathrm {exp} (– x^{–2})$ near $x = 0$, before becoming linear. This quadratic map has a non-recurrent critical point, so it is non-uniformly hyperbolic in a strong sense.
Dynamical systems and ergodic theory
2725
2761
10.4171/JEMS/569
http://www.ems-ph.org/doi/10.4171/JEMS/569
Singular localization of $\mathfrak{g}$-modules and applications to representation theory
Erik
Backelin
Universidad de los Andes, BOGOTA, COLOMBIA
Kobi
Kremnitzer
University of Oxford, OXFORD, UNITED KINGDOM
Lie algebra, Beilinson–Bernstein localization, category O
We prove a singular version of Beilinson–Bernstein localization for a complex semi-simple Lie algebra following ideas from the positive characteristic case settled by [BMR06]. We apply this theory to translation functors, singular blocks in the Bernstein–Gelfand–Gelfand category O and Whittaker modules.
Nonassociative rings and algebras
Algebraic geometry
2763
2787
10.4171/JEMS/570
http://www.ems-ph.org/doi/10.4171/JEMS/570
Symmetry of minimizers with a level surface parallel to the boundary
Giulio
Ciraolo
Università di Palermo, PALERMO, ITALY
Rolando
Magnanini
Università di Firenze, FIRENZE, ITALY
Shigeru
Sakaguchi
Tohoku University, SENDAI, JAPAN
Overdetermined problems, Minimizers of integral functionals
We consider the functional $$\mathcal I_{\Omega} (v) = \int_{\Omega} [f(|Dv|) - v] dx,$$ where $\Omega$ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$, Crasta [Cr1] has shown that if $\mathcal I_{\Omega}$ admits a minimizer in $W_0^{1,1}(\Omega)$ depending only on the distance from the boundary of $\Omega$, then $\Omega$ must be a ball. With some restrictions on $f$, we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss how these results extend to more general settings, in particular to functionals that are not differentiable and to solutions of fully nonlinear elliptic and parabolic equations.
Partial differential equations
Calculus of variations and optimal control; optimization
2789
2804
10.4171/JEMS/571
http://www.ems-ph.org/doi/10.4171/JEMS/571
Anti-self-dual orbifolds with cyclic quotient singularities
Michael
Lock
The University of Texas at Austin, AUSTIN, UNITED STATES
Jeff
Viaclovsky
University of Wisconsin, MADISON, UNITED STATES
Anti-self-dual metrics, index theory, orbifolds
An index theorem for the anti-self-dual deformation complex on anti-self-dual orbifolds with cyclic quotient singularities is proved. We present two applications of this theorem. The first is to compute the dimension of the deformation space of the Calderbank–Singer scalar-flat Kähler toric ALE spaces. A corollary of this is that, except for the Eguchi–Hanson metric, all of these spaces admit non-toric anti-self-dual deformations, thus yielding many new examples of anti-self-dual ALE spaces. For our second application, we compute the dimension of the deformation space of the canonical Bochner-Kähler metric on any weighted projective space $\mathbb {CP}^2_{(r,q,p)}$ for relatively prime integers $1 < r < q < p$. A corollary of this is that, while these metrics are rigid as Bochner–Kähler metrics, infinitely many of these admit non-trivial self-dual deformations, yielding a large class of new examples of self-dual orbifold metrics on certain weighted projective spaces.
Differential geometry
Global analysis, analysis on manifolds
2805
2842
10.4171/JEMS/572
http://www.ems-ph.org/doi/10.4171/JEMS/572
The Roquette category of finite $p$-groups
Serge
Bouc
Université de Picardie - Jules Verne, AMIENS CEDEX, FRANCE
$p$-group, Roquette, rational, biset, genetic
Let $p$ be a prime number. This paper introduces the Roquette category $\mathcal{R}_p$ of finite $p$-groups, which is an additive tensor category containing all finite $p$-groups among its objects. In $\mathcal{R}_p$, every finite $p$-group $P$ admits a canonical direct summand $\partial P$, called the edge of $P$. Moreover $P$ splits uniquely as a direct sum of edges of Roquette $p$-groups, and the tensor structure of $\mathcal{R}_p$ can be described in terms of such edges. The main motivation for considering this category is that the additive functors from $\mathcal{R}_p$ to abelian groups are exactly the rational $p$-biset functors. This yields in particular very efficient ways of computing such functors on arbitrary $p$-groups: this applies to the representation functors $R_K$, where $K$ is any field of characteristic 0, but also to the functor of units of Burnside rings, or to the torsion part of the Dade group.
Category theory; homological algebra
$K$-theory
Group theory and generalizations
2843
2886
10.4171/JEMS/573
http://www.ems-ph.org/doi/10.4171/JEMS/573
Cubic moments of Fourier coefficients and pairs of diagonal quartic forms
Jörg
Brüdern
Universität Göttingen, GÖTTINGEN, GERMANY
Trevor
Wooley
University of Bristol, BRISTOL, UNITED KINGDOM
Quartic Diophantine equations, Hardy–Littlewood method
We establish the non-singular Hasse principle for pairs of diagonal quartic equations in 22 or more variables. Our methods involve the estimation of a certain entangled two-dimensional 21st moment of quartic smooth Weyl sums via a novel cubic moment of Fourier coefficients.
Number theory
2887
2901
10.4171/JEMS/574
http://www.ems-ph.org/doi/10.4171/JEMS/574
Amenable hyperbolic groups
Pierre-Emmanuel
Caprace
Université Catholique de Louvain, LOUVAIN-LA-NEUVE, BELGIUM
Yves
de Cornulier
Université Paris-Sud, ORSAY CEDEX, FRANCE
Nicolas
Monod
Ecole Polytechnique Fédérale de Lausanne, LAUSANNE, SWITZERLAND
Romain
Tessera
Université Paris-Sud, CNRS, ORSAY CEDEX, FRANCE
Gromov hyperbolic group, locally compact group, amenable group, contracting automorphisms, compacting automorphisms
We give a complete characterization of the locally compact groups that are non elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting automorphisms. We moreover give a description of all Gromov-hyperbolic locally compact groups with a cocompact amenable subgroup: modulo a compact normal subgroup, these turn out to be either rank one simple Lie groups, or automorphism groups of semiregular trees acting doubly transitively on the set of ends. As an application, we show that the class of hyperbolic locally compact groups with a cusp-uniform nonuniform lattice is very restricted.
Group theory and generalizations
Combinatorics
Topological groups, Lie groups
Abstract harmonic analysis
2903
2947
10.4171/JEMS/575
http://www.ems-ph.org/doi/10.4171/JEMS/575
Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities
Daniele
Castorina
Università di Roma Tor Vergata, ROMA, ITALY
Manel
Sanchón
Universitat de Barcelona, BARCELONA, SPAIN
Geometric inequalities, mean curvature of level sets, Schwarz symmetrization, $p$-Laplace equations, regularity of stable solutions
We prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semistable solutions of $–\Delta_p u= g(u)$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$. In particular, we obtain new $L^r$ and $W^{1,r}$ bounds for the extremal solution $u^\star$ when the domain is strictly convex. More precisely, we prove that $u^\star\in L^{\infty}(\Omega)$ if $n\leq p+2$ and $u^\star \in L^{\frac{np}{n-p-2}} (\Omega) \cap W^{1,p}_0 (\Omega)$ if $n > p + 2$.
Partial differential equations
General
2949
2975
10.4171/JEMS/576
http://www.ems-ph.org/doi/10.4171/JEMS/576
12
Modular dynamical systems on networks
Lee
DeVille
University of Illinois at Urbana-Champaign, URBANA, UNITED STATES
Eugene
Lerman
University of Illinois at Urbana-Champaign, URBANA, UNITED STATES
Dynamical systems, networks, modularity, graph fibrations
We propose a new framework for the study of continuous time dynamical systems on networks. We view such dynamical systems as collections of interacting control systems. We show that a class of maps between graphs called graph fibrations give rise to maps between dynamical systems on networks. This allows us to produce conjugacy between dynamical systems out of combinatorial data. In particular we show that surjective graph fibrations lead to synchrony subspaces in networks. The injective graph fibrations, on the other hand, give rise to surjective maps from large dynamical systems to smaller ones. One can view these surjections as a kind of “fast/slow” variable decompositions or as “abstractions” in the computer science sense of the word.
Dynamical systems and ergodic theory
Category theory; homological algebra
2977
3013
10.4171/JEMS/577
http://www.ems-ph.org/doi/10.4171/JEMS/577
Tempered reductive homogeneous spaces
Yves
Benoist
Université Paris-Sud, ORSAY CEDEX, FRANCE
Toshiyuki
Kobayashi
University of Tokyo, TOKYO, JAPAN
Lie groups, homogeneous spaces, tempered representations, matrix coefficients, symmetric spaces
Let $G$ be a semisimple algebraic Lie group and $H$ a reductive subgroup. We find geometrically the best even integer $p$ for which the representation of $G$ in $L^2(G/H)$ is almost $L^p$. As an application, we give a criterion which detects whether this representation is tempered.
Topological groups, Lie groups
Abstract harmonic analysis
3015
3036
10.4171/JEMS/578
http://www.ems-ph.org/doi/10.4171/JEMS/578
Combinatorial topology and the global dimension of algebras arising in combinatorics
Stuart
Margolis
Bar-Ilan University, RAMAT GAN, ISRAEL
Franco
Saliola
Université du Québec à Montréal, MONTRÉAL, QC, CANADA
Benjamin
Steinberg
City College of New York, NEW YORK, UNITED STATES
Global dimension, hereditary algebra, cohomology, classifying space, left regular band, hyperplane arrangements, order complex, Leray number, chordal graph
In a highly influential paper, Bidigare, Hanlon and Rockmore showed that a number of popular Markov chains are random walks on the faces of a hyperplane arrangement. Their analysis of these Markov chains took advantage of the monoid structure on the set of faces. This theory was later extended by Brown to a larger class of monoids called left regular bands. In both cases, the representation theory of these monoids played a prominent role. In particular, it was used to compute the spectrum of the transition operators of the Markov chains and to prove diagonalizability of the transition operators. In this paper, we establish a close connection between algebraic and combinatorial invariants of a left regular band: we show that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. For instance, we show that the global dimension of these algebras is bounded above by the Leray number of the associated order complex. Conversely, we associate to every flag complex a left regular band whose algebra has global dimension precisely the Leray number of the flag complex.
Combinatorics
Commutative rings and algebras
3037
3080
10.4171/JEMS/579
http://www.ems-ph.org/doi/10.4171/JEMS/579
Two-dimensional curvature functionals with superquadratic growth
Ernst
Kuwert
Universität Freiburg, FREIBURG, GERMANY
Tobias
Lamm
Karlsruhe Institute of Technology (KIT), KARLSRUHE, GERMANY
Yuxiang
Li
Tsinghua University, BEIJING, CHINA
Curvature functionals, Palais–Smale condition
For two-dimensional, immersed closed surfaces $f:\Sigma \to \mathbb R^n$, we study the curvature functionals $\mathcal{E}^p(f)$ and $\mathcal{W}^p(f)$ with integrands $(1+|A|^2)^{p/2}$ and $(1+|H|^2)^{p/2}$, respectively. Here $A$ is the second fundamental form, $H$ is the mean curvature and we assume $p > 2$. Our main result asserts that $W^{2,p}$ critical points are smooth in both cases. We also prove a compactness theorem for $\mathcal{W}^p$-bounded sequences. In the case of $\mathcal{E}^p$ this is just Langer's theorem [16], while for $\mathcal{W}^p$ we have to impose a bound for the Willmore energy strictly below $8\pi$ as an additional condition. Finally, we establish versions of the Palais–Smale condition for both functionals.
Differential geometry
Partial differential equations
3081
3111
10.4171/JEMS/580
http://www.ems-ph.org/doi/10.4171/JEMS/580
Self-similar Lie algebras
Laurent
Bartholdi
Georg-August-Universität Göttingen, GÖTTINGEN, GERMANY
Groups acting on trees, Lie algebras, wreath products
We give a general definition of branched, self-similar Lie algebras, and show that important examples of Lie algebras fall into that class. We give sufficient conditions for a self-similar Lie algebra to be nil, and prove in this manner that the self-similar algebras associated with Grigorchuk’s and Gupta–Sidki’s torsion groups are nil as well as self-similar.We derive the same results for a class of examples constructed by Petrogradsky, Shestakov and Zelmanov
Group theory and generalizations
Associative rings and algebras
Nonassociative rings and algebras
3113
3151
10.4171/JEMS/581
http://www.ems-ph.org/doi/10.4171/JEMS/581
How to produce a Ricci flow via Cheeger–Gromoll exhaustion
Esther
Cabezas-Rivas
J. W. Goethe-Universität, FRANKFURT A.M., GERMANY
Burkhard
Wilking
Universität Münster, MÜNSTER, GERMANY
Ricci flow, short time existence, Cheeger–Gromoll exhaustion, complex sectional curvature
We prove short time existence for the Ricci flow on open manifolds of non-negative complex sectional curvature without requiring upper curvature bounds. By considering the doubling of convex sets contained in a Cheeger–Gromoll convex exhaustion and solving the singular initial value problem for the Ricci flow on these closed manifolds, we obtain a sequence of closed solutions of the Ricci flow with non-negative complex sectional curvature which subconverge to a Ricci flow on the open manifold. Furthermore, we find an optimal volume growth condition which guarantees long time existence, and give an analysis of the long time behavior of the Ricci flow. We also construct an explicit example of an immortal non-negatively curved Ricci flow with unbounded curvature for all time.
Differential geometry
Partial differential equations
Global analysis, analysis on manifolds
3153
3194
10.4171/JEMS/582
http://www.ems-ph.org/doi/10.4171/JEMS/582