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European Mathematical Society Publishing House
2024-03-28 12:38:13
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=18&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
18
2016
1
Resolutions of moduli spaces and homological stability
Oscar
Randal-Williams
University of Cambridge, CAMBRIDGE, UNITED KINGDOM
Homological stability, mapping class groups, moduli spaces
We describe partial semi-simplicial resolutions of moduli spaces of surfaces with tangential structure. This allows us to prove a homological stability theorem for these moduli spaces, which often improves the known stability ranges and gives explicit stability ranges in many new cases. In each of these cases the stable homology can be identified using the methods of Galatius, Madsen, Tillmann and Weiss.
Algebraic topology
Group theory and generalizations
Manifolds and cell complexes
1
81
10.4171/JEMS/583
http://www.ems-ph.org/doi/10.4171/JEMS/583
The spacetime positive mass theorem in dimensions less than eight
Michael
Eichmair
Universität Wien, WIEN, AUSTRIA
Lan-Hsuan
Huang
University of Connecticut, STORRS, UNITED STATES
Dan
Lee
CUNY, QUEENS, UNITED STATES
Richard
Schoen
University of California, Irvine, IRVINE, UNITED STATES
Positive mass theorem, marginally outer trapped surfaces
We prove the spacetime positive mass theorem in dimensions less than eight. This theorem asserts that for any asymptotically flat initial data set that satisfies the dominant energy condition, the inequality $E ≥ | P |$ holds, where $(E, P)$ is the ADM energy-momentum vector. Previously, this theorem was only known for spin manifolds [38]. Our approach is a modification of the minimal hypersurface technique that was used by the last named author and S.-T. Yau to establish the time-symmetric case of this theorem [30, 27]. Instead of minimal hypersurfaces, we use marginally outer trapped hypersurfaces (MOTS) whose existence is guaranteed by earlier work of the first named author [14]. An important part of our proof is to introduce an appropriate substitute for the area functional that is used in the time-symmetric case to single out certain minimal hyper surfaces. We also establish a density theorem of independent interest and use it to reduce the general case of the spacetime positive mass theorem to the special case of initial data that has harmonic asymptotics and satisfies the strict dominant energy condition.
Differential geometry
Relativity and gravitational theory
83
121
10.4171/JEMS/584
http://www.ems-ph.org/doi/10.4171/JEMS/584
Real zeros of holomorphic Hecke cusp forms and sieving short intervals
Kaisa
Matomäki
University of Turku, TURKU, FINLAND
Cusp forms, real zeros, sieving short intervals
We study so-called real zeros of holomorphic Hecke cusp forms, that is, zeros on three geodesic segments on which the cusp form (or a multiple of it) takes real values. Ghosh and Sarnak, who were the first to study this problem, showed the existence of many such zeros if many short intervals contain numbers whose prime factors all belong to a certain subset of the primes.We prove new results concerning this sieving problem which leads to improved lower bounds for the number of real zeros.
Number theory
123
146
10.4171/JEMS/585
http://www.ems-ph.org/doi/10.4171/JEMS/585
The geometry of dented pentagram maps
Boris
Khesin
University of Toronto, TORONTO, ONTARIO, CANADA
Fedor
Soloviev
University of Toronto, TORONTO, ONTARIO, CANADA
Pentagram maps, space polygons, Lax representation, discrete integrable system, KdV hierarchy, Boussinesq equation, algebraic-geometric integrability
We propose a new family of natural generalizations of the pentagram map from 2D to higher dimensions and prove their integrability on generic twisted and closed polygons. In dimension $d$ there are $d–1$ such generalizations called dented pentagram maps, and we describe their geometry, continuous limit, and Lax representations with a spectral parameter. We prove algebraic-geometric integrability of the dented pentagram maps in the 3D case and compare the dimensions of invariant tori for the dented maps with those for the higher pentagram maps constructed with the help of short diagonal hyperplanes. When restricted to corrugated polygons, the dented pentagram maps coincide with one another and with the corresponding corrugated pentagram map. Finally, we prove integrability for a variety of pentagram maps for generic and partially corrugated polygons in higher dimensions.
Dynamical systems and ergodic theory
Differential geometry
147
179
10.4171/JEMS/586
http://www.ems-ph.org/doi/10.4171/JEMS/586
On the topology of polynomials with bounded integer coefficients
De-Jun
Feng
The Chinese University of Hong Kong, SHATIN, HONG KONG, CHINA
Pisot numbers, iterated function systems
For a real number $q>1$ and a positive integer $m$, let $$ Y_m(q):=\left\{\sum_{i=0}^n\epsilon_i q^i:\; \epsilon_i\in \{0, \pm 1,\ldots, \pm m\},\; n=0, 1,\ldots \right\}. $$ In this paper, we show that $Y_m(q)$ is dense in $\mathbb R$ if and only if $q < m + 1$ and $q$ is not a Pisot number. This completes several previous results and answers an open question raised by Erdős, Joó and Komornik [8].
Number theory
Measure and integration
181
193
10.4171/JEMS/587
http://www.ems-ph.org/doi/10.4171/JEMS/587
Group actions on monotone skew-product semiflows with applications
Feng
Cao
Nanjing University of Aeronautics and Astronautics, NANJING, JIANGSU, CHINA
Mats
Gyllenberg
University of Helsinki, HELSINKI, FINLAND
Yi
Wang
University of Scinece and Technology of China, HEFEI, ANHUI, CHINA
Monotone skew-product semiflows, group actions, rotational symmetry, reaction-diffusion equations, traveling waves
We discuss a general framework of monotone skew-product semiflows under a connected group action. In a prior work, a compact connected group $G$-action has been considered on a strongly monotone skew-product semiflow. Here we relax the strong monotonicity and compactness requirements, and establish a theory concerning symmetry or monotonicity properties of uniformly stable 1-cover minimal sets. We then apply this theory to show rotational symmetry of certain stable entire solutions for a class of nonautonomous reaction-diffusion equations on $\mathbb R^n$, as well as monotonicity of stable traveling waves of some nonlinear diffusion equations in time-recurrent structures including almost periodicity and almost automorphy.
Dynamical systems and ergodic theory
Partial differential equations
195
223
10.4171/JEMS/588
http://www.ems-ph.org/doi/10.4171/JEMS/588
2
The KSBA compactification for the moduli space of degree two $K$3 pairs
Radu
Laza
Stony Brook University, STONY BROOK, UNITED STATES
$K$3 surfaces, moduli space of K3 surfaces, KSBA
Inspired by the ideas of the minimal model program, Shepherd-Barron, Kollár, and Alexeev have constructed a geometric compactification for the moduli space of surfaces of log general type. In this paper, we discuss one of the simplest examples that fits into this framework: the case of pairs $(X,H)$ consisting of a degree two $K3$ surface $X$ and an ample divisor $H$. Specifically, we construct and describe explicitly a geometric compactification $\overline{\mathcal P}_2$ for the moduli of degree two $K$3 pairs. This compactification has a natural forgetful map to the Baily–Borel compactification of the moduli space $\mathcal F_2$ of degree two $K$3 surfaces. Using this map and the modular meaning of $\overline{\mathcal P}_2$, we obtain a better understanding of the geometry of the standard compactifications of $\mathcal F_2$.
Algebraic geometry
225
279
10.4171/JEMS/589
http://www.ems-ph.org/doi/10.4171/JEMS/589
Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers
Robert
Lipshitz
Columbia University, NEW YORK, UNITED STATES
David
Treumann
Boston College, CHESTNUT HILL, UNITED STATES
Hochschild homology, localization, Smith theory, Heegaard Floer homology
Let $A$ be a dg algebra over $\mathbb F_2$ and let $M$ be a dg $A$-bimodule. We show that under certain technical hypotheses on $A$, a noncommutative analog of the Hodge-to-de Rham spectral sequence starts at the Hochschild homology of the derived tensor product $M \otimes_A^L M$ and converges to the Hochschild homology of $M$. We apply this result to bordered Heegaard Floer theory, giving spectral sequences associated to Heegaard Floer homology groups of certain branched and unbranched double covers.
Manifolds and cell complexes
Associative rings and algebras
Differential geometry
281
325
10.4171/JEMS/590
http://www.ems-ph.org/doi/10.4171/JEMS/590
Sets of $\beta$-expansions and the Hausdorff measure of slices through fractals
Tom
Kempton
University of Utrecht, UTRECHT, NETHERLANDS
Bernoulli convolution, $\beta$ expansion, slicing fractals, conditional measures
We study natural measures on sets of $\beta$-expansions and on slices through self similar sets. In the setting of $\beta$-expansions, these allow us to better understand the measure of maximal entropy for the random $\beta$-transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing, leading to conditions under which almost every slice through a self similar set has positive Hausdorff measure, generalising long known results about almost everywhere values of the Hausdorff dimension.
Measure and integration
Number theory
Dynamical systems and ergodic theory
Fourier analysis
327
351
10.4171/JEMS/591
http://www.ems-ph.org/doi/10.4171/JEMS/591
Complexity of intersections of real quadrics and topology of symmetric determinantal varieties
Antonio
Lerario
SISSA, TRIESTE, ITALY
Real algebraic geometry, real quadrics, homological compexity, determinantal varieties
Let $W$ be a linear system of quadrics on the real projective space $\mathbb R P^n$ and $X$ be the base locus of that system (i.e. the common zero set of the quadrics in $W$). We prove a formula relating the topology of $X$ to the one of the discriminant locus $\Sigma_W$ (i.e. the set of singular quadrics in $W$). The set $\Sigma_W$ equals the intersection of $W$ with the discriminant hypersurface for quadrics; its singularities are unavoidable (they might persist after a small perturbation of $W$) and we set $\{\Sigma_W^{(r)}\}_{r\geq 1}$ for its singular point stratification, i.e. $\Sigma_W^{(1)}=\Sigma_W$ and $\Sigma_W^{(r)}=\textrm{Sing}\big( \Sigma_W^{(r-1)}\big)$. With this notation, for a generic $W$ the mentioned formula writes: $$b(X) \leq b(\mathbb R P^n)+ \sum_{r \geq 1}b(\mathbb{P}\Sigma_W^{(r)}).$$ In the general case a similar formula holds, but we have to replace each $b(\mathbb{P}\Sigma_W^{(r)})$ with $\frac{1}{2}b(\Sigma_\epsilon^{(r)})$, where $\Sigma_\epsilon$ equals the intersection of the discriminant hypersurface with the unit sphere on the translation of $W$ in the direction of a small negative definite form. Each $\Sigma_\epsilon^{(r)}$ is a determinantal variety on the sphere $S^{k-1}$ defined by equations of degree at most $n+1$ (here $k$ denotes the dimension of $W$); we refine Milnor's bound, proving that for such affine varieties $b(\Sigma_\epsilon^{(r)})\leq O(n)^{k-1}$. Since the sum in the above formulas contains at most $O(k)^{1/2}$ terms, as a corollary we prove that if $X$ is any intersection of $k$ quadrics in $\mathbb R P^n$ then the following sharp estimate holds: $$ b(X) \leq O(n)^{k-1}.$$ This bound refines Barvinok's style estimates (recall that the best previously known bound, due to Basu, has the shape $O(n)^{2k+2}$).
Algebraic geometry
Algebraic topology
353
379
10.4171/JEMS/592
http://www.ems-ph.org/doi/10.4171/JEMS/592
Boundary estimates for certain degenerate and singular parabolic equations
Benny
Avelin
Uppsala Universitet, UPPSALA, SWEDEN
Ugo
Gianazza
Università di Pavia, PAVIA, ITALY
Sandro
Salsa
Politecnico di Milano, MILANO, ITALY
Degenerate and singular parabolic equations, Harnack estimates, boundary Harnack inequality, Carleson estimate
We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic $p$-Laplacian equation. Assuming that such solutions continuously vanish on some distinguished part of the lateral part $S_T$ of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of degenerate/singular parabolic equations of porous medium type.
Partial differential equations
381
424
10.4171/JEMS/593
http://www.ems-ph.org/doi/10.4171/JEMS/593
Equidistribution estimates for Fekete points on complex manifolds
Nir
Lev
Bar-Ilan University, RAMAT GAN, ISRAEL
Joaquim
Ortega-Cerdà
Universitat de Barcelona, BARCELONA, SPAIN
Beurling–Landau density, Fekete points, holomorphic line bundles
We study the equidistribution of Fekete points in a compact complex manifold. These are extremal point configurations defined through sections of powers of a positive line bundle. Their equidistribution is a known result. The novelty of our approach is that we relate them to the problem of sampling and interpolation on line bundles, which allows us to estimate the equidistribution of the Fekete points quantitatively. In particular we estimate the Kantorovich–Wasserstein distance of the Fekete points to the limiting measure. The sampling and interpolation arrays on line bundles are a subject of independent interest, and we provide necessary density conditions through the classical approach of Landau, that in this context measures the local dimension of the space of sections of the line bundle. We obtain a complete geometric characterization of sampling and interpolation arrays in the case of compact manifolds of dimension one, and we prove that there are no arrays of both sampling and interpolation in the more general setting of semipositive line bundles.
Several complex variables and analytic spaces
425
464
10.4171/JEMS/594
http://www.ems-ph.org/doi/10.4171/JEMS/594
3
The logarithmic delay of KPP fronts in a periodic medium
François
Hamel
Université d'Aix-Marseille, MARSEILLE CEDEX 13, FRANCE
James
Nolen
Duke University, DURHAM, UNITED STATES
Jean-Michel
Roquejoffre
Université Paul Sabatier, TOULOUSE CEDEX, FRANCE
Lenya
Ryzhik
Stanford University, STANFORD, UNITED STATES
Reaction-diffusion equations, periodic media, pulsating traveling fronts, Cauchy problem, asymptotic behavior, logarithmic shift
We extend, to parabolic equations of the KPP type in periodic media, a result of Bramson which asserts that, in the case of a spatially homogeneous reaction rate, the time lag between the position of an initially compactly supported solution and that of a traveling wave grows logarithmically in time.
Partial differential equations
465
505
10.4171/JEMS/595
http://www.ems-ph.org/doi/10.4171/JEMS/595
Exceptional collections on isotropic Grassmannians
Alexander
Kuznetsov
Steklov Mathematical Institute, MOSCOW, RUSSIAN FEDERATION
Alexander
Polishchuk
University of Oregon, EUGENE, UNITED STATES
Exceptional collection, derived category of sheaves, isotropic Grassmannian
We introduce a new construction of exceptional objects in the derived category of coherent sheaves on a compact homogeneous space of a semisimple algebraic group and show that it produces exceptional collections of the length equal to the rank of the Grothendieck group on homogeneous spaces of all classical groups.
Algebraic geometry
Category theory; homological algebra
507
574
10.4171/JEMS/596
http://www.ems-ph.org/doi/10.4171/JEMS/596
A large data regime for nonlinear wave equations
Jinhua
Wang
Max-Planck-Institute for Gravitational Physics, GOLM, GERMANY
Pin
Yu
Tsinghua University, BEIJING, CHINA
Large data problem, wave equation, null form
For semi-linear wave equations with null form nonlinearities on $\mathbb R^{3+1}$, we exhibit an open set of initial data which are allowed to be large in energy spaces, yet we can still obtain global solutions in the future. We also exhibit a set of localized data for which the corresponding solutions are strongly focused, which in geometric terms means that a wave travels along an speci fic incoming null geodesic in such a way that almost all of the energy is concentrated in a tubular neighborhood of the geodesic and almost no energy radiates out of this neighborhood.
Partial differential equations
575
622
10.4171/JEMS/597
http://www.ems-ph.org/doi/10.4171/JEMS/597
Noncommutative numerical motives, Tannakian structures, and motivic Galois groups
Matilde
Marcolli
California Institute of Technology, PASADENA, UNITED STATES
Gonçalo
Tabuada
Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES
Noncommutative algebraic geometry, noncommutative motives, periodic cyclic homology, Tannakian formalism, motivic Galois groups
In this article we further the study of noncommutative numerical motives, initiated in [30, 31]. By exploring the change-of-coefficients mechanism, we start by improving some of the main results of [30]. Then, making use of the notion of Schur-finiteness, we prove that the category NNum$(k)_F$ of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum$(k)_F$ is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined symmetric monoidal functor $\overline{HP_\ast}$ on the category of noncommutative Chow motives. This allows us to introduce the correct noncommutative analogues $C_{NC}$ and $D_{NC}$ of Grothendieck's standard conjectures $C$ and $D$. Assuming $C_{NC}$, we prove that NNum$(k)_F$ can be made into a Tannakian category NNum$^\dagger(k)_F$ by modifying its symmetry isomorphism constraints. By further assuming $D_{NC}$, we neutralize the Tannakian category Num$^\dagger(k)_F$ using $\overline{HP_\ast}$. Via the (super-)Tannakian formalism, we then obtain well-defined noncommutative motivic Galois (super-)groups. Finally, making use of Deligne-Milne's theory of Tate triples, we construct explicit morphisms relating these noncommutative motivic Galois (super-)groups with the classical ones as suggested by Kontsevich.
Algebraic geometry
Category theory; homological algebra
$K$-theory
623
655
10.4171/JEMS/598
http://www.ems-ph.org/doi/10.4171/JEMS/598
Degree three cohomological invariants of semisimple groups
Alexander
Merkurjev
University of California, LOS ANGELES, UNITED STATES
Semisimple groups, cohomological invariants, torsors, classifying space
We study the degree 3 cohomological invariants with coefficients in $\mathbb Q/\mathbb Z(2)$ of a semisimple group over an arbitrary field. A list of all invariants of adjoint groups of inner type is given.
Field theory and polynomials
Number theory
Algebraic geometry
657
680
10.4171/JEMS/599
http://www.ems-ph.org/doi/10.4171/JEMS/599
4
Matroids over a ring
Alex
Fink
Queen Mary University of London, LONDON, UNITED KINGDOM
Luca
Moci
Université Paris-Diderot Paris 7, PARIS CEDEX 13, FRANCE
Matroid, module over Dedekind ring, arithmetic matroid, valuated matroid, arithmetic Tutte polynomial, tropical flag Dressian, Tutte–Grothendieck ring
We introduce the notion of a matroid $M$ over a commutative ring $R$, assigning to every subset of the ground set an $R$-module according to some axioms. When $R$ is a field, we recover matroids. When $R = \mathbb Z$, and when $R$ is a DVR, we get (structures which contain all the data of) quasi-arithmetic matroids, and valuated matroids, i.e. tropical linear spaces, respectively. More generally, whenever $R$ is a Dedekind domain, we extend all the usual properties and operations holding for matroids (e.g., duality), and we explicitly describe the structure of the matroids over $R$. Furthermore, we compute the Tutte–Grothendieck ring of matroids over $R$. We also show that the Tutte quasi-polynomial of a matroid over $\mathbb Z$ can be obtained as an evaluation of the class of the matroid in the Tutte–Grothendieck ring.
Combinatorics
Commutative rings and algebras
681
731
10.4171/JEMS/600
http://www.ems-ph.org/doi/10.4171/JEMS/600
Betti numbers of random real hypersurfaces and determinants of random symmetric matrices
Damien
Gayet
Université Joseph Fourier Grenoble 1, SAINT-MARTIN D'HERES CEDEX, FRANCE
Jean-Yves
Welschinger
Université Lyon 1, VILLEURBANNE CEDEX, FRANCE
Real projective manifold, ample line bundle, random matrix, random polynomial
We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the Kählerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of given index. In particular, for large dimensions, these coefficients get exponentially small away from mid-dimensional Betti numbers. In order to get these results, we first establish the equidistribution of the critical points of a given Morse function restricted to the random real hypersurfaces.
Algebraic geometry
Several complex variables and analytic spaces
Probability theory and stochastic processes
733
772
10.4171/JEMS/601
http://www.ems-ph.org/doi/10.4171/JEMS/601
Coherent randomness tests and computing the $K$-trivial sets
Laurent
Bienvenu
Université Paris Diderot - Paris 7, PARIS CEDEX 13, FRANCE
Noam
Greenberg
Victoria University of Wellington, WELLINGTON, NEW ZEALAND
Antonín
Kučera
Charles University, PRAGUE 1, CZECH REPUBLIC
André
Nies
University of Auckland, AUCKLAND, NEW ZEALAND
Dan
Turetsky
Universität Wien, WIEN, AUSTRIA
Coherent randomness tests, $K$-trivial sets
We introduce Oberwolfach randomness, a notion within Demuth's framework of statistical tests with moving components; here the components' movement has to be coherent across levels. We show that a ML-random set computes all $K$-trivial sets if and only if it is not Oberwolfach random, and indeed that there is a $K$-trivial set which is not computable from any Oberwolfach random set. We show that Oberwolfach random sets satisfy e ffective versions of almost-everywhere theorems of analysis, such as the Lebesgue density theorem and Doob's martingale convergence theorem. We also show that random sets which are not Oberwolfach random satisfy highness properties (such as LR-hardness) which mean they are close to computing the halting problem. A consequence of these results is that a ML-random set failing the e ffective version of Lebesgue's density theorem for closed sets must compute all $K$-trivial sets. Combined with a recent result by Day and Miller, this gives a positive solution to the ML-covering problem of algorithmic randomness. On the other hand these results settle stronger variants of the covering problem in the negative: no low ML-random set computes all $K$-trivial sets, and not every $K$-trivial set is computable from both halves of a random set.
Mathematical logic and foundations
General
773
812
10.4171/JEMS/602
http://www.ems-ph.org/doi/10.4171/JEMS/602
Simple zeros of degree 2 $L$-functions
Andrew
Booker
University of Bristol, BRISTOL, UNITED KINGDOM
$L$-functions, modular forms, simple zeros
We prove that the complete $L$-functions of classical holomorphic newforms have infi nitely many simple zeros.
Number theory
General
813
823
10.4171/JEMS/603
http://www.ems-ph.org/doi/10.4171/JEMS/603
The geometric genus of hypersurface singularities
András
Némethi
Hungarian Academy of Sciences, BUDAPEST, HUNGARY
Baldur
Sigurdsson
Hungarian Academy of Sciences, BUDAPEST, HUNGARY
Normal surface singularities, hypersurface singularities, links of singularities, Newton non-degenerate singularities, geometric genus, plumbing graphs, $\mathbb Q$-homology spheres, lattice cohomology, path lattice cohomology, Heegaard–Floer homology, Seiberg–Witten invariant
Using the path lattice cohomology we provide a conceptual topological characterization of the geometric genus for certain complex normal surface singularities with rational homology sphere links, which is uniformly valid for all superisolated and Newton non-degenerate hypersurface singularities.
Several complex variables and analytic spaces
Algebraic geometry
Algebraic topology
Manifolds and cell complexes
825
851
10.4171/JEMS/604
http://www.ems-ph.org/doi/10.4171/JEMS/604
Fine scales of decay of operator semigroups
Charles J.K.
Batty
University of Oxford, OXFORD, UNITED KINGDOM
Ralph
Chill
Technische Universität Dresden, DRESDEN, GERMANY
Yuri
Tomilov
Nicolaus Copernicus University, TORUŃ, POLAND
Motivated by potential applications to partial differential equations, we develop a theory of fine scales of decay rates for operator semigroups. The theory contains, unifies, and extends several notable results in the literature on decay of operator semigroups and yields a number of new ones. Its core is a new operator-theoretical method of deriving rates of decay combining ingredients from functional calculus and complex, real and harmonic analysis. It also leads to several results of independent interest.
Operator theory
Ordinary differential equations
853
929
10.4171/JEMS/605
http://www.ems-ph.org/doi/10.4171/JEMS/605
5
Scaling limit and cube-root fluctuations in SOS surfaces above a wall
Pietro
Caputo
Università degli studi Roma Tre, ROMA, ITALY
Eyal
Lubetzky
New York University, NEW YORK, UNITED STATES
Fabio
Martinelli
Università degli Studi Roma Tre, ROMA, ITALY
Allan
Sly
University of California at Berkeley, BERKELEY, UNITED STATES
Fabio Lucio
Toninelli
École Normale Supérieure de Lyon, LYON CEDEX 07, FRANCE
SOS model, scaling limits, loop ensembles, random surface models
Consider the classical $(2+1)$-dimensional Solid-On-Solid model above a hard wall on an $L \times L$ box of $\mathbb Z^2$. The model describes a crystal surface by assigning a non-negative integer height $\eta_x$ to each site $x$ in the box and 0 heights to its boundary. The probability of a surface configuration $\eta$ is proportional to $\mathrm {exp}(-\beta \mathcal{H}(\eta))$, where $\beta$ is the inverse-temperature and $\mathcal{H}(\eta)$ sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough temperatures. First we show that with high probability (w.h.p.) the height of almost all sites is concentrated on two levels, $H(L) = \lfloor (1/4\beta)\mathrm {log} L \rfloor$ and $H(L)-1$. Moreover, for most values of $L$ the height is concentrated on the single value $H(L)$. Next, we study the ensemble of level lines corresponding to the heights $(H(L),H(L)-1,\ldots)$. We prove that w.h.p.\ there is a unique macroscopic level line for each height. Furthermore, when taking a diverging sequence of system sizes $L_k$, the rescaled macroscopic level line at height $H(L_k)-n$ has a limiting shape if the fractional parts of $(1/4\beta)\mathrm {log} L_k$ converge to a noncritical value. The scaling limit is an explicit convex subset of the unit square $Q$ and its boundary has a flat component on the boundary of $Q$. Finally, the highest macroscopic level line has $L_k^{1/3+o(1)}$ fluctuations along the flat part of the boundary of its limiting shape.
Probability theory and stochastic processes
Statistical mechanics, structure of matter
931
995
10.4171/JEMS/606
http://www.ems-ph.org/doi/10.4171/JEMS/606
Upper bounds for the number of resonances on geometrically finite hyperbolic manifolds
David
Borthwick
Emory University, ATLANTA, UNITED STATES
Colin
Guillarmou
Ecole Normale Superieure, PARIS CEDEX 05, FRANCE
Spectral geometry, hyperbolic manifolds, resonances
On geometrically finite hyperbolic manifolds $\Gamma\backslash\mathbb H^{d}$, including those with non-maximal rank cusps, we give upper bounds on the number $N(R)$ of resonances of the Laplacian in disks of size $R$ as $R \to \infty$. In particular, if the parabolic subgroups of $\Gamma$ satisfy a certain Diophantine condition, the bound is $N(R)=\mathcal O(R^d (\mathrm {log} R)^{d+1})$.
Global analysis, analysis on manifolds
Partial differential equations
997
1041
10.4171/JEMS/607
http://www.ems-ph.org/doi/10.4171/JEMS/607
Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains
Yannick
Privat
Université Pierre et Marie Curie (Paris 6), PARIS CEDEX 05, FRANCE
Emmanuel
Trélat
Université Pierre et Marie Curie (Paris 6), PARIS CEDEX 05, FRANCE
Enrique
Zuazua
Universidad Autónoma de Madrid, MADRID, SPAIN
Wave equation, Schrödinger equation, observability inequality, optimal design, spectral decomposition, ergodic properties, quantum ergodicity
We consider the wave and Schrödinger equations on a bounded open connected subset $\Omega$ of a Riemannian manifold, with Dirichlet, Neumann or Robin boundary conditions whenever its boundary is nonempty. We observe the restriction of the solutions to a measurable subset $\omega$ of $\Omega$ during a time interval $[0, T]$ with $T>0$. It is well known that, if the pair $(\omega,T)$ satisfies the Geometric Control Condition ($\omega$ being an open set), then an observability inequality holds guaranteeing that the total energy of solutions can be estimated in terms of the energy localized in $\omega \times (0, T)$. We address the problem of the optimal location of the observation subset $\omega$ among all possible subsets of a given measure or volume fraction. A priori this problem can be modeled in terms of maximizing the observability constant, but from the practical point of view it appears more relevant to model it in terms of maximizing an average either over random initial data or over large time. This leads us to define a new notion of observability constant, either randomized, or asymptotic in time. In both cases we come up with a spectral functional that can be viewed as a measure of eigenfunction concentration. Roughly speaking, the subset $\omega$ has to be chosen so to maximize the minimal trace of the squares of all eigenfunctions. Considering the convexified formulation of the problem, we prove a no-gap result between the initial problem and its convexified version, under appropriate quantum ergodicity assumptions, and compute the optimal value. Our results reveal intimate relations between shape and domain optimization, and the theory of quantum chaos (more precisely, quantum ergodicity properties of the domain $\Omega$). We prove that in 1D a classical optimal set exists only for exceptional values of the volume fraction, and in general one expects relaxation to occur and therefore classical optimal sets not to exist. We then provide spectral approximations and present some numerical simulations that fully confirm the theoretical results in the paper and support our conjectures. Finally, we provide several remedies to nonexistence of an optimal domain. We prove that when the spectral criterion is modified to consider a weighted one in which the high frequency components are penalized, the problem has then a unique classical solution determined by a finite number of low frequency modes. In particular the maximizing sequence built from spectral approximations is stationary.
Partial differential equations
Calculus of variations and optimal control; optimization
Global analysis, analysis on manifolds
Systems theory; control
1043
1111
10.4171/JEMS/608
http://www.ems-ph.org/doi/10.4171/JEMS/608
A cluster algebra approach to $q$-characters of Kirillov–Reshetikhin modules
David
Hernandez
Université Paris Diderot – Paris 7, Paris Rive Gauche UMR 7586, PARIS CEDEX 13, FRANCE
Bernard
Leclerc
Université de Caen, CAEN CEDEX, FRANCE
Quantum affine algebra, cluster algebras, $q$-characters, Kirillov–Reshetikhin modules, geometric character formula
We describe a cluster algebra algorithm for calculating $q$-characters of Kirillov–Reshetikhin modules for any untwisted quantum affine algebra $U_q(\widehat{\mathfrak{g}})$. This yields a geometric $q$-character formula for tensor products of Kirillov–Reshetikhin modules. When $\mathfrak g$ is of type $A, D, E$, this formula extends Nakajima's formula for $q$-characters of standard modules in terms of homology of graded quiver varieties.
Nonassociative rings and algebras
Commutative rings and algebras
1113
1159
10.4171/JEMS/609
http://www.ems-ph.org/doi/10.4171/JEMS/609
6
On the Klainerman–Machedon conjecture for the quantum BBGKY hierarchy with self-interaction
Xuwen
Chen
University of Rochester, ROCHESTER, UNITED STATES
Justin
Holmer
Brown University, PROVIDENCE, UNITED STATES
BBGKY Hierarchy, $n$-particle Schrödinger Equation, Klainerman–Machedon space-time Bound, quantum Kac program
We consider the 3D quantum BBGKY hierarchy which corresponds to the $N$-particle Schrödinger equation. We assume the pair interaction is $N^{3\beta – 1} V (B^\beta)$. For the interaction parameter $\beta \in (0, 2/3)$, we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the $N \to \infty$ limit points satisfy the space-time bound conjectured by S. Klainerman and M. Machedon [45] in 2008. The energy bound was proven to hold for $\beta \in (0, 3/5)$ in [28]. This allows, in the case $\beta \in (0, 3/5)$, for the application of the Klainerman–Machedon uniqueness theorem and hence implies that the $N \to \infty$ limit of BBGKY is uniquely determined as a tensor product of solutions to the Gross–Pitaevskii equation when the $N$-body initial data is factorized. The first result in this direction in 3D was obtained by T. Chen and N. Pavlović [11] for $\beta \in (0, 1/4)$ and subsequently by X. Chen [15] for $\beta \in (0, 2/7)$. We build upon the approach of X. Chen but apply frequency localized Klainerman–Machedon collapsing estimates and the endpoint Strichartz estimate in the estimate of the “potential part” to extend the range to $\beta \in (0, 2/3)$. Overall, this provides an alternative approach to the mean-field program by L. Erdős, B. Schlein, and H.-T. Yau [28], whose uniqueness proof is based upon Feynman diagram combinatorics.
Partial differential equations
Quantum theory
1161
1200
10.4171/JEMS/610
http://www.ems-ph.org/doi/10.4171/JEMS/610
Representation stability for syzygies of line bundles on Segre–Veronese varieties
Claudiu
Raicu
University of Notre Dame, NOTRE DAME, UNITED STATES
Syzygies, representation stability, Segre varieties, Veronese varieties, chessboard complexes, matching complexes, packing complexes, asymptotic vanishing
The rational homology groups of packing complexes are important in algebraic geometry since they control the syzygies of line bundles on projective embeddings of products of projective spaces (Segre–Veronese varieties). These complexes are a common generalization of the multidimensional chessboard complexes and of the matching complexes of complete uniform hypergraphs, whose study has been a topic of interest in combinatorial topology. We prove that the multivariate version of representation stability, a notion recently introduced and studied by Church and Farb, holds for the homology groups of packing complexes. This allows us to deduce stability properties for the syzygies of line bundles on Segre–Veronese varieties. We provide bounds for when stabilization occurs and show that these bounds are sometimes sharp by describing the linear syzygies for a family of line bundles on Segre varieties. As a motivation for our investigation, we show in an appendix that Ein and Lazarsfeld’s conjecture on the asymptotic vanishing of syzygies of coherent sheaves on arbitrary projective varieties reduces to the case of line bundles on a product of (at most three) projective spaces.
Commutative rings and algebras
Combinatorics
Algebraic geometry
Algebraic topology
1201
1231
10.4171/JEMS/611
http://www.ems-ph.org/doi/10.4171/JEMS/611
Metric Diophantine approximation on the middle-third Cantor set
Yann
Bugeaud
Université de Strasbourg, STRASBOURG CEDEX, FRANCE
Arnaud
Durand
Université Paris-Sud, ORSAY CEDEX, FRANCE
Diophantine approximation, Hausdorff dimension, irrationality exponent, Cantor set, Mahler’s problem
Let $\mu \geq 2$ be a real number and let $\mathcal{M} (\mu)$ denote the set of real numbers approximable at order at least $\mu$ by rational numbers. More than eighty years ago, Jarník and, independently, Besicovitch established that the Hausdorff dimension of $\mathcal{M} (\mu)$ is equal to $2 / \mu$. We investigate the size of the intersection of $\mathcal{M} (\mu)$ with Ahlfors regular compact subsets of the interval $[0, 1]$. In particular, we propose a conjecture for the exact value of the dimension of $\mathcal{M} (\mu)$ intersected with the middle-third Cantor set and give several results supporting this conjecture. We show in particular that the conjecture holds for a natural probabilistic model that is intended to mimic the distribution of the rationals. The core of our study relies heavily on dimension estimates concerning the set of points lying in an Ahlfors regular set and approximated at a given rate by a system of random points.
Number theory
Measure and integration
Probability theory and stochastic processes
1233
1272
10.4171/JEMS/612
http://www.ems-ph.org/doi/10.4171/JEMS/612
Integrating central extensions of Lie algebras via Lie 2-groups
Christoph
Wockel
Universität Hamburg, HAMBURG, GERMANY
Chenchang
Zhu
Georg-August-Universität Göttingen, GÖTTINGEN, GERMANY
Infinite-dimensional Lie group, central extension, smooth group cohomology, group stack, Lie 2-group, integration of cocycles, Lie’s Third Theorem, 2-connected cover
The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups in the sense of [Get09, Hen08]. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of $\pi_ 2$ for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated to each finite-dimensional Lie algebra. In infinite dimensions, there is an obstruction for a central extension of Lie algebras to integrate to a central extension of Lie groups. This obstruction comes from non-trivial $\pi_2$2 for general Lie groups. We show that this obstruction may be overcome by integrating central extensions of Lie algebras not to Lie groups but to central extensions of étale Lie 2-groups. As an application, we obtain a generalization of Lie’s Third Theorem to infinite-dimensional Lie algebras.
Topological groups, Lie groups
Algebraic topology
Global analysis, analysis on manifolds
1273
1320
10.4171/JEMS/613
http://www.ems-ph.org/doi/10.4171/JEMS/613
Estimates on elliptic equations that hold only where the gradient is large
Cyril
Imbert
Université Paris-Est Créteil Val de Marne, CRÉTEIL CEDEX, FRANCE
Luis
Silvestre
University of Chicago, CHICAGO, UNITED STATES
Degenerate elliptic equations, regularity, viscosity solutions
We consider a function which is a viscosity solution of a uniformly elliptic equation only at those points where the gradient is large. We prove that the Hölder estimates and the Harnack inequality, as in the theory of Krylov and Safonov, apply to these functions.
Partial differential equations
1321
1338
10.4171/JEMS/614
http://www.ems-ph.org/doi/10.4171/JEMS/614
The freeness of ideal subarrangements of Weyl arrangements
Takuro
Abe
Kyushu University, FUKUOKA, JAPAN
Mohamed
Barakat
Universität Siegen, SIEGEN, GERMANY
Michael
Cuntz
Leibniz Universität Hannover, HANNOVER, GERMANY
Torsten
Hoge
Leibniz Universität Hannover, HANNOVER, GERMANY
Hiroaki
Terao
Hokkaido University, SAPPORO, JAPAN
Arrangement of hyperplanes, root system,Weyl arrangement, free arrangement, ideals, dual partition theorem
A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers–Tymoczko. In particular, when an ideal subarrangement is equal to the entireWeyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald. The proof of the main theorem is classification-free. It heavily depends on the theory of free arrangements and thus greatly differs from the earlier proofs of the formula
Several complex variables and analytic spaces
Combinatorics
Nonassociative rings and algebras
1339
1348
10.4171/JEMS/615
http://www.ems-ph.org/doi/10.4171/JEMS/615
Twists and resonance of $L$-functions, I
Jerzy
Kaczorowski
Adam Mickiewicz University, POZNAN, POLAND
Alberto
Perelli
Università di Genova, GENOVA, ITALY
$L$-functions, Selberg class, twists, resonance
We obtain the basic analytic properties, i.e. meromorphic continuation, polar structure and bounds for the order of growth, of all the nonlinear twists with exponents $≤ 1 / d$ of the $L$-functions of any degree $d ≥ 1$ in the extended Selberg class. In particular, this solves the resonance problem in all such cases.
Number theory
General
1349
1389
10.4171/JEMS/616
http://www.ems-ph.org/doi/10.4171/JEMS/616
Finite orbit decomposition of real flag manifolds
Bernhard
Krötz
Universität Paderborn, PADERBORN, GERMANY
Henrik
Schlichtkrull
University of Copenhagen, COPENHAGEN Ø, DENMARK
Flag manifold, orbit decomposition, spherical subgroup
Let $G$ be a connected real semi-simple Lie group and $H$ a closed connected subgroup. Let $P$ be a minimal parabolic subgroup of $G$. It is shown that $H$ has an open orbit on the flag manifold $G/P$ if and only if it has fi nitely many orbits on $G/P$. This confi rms a conjecture by T. Matsuki.
Topological groups, Lie groups
Algebraic geometry
1391
1403
10.4171/JEMS/617
http://www.ems-ph.org/doi/10.4171/JEMS/617
7
Modular generalized Springer correspondence I: the general linear group
Pramod
Achar
Louisiana State University, BATON ROUGE, UNITED STATES
Anthony
Henderson
University of Sydney, SYDNEY NSW, AUSTRALIA
Daniel
Juteau
Université de Caen Basse-Normandie, CAEN CEDEX, FRANCE
Simon
Riche
Université Blaise Pascal, AUBIÈRE CEDEX, FRANCE
Springer correspondence, nilpotent cone, nilpotent orbits, perverse sheaves, cuspidal pairs, recollement, Fourier transform, modular reduction
We define a generalized Springer correspondence for the group GL($n$) over any field. We also determine the cuspidal pairs, and compute the correspondence explicitly. Finally we define a stratification of the category of equivariant perverse sheaves on the nilpotent cone of GL($n$) satisfying the ‘recollement’ properties, and with subquotients equivalent to categories of representations of a product of symmetric groups.
Nonassociative rings and algebras
Algebraic geometry
Group theory and generalizations
1405
1436
10.4171/JEMS/618
http://www.ems-ph.org/doi/10.4171/JEMS/618
$L^p$ norms of higher rank eigenfunctions and bounds for spherical functions
Simon
Marshall
University of Wisconsin, MADISON, UNITED STATES
$L^p$ norms, symmetric spaces, spherical functions
We prove almost sharp upper bounds for the $L^p$ norms of eigenfunctions of the full ring of invariant differential operators on a compact locally symmetric space, as well as their restrictions to maximal flat subspaces. Our proof combines techniques from semiclassical analysis with harmonic theory on reductive groups, and makes use of new asymptotic bounds for spherical functions that are of independent interest.
Topological groups, Lie groups
Partial differential equations
1437
1493
10.4171/JEMS/619
http://www.ems-ph.org/doi/10.4171/JEMS/619
Sums of squares of polynomials with rational coefficients
Claus
Scheiderer
Universität Konstanz, KONSTANZ, GERMANY
Sums of squares, rational coefficients, Hilbert's 17th problem, real plane quartics, exact positivity certificates, semidefinite programming
We construct families of explicit (homogeneous) polynomials $f$ over $\mathbb Q$ that are sums of squares of polynomials over $\mathbb R$, but not over $\mathbb Q$. Whether or not such examples exist was an open question originally raised by Sturmfels. In the case of ternary quartics we prove that our construction yields all possible examples. We also study representations of the $f$ we construct as sums of squares of rational functions over $\mathbb Q$, proving lower bounds for the possible degrees of denominators. For deg$(f) = 4$, or for ternary sextics, we obtain explicit such representations with the minimum degree of the denominators.
Algebraic geometry
Number theory
Operations research, mathematical programming
1495
1513
10.4171/JEMS/620
http://www.ems-ph.org/doi/10.4171/JEMS/620
Symplectic fillings of lens spaces as Lefschetz fibrations
Mohan
Bhupal
Middle East Technical University, ANKARA, TURKEY
Burak
Ozbagci
Koç University, ISTANBUL, TURKEY
Symplectic fillings, lens spaces, canonical contact structure, Lefschetz fibrations
We construct a positive allowable Lefschetz fibration over the disk on any minimal (weak) symplectic filling of the canonical contact structure on a lens space. Using this construction we prove that any minimal symplectic filling of the canonical contact structure on a lens space is obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding complex two-dimensional cyclic quotient singularity.
Manifolds and cell complexes
Several complex variables and analytic spaces
Differential geometry
1515
1535
10.4171/JEMS/621
http://www.ems-ph.org/doi/10.4171/JEMS/621
Affine cones over smooth cubic surfaces
Ivan
Cheltsov
Edinburgh University, EDINBURGH, UNITED KINGDOM
Jihun
Park
Institute for Basic Sciences (IBS), GYEONGBUK, SOUTH KOREA
Joonyeong
Won
KIAS, SEOUL, SOUTH KOREA
affine cone, $\alpha$-invariant, anticanonical divisor, cylinder, del Pezzo surface, $\mathbb{G}_a$-action, log canonical singularity
We show that affine cones over smooth cubic surfaces do not admit non-trivial $\mathbb{G}_{a}$-actions.
Algebraic geometry
1537
1564
10.4171/JEMS/622
http://www.ems-ph.org/doi/10.4171/JEMS/622
Proof of the cosmic no-hair conjecture in the $\mathbb T^3$-Gowdy symmetric Einstein–Vlasov setting
Håkan
Andréasson
Chalmers University of Technology, GÖTEBORG, SWEDEN
Hans
Ringström
KTH Royal Institute of Technology, STOCKHOLM, SWEDEN
Einstein–Vlasov system, cosmic no-hair conjecture, Gowdy symmetry
The currently preferred models of the universe undergo accelerated expansion induced by dark energy. One model for dark energy is a positive cosmological constant. It is consequently of interest to study Einstein’s equations with a positive cosmological constant coupled to matter satisfying the ordinary energy conditions: the dominant energy condition etc. Due to the difficulty of analysing the behaviour of solutions to Einstein’s equations in general, it is common to either study situations with symmetry, or to prove stability results. In the present paper, we do both. In fact, we analyse, in detail, the future asymptotic behaviour of $\mathbb T^3$-Gowdy symmetric solutions to the Einstein–Vlasov equations with a positive cosmological constant. In particular, we prove the cosmic no-hair conjecture in this setting. However, we also prove that the solutions are future stable (in the class of all solutions). Some of the results hold in a more general setting. In fact, we obtain conclusions concerning the causal structure of $\mathbb T^2$-symmetric solutions, assuming only the presence of a positive cosmological constant, matter satisfying various energy conditions and future global existence. Adding the assumption of $\mathbb T^3$-Gowdy symmetry to this list of requirements, we obtain $C^0$-estimates for all but one of the metric components. There is consequently reason to expect that many of the results presented in this paper can be generalised to other types of matter.
Partial differential equations
Relativity and gravitational theory
1565
1650
10.4171/JEMS/623
http://www.ems-ph.org/doi/10.4171/JEMS/623
8
Periodic knots and Heegaard Floer correction terms
Stanislav
Jabuka
University of Nevada Reno, RENO, UNITED STATES
Swatee
Naik
University of Nevada Reno, RENO, UNITED STATES
Knot, periodic, Heegaard Floer, correction terms
We derive new obstructions to periodicity of classical knots by employing the Heegaard Floer correction terms of the finite cyclic branched covers of the knots. Applying our results to 2-fold covers, we demonstrate through numerous examples that our obstructions are successful where many existing periodicity obstructions fail. A combination of previously known periodicity obstructions and the results presented here leads to a nearly complete (with the exception of a single knot) classification of alternating, periodic, 12-crossing knots with odd prime periods. For the case of alternating knots with 13, 14 and 15 crossings, we give a complete list of all periodic knots with odd prime periods $q > 3$.
Manifolds and cell complexes
1651
1674
10.4171/JEMS/624
http://www.ems-ph.org/doi/10.4171/JEMS/624
Complete bounded embedded complex curves in $\mathbb C^2$
Antonio
Alarcón
Universidad de Granada, GRANADA, SPAIN
Francisco
López
Universidad de Granada, GRANADA, SPAIN
Riemann surfaces, complex curves, complete holomorphic embeddings
We prove that any convex domain of $\mathbb C^2$ carries properly embedded complete complex curves. In particular, we give the first examples of complete bounded embedded complex curves in $\mathbb C^2$.
Several complex variables and analytic spaces
1675
1705
10.4171/JEMS/625
http://www.ems-ph.org/doi/10.4171/JEMS/625
Local profile of fully bubbling solutions to $\mathrm {SU} (n+1)$ Toda systems
Chang-Shou
Lin
National Taiwan University, TAIPEI, TAIWAN
Juncheng
Wei
University of British Columbia, VANCOUVER, CANADA
Lei
Zhang
University of Florida, GAINESVILLE, UNITED STATES
$\mathrm {SU} (n+1)$ Toda system, non-degeneracy, a priori estimate, classification theorem, fully bubbling, blowup solutions
In this article we prove that for locally defined singular $\mathrm {SU} (n+1)$ Toda systems in $\mathbb R^2$, the profile of fully bubbling solutions near the singular source can be accurately approximated by global solutions. The main ingredients of our new approach are the classification theorem of Lin–Wei–Ye [22] and the non-degeneracy of the linearized Toda system [22], which let us overcome the difficulties that come from lack of symmetry and the singular source.
Partial differential equations
1707
1728
10.4171/JEMS/626
http://www.ems-ph.org/doi/10.4171/JEMS/626
Null structure and local well-posedness in the energy class for the Yang–Mills equations in Lorenz gauge
Sigmund
Selberg
University of Bergen, BERGEN, NORWAY
Achenef
Tesfahun
University of Bergen, BERGEN, NORWAY
Yang–Mills equations, well-posedness, Lorenz gauge, null structure
We demonstrate null structure in the Yang–Mills equations in Lorenz gauge. Such structure was found in Coulomb gauge by Klainerman and Machedon, who used it to prove global wellposedness for finite-energy data in the temporal gauge by passing to local Coulomb gauges via Uhlenbeck’s Lemma. Compared with the Coulomb gauge, the Lorenz gauge has the advantage – shared with the temporal gauge – that it can be imposed globally in space even for large solutions. Using the null structure and bilinear space-time estimates, we also prove local-in-time wellposedness of the Yang–Mills equations in Lorenz gauge for data with finite energy, with a time of existence depending on the initial energy and on the $H^s \times H^{s–1}$-norm of the initial gauge potential, for some choice of $s < 1$ sufficiently close to 1.
Partial differential equations
1729
1752
10.4171/JEMS/627
http://www.ems-ph.org/doi/10.4171/JEMS/627
Stable lattices and the diagonal group
Uri
Shapira
Technion - Israel Institute of Technology, HAIFA, ISRAEL
Barak
Weiss
Tel Aviv University, TEL AVIV, ISRAEL
Lattices, stable, diagonal group
Inspired by work of McMullen, we show that any orbit of the diagonal group in the space of lattices accumulates on the set of stable lattices. As consequences, we settle a conjecture of Ramharter concerning the asymptotic behavior of the Mordell constant, and reduce Minkowski’s conjecture on products of linear forms to a geometric question, yielding two new proofs of the conjecture in dimensions up to 7.
Dynamical systems and ergodic theory
Number theory
1753
1767
10.4171/JEMS/628
http://www.ems-ph.org/doi/10.4171/JEMS/628
Rational Pontryagin classes and functor calculus
Rui
Reis
Universität Münster, MÜNSTER, GERMANY
Michael
Weiss
Universität Münster, MÜNSTER, GERMANY
Pontryagin classes, smoothing theory, functor calculus
It is known that in the integral cohomology of $B \mathrm {SO}(2m)$, the square of the Euler class is the same as the Pontryagin class in degree $4m$. Given that the Pontryagin classes extend rationally to the cohomology of $B$STOP($2m$), it is reasonable to ask whether the same relation between the Euler class and the Pontryagin class in degree $4m$ is still valid in the rational cohomology of $B$STOP($2m$). In this paper we reformulate the hypothesis as a statement in differential topology, and also in a functor calculus setting.
Manifolds and cell complexes
Algebraic topology
1769
1811
10.4171/JEMS/629
http://www.ems-ph.org/doi/10.4171/JEMS/629
Ricci flow on quasiprojective manifolds II
John
Lott
University of California, BERKELEY, UNITED STATES
Zhou
Zhang
University of Sydney, SYDNEY, AUSTRALIA
Ricci flow, Kähler, quasiprojective
We study the Ricci flow on complete Kähler metrics that live on the complement of a divisor in a compact complex manifold. In earlier work, we considered finite-volume metrics which, at spatial infinity, are transversely hyperbolic. In the present paper we consider three different types of spatial asymptotics: cylindrical, bulging and conical. We show that in each case, the asymptotics are preserved by the Kähler–Ricci flow.We address long-time existence, parabolic blowdown limits and the role of the Kähler–Ricci flow on the divisor.
Differential geometry
Several complex variables and analytic spaces
1813
1854
10.4171/JEMS/630
http://www.ems-ph.org/doi/10.4171/JEMS/630
Classification of higher rank orbit closures in ${\mathcal H^{\mathrm{odd}}(4)}$
David
Aulicino
University of Chicago, CHICAGO, UNITED STATES
Duc-Manh
Nguyen
Université de Bordeaux I, TALENCE CEDEX, FRANCE
Alex
Wright
Stanford University, STANFORD, UNITED STATES
Translation surface, Abelian differential, Teichmüller dynamics, affine invariant submanifold, orbit closure, Prym locus, Teichmüller curves
The moduli space of genus 3 translation surfaces with a single zero has two connected components. We show that in the odd connected component ${\mathcal H^{\mathrm{odd}}(4)}$ the only ${GL^+(2,\mathbb R)}$ orbit closures are closed orbits, the Prym locus ${\tilde{\mathcal{Q}}(3,-1^3)}$, and ${\mathcal H^{\mathrm{odd}}(4)}$. Together with work of Matheus–Wright, this implies that there are only finitely many non-arithmetic closed orbits (Teichmüller curves) in $\mathcal H^{\mathrm{odd}}(4)$ outside of the Prym locus.
Dynamical systems and ergodic theory
Several complex variables and analytic spaces
1855
1872
10.4171/JEMS/631
http://www.ems-ph.org/doi/10.4171/JEMS/631
9
Structure of classical (finite and affine) $\mathcal W$-algebras
Alberto
De Sole
Università di Roma La Sapienza, ROMA, ITALY
Victor
Kac
Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES
Daniele
Valeri
SISSA, TRIESTE, ITALY
$\mathcal W$-algebra, Poisson algebra, Poisson vertex algebra, Slodowy slice, Hamiltonian reduction, Zhu algebra, Miura map
First, we derive an explicit formula for the Poisson bracket of the classical finite $\mathcal W$-algebra $\mathcal W^{\mathrm {fin}}(\mathfrak g,f)$, the algebra of polynomial functions on the Slodowy slice associated to a simple Lie algebra $\mathfrak g$ and its nilpotent element $f$. On the other hand, we produce an explicit set of generators and we derive an explicit formula for the Poisson vertex algebra structure of the classical affine $\mathcal W$-algebra $\mathcal W(\mathfrak g,f)$. As an immediate consequence, we obtain a Poisson algebra isomorphism between $\mathcal W^{\mathrm {fin}}(\mathfrak g,f)$ and the Zhu algebra of $\mathcal W(\mathfrak g,f)$. We also study the generalized Miura map for classical $\mathcal W$-algebras.
Nonassociative rings and algebras
Dynamical systems and ergodic theory
1873
1908
10.4171/JEMS/632
http://www.ems-ph.org/doi/10.4171/JEMS/632
The strong profinite genus of a finitely presented group can be infinite
Martin
Bridson
University of Oxford, OXFORD, UNITED KINGDOM
Profinite completion, profinite genus, Grothendieck pairs
We construct the first examples of finitely-presented, residually-finite groups $\Gamma$ that contain an infinite sequence of non-isomorphic finitely-presented subgroups $P_n \hookrightarrow \Gamma$ such that the inclusion maps induce isomorphisms of profinite completions $\widehat{P}_n \cong \widehat {\Gamma}$.
Group theory and generalizations
1909
1918
10.4171/JEMS/633
http://www.ems-ph.org/doi/10.4171/JEMS/633
Pseudo-holomorphic functions at the critical exponent
Laurent
Baratchart
INRIA, SOPHIA ANTIPOLIS CEDEX, FRANCE
Alexander
Borichev
Aix Marseille Université, MARSEILLE CEDEX 13, FRANCE
Slah
Chaabi
INRIA, SOPHIA ANTIPOLIS CEDEX, FRANCE
Pseudo-holomorphic functions, Hardy spaces, conjugate Beltrami equation, nonstrictly elliptic equations, Dirichlet problem
We study Hardy classes on the disk associated to the equation $\bar \partial w= \alpha \bar w$ for $\alpha \in L^r$ with $2 \leq r < \infty$. The paper seems to be the first to deal with the case $r=2$. We prove an analog of the M. Riesz theorem and a topological converse to the Bers similarity principle. Using the connection between pseudo-holomorphic functions and conjugate Beltrami equations, we deduce well-posedness on smooth domains of the Dirichlet problem with weighted $L^p$ boundary data for 2D isotropic conductivity equations whose coefficients have logarithm in $W^{1,2}$. In particular these are not strictly elliptic. Our results depend on a new multiplier theorem for $W^{1,2}_0$-functions.
Functions of a complex variable
Partial differential equations
Functional analysis
1919
1960
10.4171/JEMS/634
http://www.ems-ph.org/doi/10.4171/JEMS/634
Optimal decay estimates for the general solution to a class of semi-linear dissipative hyperbolic equations
Marina
Ghisi
Università degli Studi di Pisa, PISA, ITALY
Massimo
Gobbino
Università di Pisa, PISA, ITALY
Alain
Haraux
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
Semi-linear hyperbolic equation, dissipative hyperbolic equations, slow solutions, decay estimates, energy estimates
We consider a class of semi-linear dissipative hyperbolic equations in which the operator associated to the linear part has a nontrivial kernel. Under appropriate assumptions on the nonlinear term, we prove that all solutions decay to 0, as $t \to \infty$, at least as fast as a suitable negative power of $t$. Moreover, we prove that this decay rate is optimal in the sense that there exists a nonempty open set of initial data for which the corresponding solutions decay exactly as that negative power of $t$. Our results are stated and proved in an abstract Hilbert space setting, and then applied to partial differential equations.
Partial differential equations
1961
1982
10.4171/JEMS/635
http://www.ems-ph.org/doi/10.4171/JEMS/635
Forking and JSJ decompositions in the free group
Chloé
Perin
Université de Strasbourg, STRASBOURG CEDEX, FRANCE
Rizos
Sklinos
Université Claude Bernard Lyon 1, VILLEURBANNE CEDEX, FRANCE
Forking independence, torsion-free hyperbolic groups, curve complex
We give a description of the model-theoretic relation of forking independence in terms of the notion of JSJ decompositions in non-abelian free groups
Mathematical logic and foundations
Group theory and generalizations
1983
2017
10.4171/JEMS/636
http://www.ems-ph.org/doi/10.4171/JEMS/636
A minimization approach to hyperbolic Cauchy problems
Enrico
Serra
Politecnico di Torino, TORINO, ITALY
Paolo
Tilli
Politecnico di Torino, TORINO, ITALY
Nonlinear hyperbolic equations, mimimization, a priori estimates
Developing an original idea of De Giorgi, we introduce a new and purely variational approach to the Cauchy Problem for a wide class of defocusing hyperbolic equations. The main novel feature is that the solutions are obtained as limits of functions that minimize suitable functionals in space-time (where the initial data of the Cauchy Problem serve as prescribed boundary conditions). This opens up the way to new connections between the hyperbolic world and that of the calculus of variations. Also dissipative equations can be treated. Finally, we discuss several examples of equations that fit in this framework, including nonlocal equations, in particular equations with the fractional Laplacian.
Partial differential equations
Calculus of variations and optimal control; optimization
2019
2044
10.4171/JEMS/637
http://www.ems-ph.org/doi/10.4171/JEMS/637
Bounds on the Bondi energy by a flux of curvature
Spyros
Alexakis
University of Toronto, TORONTO, ONTARIO, CANADA
Arick
Shao
University of Toronto, TORONTO, ONTARIO, CANADA
Bondi, mass, energy, angular, momentum, curvature, flux, asymptotic, round, uniformization
We consider smooth null cones in a vacuum spacetime that extend to future null infinity. For such cones that are perturbations of shear-free outgoing null cones in Schwarzschild spacetimes, we prove bounds for the Bondi energy, momentum, and rate of energy loss. The bounds depend on the closeness between the given cone and a corresponding cone in a Schwarzschild spacetime, measured purely in terms of the differences between certain weighted $L^2$-norms of the spacetime curvature on the cones, and of the geometries of the spheres from which they emanate. This paper relies on the results in [1], which uniformly control the geometry of the given null cone up to infinity, as well as those of [18], which establish machinery for dealing with low regularities. A key step in this paper is the construction of a family of asymptotically round cuts of our cone, relative to which the Bondi energy is measured.
Relativity and gravitational theory
Partial differential equations
Differential geometry
2045
2106
10.4171/JEMS/638
http://www.ems-ph.org/doi/10.4171/JEMS/638
Balanced Viscosity (BV) solutions to infinite-dimensional rate-independent systems
Alexander
Mielke
Angewandte Analysis und Stochastik, BERLIN, GERMANY
Riccarda
Rossi
Università degli Studi di Brescia, BRESCIA, ITALY
Giuseppe
Savaré
Università di Pavia, PAVIA, ITALY
Rate-independent systems, energetic solutions, BV solutions, existence results, vanishing viscosity, time discretization
Balanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-independent flows by adding a superlinear vanishing-viscosity dissipation. We address the main issue of proving the existence of such limits for infinite-dimensional systems and of characterizing them by a couple of variational properties that combine a local stability condition and a balanced energy-dissipation identity. A careful description of the jump behavior of the solutions, of their differentiability properties, and of their equivalent representation by time rescaling is also presented. Our techniques rely on a suitable chain-rule inequality for functions of bounded variation in Banach spaces, on refined lower-semicontinuity compactness arguments, and on new BV-estimates that are of independent interest.
Partial differential equations
Ordinary differential equations
2107
2165
10.4171/JEMS/639
http://www.ems-ph.org/doi/10.4171/JEMS/639
10
Symplectomorphism group relations and degenerations of Landau–Ginzburg models
Colin
Diemer
University of Miami, CORAL GABLES, UNITED STATES
Ludmil
Katzarkov
University of Miami, CORAL GABLES, UNITED STATES
Gabriel
Kerr
Kansas State University, MANHATTAN, UNITED STATES
Homological mirror symmetry, symplectomorphisms, Landau–Ginzburg models, minimal model program, toric varieties
We describe explicit relations in the symplectomorphism groups of hypersurfaces in toric stacks. To define the elements involved, we construct a proper stack of these hypersurfaces whose boundary represents stable pair degenerations. Our relations arise through the study of the one-dimensional strata of this stack. The results are then examined from the perspective of homological mirror symmetry where we view sequences of relations as maximal degenerations of Landau–Ginzburg models. We then study the $B$-model mirror to these degenerations, which gives a new mirror symmetry approach to the minimal model program.
Differential geometry
2167
2271
10.4171/JEMS/640
http://www.ems-ph.org/doi/10.4171/JEMS/640
A categorification of non-crossing partitions
Andrew
Hubery
Universität Bielefeld, BIELEFELD, GERMANY
Henning
Krause
Universität Bielefeld, BIELEFELD, GERMANY
Non-crossing partition, hereditary algebra, Grothendieck group, Weyl group, Coxeter group, exceptional sequence, symmetrisable generalised Cartan matrix, perpendicular category
We present a categorification of the non-crossing partitions given by crystallographic Coxeter groups. This involves a category of certain bilinear lattices, which are essentially determined by a symmetrisable generalised Cartan matrix together with a particular choice of a Coxeter element. Examples arise from Grothendieck groups of hereditary artin algebras.
Associative rings and algebras
Combinatorics
Group theory and generalizations
2273
2313
10.4171/JEMS/641
http://www.ems-ph.org/doi/10.4171/JEMS/641
Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem
Jacques
Féjoz
IMCCE, PARIS, FRANCE
Marcel
Guàrdia
Université Paris 7 Denis Diderot, PARIS CEDEX 13, FRANCE
Vadim
Kaloshin
University of Maryland, COLLEGE PARK, UNITED STATES
Pablo
Roldán
Escola Tècnica Superior d'Enginyeria Industrial de Barcelona, BARCELONA, SPAIN
Three-body problem, instability, resonance, hyperbolicity, Mather mechanism, Arnol’d diffusion, Solar System, Asteroid Belt, Kirkwood gap
We study the dynamics of the restricted planar three-body problem near mean motion resonances, i.e. a resonance involving the Keplerian periods of the two lighter bodies revolving around the most massive one. This problem is often used to model Sun–Jupiter–asteroid systems. For the primaries (Sun and Jupiter), we pick a realistic mass ratio $\mu=10^{-3}$ and a small eccentricity $e_0>0$. The main result is a construction of a variety of non local diffusing orbits which show a drastic change of the osculating (instant) eccentricity of the asteroid, while the osculating semi major axis is kept almost constant. The proof relies on the careful analysis of the circular problem, which has a hyperbolic structure, but for which diffusion is prevented by KAM tori. In the proof we verify certain non-degeneracy conditions numerically. Based on the work of Treschev, it is natural to conjecture that the time of diffusion for this problem is $\sim \frac{-\ln (\mu e_0)}{\mu^{3/2} e_0}$. We expect our instability mechanism to apply to realistic values of $e_0$ and we give heuristic arguments in its favor. If so, the applicability of Nekhoroshev theory to the three-body problem as well as the long time stability become questionable. It is well known that, in the Asteroid Belt, located between the orbits of Mars and Jupiter, the distribution of asteroids has the so-called Kirkwood gaps exactly at mean motion resonances of low order. Our mechanism gives a possible explanation of their existence. To relate the existence of Kirkwood gaps with Arnol'd diffusion, we also state a conjecture on its existence for a typical $\epsilon$-perturbation of the product of the pendulum and the rotator. Namely, we predict that a positive conditional measure of initial conditions concentrated in the main resonance exhibits Arnol’d diffusion on time scales $\frac{- \ln \epsilon }{\epsilon^{2}}$.
Mechanics of particles and systems
Dynamical systems and ergodic theory
2315
2403
10.4171/JEMS/642
http://www.ems-ph.org/doi/10.4171/JEMS/642
Overconvergent subanalytic subsets in the framework of Berkovich spaces
Florent
Martin
Universität Regensburg, REGENSBURG, GERMANY
Berkovich spaces, semianalytic sets, subanalytic sets, overconvergent
We study the class of overconvergent subanalytic subsets of a $k$-affinoid space $X$ when $k$ is a non-archimedean field. These are the images along the projection $X \times \mathbb B^n \to X$ of subsets defined with inequalities between functions of $X \times \mathbb B^n$ which are overconvergent in the variables of $\mathbb B^n$. In particular, we study the local nature, with respect to $X$, of overconvergent subanalytic subsets. We show that they behave well with respect to the Berkovich topology, but not to the $G$-topology. This gives counter-examples to previous results on the subject, and a way to correct them. Moreover, we study the case dim$(X)=2$, for which a simpler characterisation of overconvergent subanalytic subsets is proven.
Algebraic geometry
Mathematical logic and foundations
Field theory and polynomials
Several complex variables and analytic spaces
2405
2457
10.4171/JEMS/643
http://www.ems-ph.org/doi/10.4171/JEMS/643
11
On subvarieties with ample normal bundle
John Christian
Ottem
University of Oslo, OSLO, NORWAY
Ample normal bundles, Hartshorne’s conjecture, positive cycles
We show that a pseudoeffective $\mathbb R$-divisor has numerical dimension 0 if it is numerically trivial on a subvariety with ample normal bundle. This implies that the cycle class of a curve with ample normal bundle is big, which gives an affirmative answer to a conjecture of Peternell. We also give other positivity properties of such subvarieties.
Algebraic geometry
2459
2468
10.4171/JEMS/644
http://www.ems-ph.org/doi/10.4171/JEMS/644
Curves in $\mathbb R^d$ intersecting every hyperplane at most $d+1$ times
Imre
Bárány
Hungarian Academy of Sciences, BUDAPEST, HUNGARY
Jiří
Matoušek
Charles University, PRAHA 1, CZECH REPUBLIC
Attila
Pór
Western Kentucky University, BOWLING GREEN, UNITED STATES
Ramsey function, order type, convex curve, moment curve, Chebyshev system
By a curve in $\mathbb R^d$ we mean a continuous map $\gamma\:I\to\mathbb R^d$, where $I\subset\mathbb R$ is a closed interval. We call a curve $\gamma$ in $\mathbb R^d\: (≤ k)$-crossing if it intersects every hyperplane at most $k$ times (counted with multiplicity). The $(≤ d)$-crossing curves in $\mathbb R^d$ are often called convex curves and they form an important class; a primary example is the moment curve $\{(t,t^2,\ldots,t^d):t\in[0,1]\}$. They are also closely related to Chebyshev systems, which is a notion of considerable importance, e.g., in approximation theory. Our main result is that for every $d$ there is $M=M(d)$ such that every $(≤ d+1)$-crossing curve in $\mathbb R^d$ can be subdivided into at most $M\: (≤ d)$-crossing curve segments. As a consequence, based on the work of Eliáš, Roldán, Safernová, and the second author, we obtain an essentially tight lower bound for a geometric Ramsey-type problem in $\mathbb R^d$ concerning order-type homogeneous sequences of points, investigated in several previous papers.
Combinatorics
Convex and discrete geometry
2469
2482
10.4171/JEMS/645
http://www.ems-ph.org/doi/10.4171/JEMS/645
A Gromov–Winkelmann type theorem for flexible varieties
Hubert
Flenner
Ruhr-Universität Bochum, BOCHUM, GERMANY
Shulim
Kaliman
University of Miami, CORAL GABLES, UNITED STATES
Mikhail
Zaidenberg
Université Grenoble I, SAINT MARTIN D'HERES CEDEX, FRANCE
Affine varieties, group actions, one-parameter subgroups, transitivity
An affi ne variety $X$ of dimension ≥ 2 is called flexible if its special automorphism group SAut($X$) acts transitively on the smooth locus $X_{\mathrm {reg}}$ [1]. Recall that SAut($X$) is the subgroup of the automorphism group Aut($X$) generated by all one-parameter unipotent subgroups [1]. Given a normal, flexible, affi ne variety $X$ and a closed subvariety $Y$ in $X$ of codimension at least 2, we show that the pointwise stabilizer subgroup of $Y$ in the group SAut($X$) acts infi nitely transitively on the complement $X \setminus Y$, that is, $m$-transitively for any $m ≥ 1$. More generally we show such a result for any quasi-affi ne variety $X$ and codimension ≥ 2 subset $Y$ of $X$. In the particular case of $X = \mathbb A^n$, $n ≥ 2$, this yields a Theorem of Gromov and Winkelmann [8], [18].
Algebraic geometry
Several complex variables and analytic spaces
2483
2510
10.4171/JEMS/646
http://www.ems-ph.org/doi/10.4171/JEMS/646
Noncritical holomorphic functions on Stein spaces
Franc
Forstnerič
University of Ljubljana, LJUBLJANA, SLOVENIA
Holomorphic functions, critical points, Stein manifolds, Stein spaces, 1-convex manifolds, stratifications
In this paper we prove that every reduced Stein space admits a holomorphic function without critical points. Furthermore, every closed discrete subset of a reduced Stein space $X$ is the critical locus of a holomorphic function on $X$. We also show that for every complex analytic strati cation with nonsingular strata on a reduced Stein space there exists a holomorphic function whose restriction to every stratum is noncritical. These result provide some information on critical loci of holomorphic functions on desingularizations of Stein spaces. In particular, every 1-convex manifold admits a holomorphic function that is noncritical outside the exceptional variety.
Several complex variables and analytic spaces
Manifolds and cell complexes
Global analysis, analysis on manifolds
2511
2543
10.4171/JEMS/647
http://www.ems-ph.org/doi/10.4171/JEMS/647
Local cohomology modules supported at determinantal ideals
Gennady
Lyubeznik
University of Minnesota, MINNEAPOLIS, UNITED STATES
Anurag
Singh
University of Utah, SALT LAKE CITY, UNITED STATES
Uli
Walther
Purdue University, WEST LAFAYETTE, UNITED STATES
Determinantal ideals, local cohomology, vanishing theorems, $\mathcal D$-modules, $\mathcal F$-modules
We provide new results on the vanishing of local cohomology modules supported at ideals of minors of matrices over arbitrary commutative Noetherian rings. In the process, we compute the local cohomology of rings of polynomials with integer coefficients – supported at generic determinantal ideals – and also obtain results on $\mathcal F$-modules and $\mathcal D$-modules that are likely to be of independent interest.
Commutative rings and algebras
Algebraic geometry
2545
2578
10.4171/JEMS/648
http://www.ems-ph.org/doi/10.4171/JEMS/648
The Calderón problem in transversally anisotropic geometries
David
Dos Santos Ferreira
Université de Lorraine, CNRS, VANDOEUVRE-LÈS-NANCY, FRANCE
Yaroslav
Kurylev
University College London, LONDON, UNITED KINGDOM
Matti
Lassas
University of Helsinki, HELSINKI, FINLAND
Mikko
Salo
University of Jyväskylä, JYVÄSKYLÄ, FINLAND
Inverse boundary value problem, Calderón problem, Riemannian manifold, complex geometrical optics solution, boundary control method
We consider the anisotropic Calderón problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work [14], it was shown that a metric in a fixed conformal class is uniquely determined by boundary measurements under two conditions: (1) the metric is conformally transversally anisotropic (CTA), and (2) the transversal manifold is simple. In this paper we will consider geometries satisfying (1) but not (2). The first main result states that the boundary measurements uniquely determine a mixed Fourier transform/attenuated geodesic ray transform (or integral against a more general semiclassical limit measure) of an unknown coefficient. In particular, one obtains uniqueness results whenever the geodesic ray transform on the transversal manifold is injective. The second result shows that the boundary measurements in an infinite cylinder uniquely determine the transversal metric. The first result is proved by using complex geometrical optics solutions involving Gaussian beam quasimodes, and the second result follows from a connection between the Calderón problem and Gel’fand’s inverse problem for the wave equation and the boundary control method.
Partial differential equations
Global analysis, analysis on manifolds
2579
2626
10.4171/JEMS/649
http://www.ems-ph.org/doi/10.4171/JEMS/649
Legendrian knots and exact Lagrangian cobordisms
Tobias
Ekholm
Uppsala Universitet, UPPSALA, SWEDEN
Ko
Honda
University of Southern California, LOS ANGELES, UNITED STATES
Tamás
Kálmán
Tokyo Institute of Technology, TOKYO, JAPAN
Contact structure, Legendrian knot, exact Lagrangian cobordism, contact homology
We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair $(X,L)$ consisting of an exact symplectic manifold $X$ and an exact Lagrangian cobordism $L\subset X$ which agrees with cylinders over Legendrian links $\Lambda_+$ and $\Lambda_-$ at the positive and negative ends induces a differential graded algebra (DGA) map from the Legendrian contact homology DGA of $\Lambda_+$ to that of $\Lambda_-$. We give a gradient flow tree description of the DGA maps for certain pairs $(X,L)$, which in turn yields a purely combinatorial description of the cobordism map for elementary cobordisms, i.e., cobordisms that correspond to certain local modifications of Legendrian knots. As an application, we find exact Lagrangian surfaces that fill a fixed Legendrian link and are not isotopic through exact Lagrangian surfaces.
Manifolds and cell complexes
Differential geometry
2627
2689
10.4171/JEMS/650
http://www.ems-ph.org/doi/10.4171/JEMS/650
12
Orthosymplectic Lie superalgebras, Koszul duality, and a complete intersection analogue of the Eagon–Northcott complex
Steven
Sam
University of California, BERKELEY, UNITED STATES
Complete intersections, minimal free resolutions, classical Lie superalgebras, Koszul duality, Howe duality
We study the ideal of maximal minors in Littlewood varieties, a class of quadratic complete intersections in spaces of matrices. We give a geometric construction for a large class of modules, including all powers of this ideal, and show that they have a linear free resolution over the complete intersection and that their Koszul dual is an infinite-dimensional irreducible representation of the orthosymplectic Lie superalgebra. We calculate the algebra of cohomology operators acting on this free resolution. We prove analogous results for powers of the ideals of maximal minors in the variety of length 2 complexes when it is a complete intersection, and show that their Koszul dual is an infinite-dimensional irreducible representation of the general linear Lie superalgebra. This generalizes work of Akin, Józefiak, Pragacz, Weyman, and the author on resolutions of determinantal ideals in polynomial rings to the setting of complete intersections and provides a new connection between representations of classical Lie superalgebras and commutative algebra. As a curious application, we prove that the cohomology of a class of reducible homogeneous bundles on symplectic and orthogonal Grassmannians and 2-step flag varieties can be calculated by an analogue of the Borel–Weil–Bott theorem.
Commutative rings and algebras
Nonassociative rings and algebras
Category theory; homological algebra
2691
2732
10.4171/JEMS/651
http://www.ems-ph.org/doi/10.4171/JEMS/651
Orbit equivalence and Borel reducibility rigidity for profinite actions with spectral gap
Adrian
Ioana
University of California, San Diego, LA JOLLA, UNITED STATES
Spectral gap, rigidity, orbit equivalence, Borel reducibility, equivalence relations, profinite actions, outer automorphism group, II$_1$ factor
We study equivalence relations $\mathcal R(\Gamma\curvearrowright G)$ that arise from left translation actions of countable groups on their profinite completions. Under the assumption that the action $\Gamma\curvearrowright G$ is free and has spectral gap, we describe precisely when $\mathcal R(\Gamma\curvearrowright G)$ is orbit equivalent or Borel reducible to another such equivalence relation $\mathcal R(\Lambda\curvearrowright H)$. As a consequence, we provide explicit uncountable families of free ergodic probability measure preserving (p.m.p.) profinite actions of $SL_2(\mathbb Z)$ and its non-amenable subgroups (e.g. $\mathbb F_n$, with $2\leqslant n\leqslant\infty$) whose orbit equivalence relations are mutually not orbit equivalent and not Borel reducible. In particular, we show that if $S$ and $T$ are distinct sets of primes, then the orbit equivalence relations associated to the actions $SL_2(\mathbb Z)\curvearrowright\prod_{p\in S}SL_2(\mathbb Z_p)$ and $SL_2(\mathbb Z)\curvearrowright\prod_{p\in T}SL_2(\mathbb Z_p)$ are neither orbit equivalent nor Borel reducible. This settles a conjecture of S. Thomas [Th01,Th06]. Other applications include the first calculations of outer automorphism groups for concrete treeable p.m.p. equivalence relations, and the first concrete examples of free ergodic p.m.p. actions of $\mathbb F_{\infty}$ whose orbit equivalence relations have trivial fundamental group.
Dynamical systems and ergodic theory
Functional analysis
2733
2784
10.4171/JEMS/652
http://www.ems-ph.org/doi/10.4171/JEMS/652
Divergent CR-equivalences and meromorphic differential equations
Ilya
Kossovskiy
Masaryk University, BRNO, CZECH REPUBLIC
Rasul
Shafikov
The University of Western Ontario, LONDON, CANADA
CR-manifolds, formal mappings, holomorphic classification
Using the analytic theory of di erential equations, we construct, in any positive CR-dimension and CR-codimension, examples of formally but not holomorphically equivalent real-analytic CR-submanifolds in complex space.
Several complex variables and analytic spaces
2785
2819
10.4171/JEMS/653
http://www.ems-ph.org/doi/10.4171/JEMS/653
On non-forking spectra
Artem
Chernikov
University of California at Los Angeles, LOS ANGELES, UNITED STATES
Itay
Kaplan
The Hebrew University of Jerusalem, JERUSALEM, ISRAEL
Saharon
Shelah
The Hebrew University of Jerusalem, JERUSALEM, ISRAEL
Forking, dividing, NIP, NTP2, circularization, Dedekind cuts, cardinal arithmetic
Non-forking is one of the most important notions in modern model theory capturing the idea of a generic extension of a type (which is a far-reaching generalization of the concept of a generic point of a variety). To a countable first-order theory we associate its non-forking spectrum – a function of two cardinals $\kappa$ and $\lambda$ giving the supremum of the possible number of types over a model of size $\lambda$ that do not fork over a sub-model of size $\kappa$. This is a natural generalization of the stability function of a theory. We make progress towards classifying the non-forking spectra. On the one hand, we show that the possible values a non-forking spectrum may take are quite limited. On the other hand, we develop a general technique for constructing theories with a prescribed non-forking spectrum, thus giving a number of examples. In particular, we answer negatively a question of Adler whether NIP is equivalent to bounded non-forking. In addition, we answer a question of Keisler regarding the number of cuts a linear order may have. Namely, we show that it is possible that ded $\kappa$ < (ded$_\kappa) ^{\omega}$.
Mathematical logic and foundations
2821
2848
10.4171/JEMS/654
http://www.ems-ph.org/doi/10.4171/JEMS/654
Segre classes as integrals over polytopes
Paolo
Aluffi
Florida State University, TALLAHASSEE, UNITED STATES
Segre classes, monomial ideals, Newton polyhedra
We express the Segre class of a monomial scheme – or, more generally, a scheme monomially supported on a set of divisors cutting out complete intersections – in terms of an integral computed over an associated body in Euclidean space. The formula is in the spirit of the classical Bernstein–Kouchnirenko theorem computing intersection numbers of equivariant divisors in a torus in terms of mixed volumes, but deals with the more refined intersection-theoretic invariants given by Segre classes, and holds in the less restrictive context of ‘r.c. monomial schemes’.
Algebraic geometry
2849
2863
10.4171/JEMS/655
http://www.ems-ph.org/doi/10.4171/JEMS/655
Uniform Hölder bounds for strongly competing systems involving the square root of the laplacian
Susanna
Terracini
Università di Torino, TORINO, ITALY
Gianmaria
Verzini
Politecnico di Milano, MILANO, ITALY
Alessandro
Zilio
Ecole des Hautes Etudes en Sciences Sociales (EHESS), PARIS CEDEX 13, FRANCE
Square root of the laplacian, spatial segregation, strongly competing systems, optimal regularity of limiting profiles, singular perturbations
For a class of competition-diffusion nonlinear systems involving the square root of the Laplacian, including the fractional Gross–Pitaevskii system \[ (-\Delta)^{1/2} u_i=\omega_i u_i^3 + \lambda_i u_i - \beta u_i\sum_{j\neq i}a_{ij}u_j^2,\qquad i=1,\dots,k, \] we prove that $L^\infty$ boundedness implies $\mathcal C^{0,\alpha}$ boundedness for every $\alpha\in[0,1/2)$, uniformly as $\beta\to +\infty$. Moreover we prove that the limiting profile is $\mathcal C^{0,1/2}$.This system arises, for instance, in the relativistic Hartree—Fock approximation theory for $k$-mixtures of Bose–Einstein condensates in different hyperfine states.
Partial differential equations
Quantum theory
Statistical mechanics, structure of matter
2865
2924
10.4171/JEMS/656
http://www.ems-ph.org/doi/10.4171/JEMS/656
Top tautological group of $\mathcal M_{g,n}$
Alexandr
Buryak
ETH Zentrum, ZÜRICH, SWITZERLAND
Sergey
Shadrin
University of Amsterdam, AMSTERDAM, NETHERLANDS
Dimitri
Zvonkine
Université Pierre et Marie Curie – Paris 6, PARIS CEDEX 05, FRANCE
Moduli space of curves, cohomology, tautological groups
We describe the structure of the top tautological group in the cohomology of the moduli space of smooth genus $g$ curves with $n$ marked points.
Algebraic geometry
Algebraic topology
2925
2951
10.4171/JEMS/657
http://www.ems-ph.org/doi/10.4171/JEMS/657
Perron's method for the porous medium equation
Juha
Kinnunen
Aalto University, AALTO UNIVERSITY, FINLAND
Peter
Lindqvist
Norwegian University of Science and Technology, TRONDHEIM, NORWAY
Teemu
Lukkari
Aalto University, AALTO UNIVERSITY, FINLAND
Perron method, Porous medium equation, comparison principle, obstacles
This work extends Perron’s method for the porous medium equation in the slow diffusion case. The main result shows that nonnegative continuous boundary functions are resolutive in a general cylindrical domain.
Partial differential equations
Potential theory
2953
2969
10.4171/JEMS/658
http://www.ems-ph.org/doi/10.4171/JEMS/658
Sharp isoperimetric inequalities via the ABP method
Xavier
Cabré
Universitat Politècnica de Catalunya, BARCELONA, SPAIN
Xavier
Ros-Oton
University of Texas at Austin, AUSTIN, UNITED STATES
Joaquim
Serra
ETH Zürich, ZÜRICH, SWITZERLAND
Isoperimetric inequalities, densities, convex cones, homogeneous weights, Wulff shapes, ABP method
Given an arbitrary convex cone of $\mathbb R^n$, we nd a geometric class of homogeneous weights for which balls centered at the origin and intersected with the cone are minimizers of the weighted isoperimetric problem in the convex cone. This leads to isoperimetric inequalities with the optimal constant that were unknown even for a sector of the plane. Our result applies to all nonnegative homogeneous weights in $\mathbb R^n$ satisfying a concavity condition in the cone. The condition is equivalent to a natural curvature-dimension bound and also to the nonnegativeness of a Bakry- Emery Ricci tensor. Even that our weights are nonradial, still balls are minimizers of the weighted isoperimetric problem. A particular important case is that of monomial weights. Our proof uses the ABP method applied to an appropriate linear Neumann problem. We also study the anisotropic isoperimetric problem in convex cones for the same class of weights. We prove that the Wulff shape (intersected with the cone) minimizes the anisotropic weighted perimeter under the weighted volume constraint. As a particular case of our results, we give new proofs of two classical results: the Wul ff inequality and the isoperimetric inequality in convex cones of Lions and Pacella.
Measure and integration
Partial differential equations
Calculus of variations and optimal control; optimization
2971
2998
10.4171/JEMS/659
http://www.ems-ph.org/doi/10.4171/JEMS/659