- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 13:58:39
11
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=14&iss=5&update_since=2024-03-29
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
14
2012
5
A tight quantitative version of Arrow’s impossibility theorem
Nathan
Keller
Weizmann Institute of Science, REHOVOT, ISRAEL
Arrow's impossibility theorem, hypercontractivity, reverse hypercontractivity, algorithmic game theory, discrete Fourier analysis
The well-known Impossibility Theorem of Arrow asserts that any generalized social welfare function (GSWF) with at least three alternatives, which satisfies Independence of Irrelevant Alternatives (IIA) and Unanimity and is not a dictatorship, is necessarily non-transitive. In 2002, Kalai asked whether one can obtain the following quantitative version of the theorem: For any $\epsilon>0$, there exists $\delta=\delta(\epsilon)$ such that if a GSWF on three alternatives satisfies the IIA condition and its probability of non-transitive outcome is at most $\delta$, then the GSWF is at most $\epsilon$-far from being a dictatorship or from breaching the Unanimity condition. In 2009, Mossel proved such quantitative version, with $\delta(\epsilon)=\exp(-C/\epsilon^{21})$, and generalized it to GSWFs with $k$ alternatives, for all $k \geq 3$. In this paper we show that the quantitative version holds with $\delta(\epsilon)=C \cdot \epsilon^3$, and that this result is tight up to logarithmic factors. Furthermore, our result (like Mossel's) generalizes to GSWFs with $k$ alternatives. Our proof is based on the works of Kalai and Mossel, but uses also an additional ingredient: a combination of the Bonami-Beckner hypercontractive inequality with a reverse hypercontractive inequality due to Borell, applied to find simultaneously upper bounds and lower bounds on the "noise correlation'' between Boolean functions on the discrete cube.
Probability theory and stochastic processes
Combinatorics
Difference and functional equations
Game theory, economics, social and behavioral sciences
1331
1355
10.4171/JEMS/334
http://www.ems-ph.org/doi/10.4171/JEMS/334
Counterexamples to the Strichartz inequalities for the wave equation in general domains with boundary
Oana
Ivanovici
Université de Nice Sophia Antipolis, NICE, FRANCE
Microlocal analysis, wave equation, Dirichlet boundary condition, Strichartz estimates, propagation and reflection of singularities, conormal waves with cusps, caustics
In this paper we consider a smooth and bounded domain $\Omega\subset\mathbb{R}^d$ of dimension $d\geq 2$ with boundary and we construct sequences of solutions to the wave equation with Dirichlet boundary condition which contradict the Strichartz estimates of the free space, providing losses of derivatives at least for a subset of the usual range of indices. This is due to microlocal phenomena such as caustics generated in arbitrarily small time near the boundary. Moreover, the result holds for microlocally strictly convex domains in $\mathbb{R}^d$.
Partial differential equations
Global analysis, analysis on manifolds
General
1357
1388
10.4171/JEMS/335
http://www.ems-ph.org/doi/10.4171/JEMS/335
Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case
Thomas
Duyckaerts
Institut Galilée, Université Paris 13, VILLETANEUSE, FRANCE
Carlos
Kenig
University of Chicago, CHICAGO, UNITED STATES
Frank
Merle
Université de Cergy-Pontoise, CERGY-PONTOISE CEDEX, FRANCE
Following our previous paper in the radial case, we consider type II blow-up solutions to the energy-critical focusing wave equation. Let W be the unique radial positive stationary solution of the equation. Up to the symmetries of the equation, under an appropriate smallness assumption, any type II blow-up solution is asymptotically a regular solution plus a rescaled Lorentz transform of $W$ concentrating at the origin.
General
1389
1454
10.4171/JEMS/336
http://www.ems-ph.org/doi/10.4171/JEMS/336
A spectral gap theorem in SU$(d)$
Jean
Bourgain
Institute for Advanced Study, PRINCETON, UNITED STATES
Alex
Gamburd
University of California at Santa Cruz, SANTA CRUZ, UNITED STATES
We establish the spectral gap property for dense subgroups of SU$(d)$ ($d\geq 2$), generated by finitely many elements with algebraic entries; this result was announced in [BG3]. The method of proof differs, in several crucial aspects, from that used in [BG] in the case of SU$(2)$.
Group theory and generalizations
General
1455
1511
10.4171/JEMS/337
http://www.ems-ph.org/doi/10.4171/JEMS/337
Pairings, duality, amenability and bounded cohomology
Jacek
Brodzki
University of Southampton, SOUTHAMPTON, UNITED KINGDOM
Graham
Niblo
University of Southampton, SOUTHAMPTON, UNITED KINGDOM
Nick
Wright
University of Southampton, SOUTHAMPTON, UNITED KINGDOM
Amenability, group cohomology, bounded cohomology, uniformly finite homology, invariant means
We give a new perspective on the homological characterizations of amenability given by Johnson & Ringrose in the context of bounded cohomology and by Block & Weinberger in the context of uniformly finite homology. We examine the interaction between their theories and explain the relationship between these characterizations. We apply these ideas to give a new proof of non-vanishing for the bounded cohomology of a free group.
Group theory and generalizations
Abstract harmonic analysis
General
1513
1518
10.4171/JEMS/338
http://www.ems-ph.org/doi/10.4171/JEMS/338
The density of representation degrees
Martin
Liebeck
Imperial College, LONDON, UNITED KINGDOM
Dan
Segal
University of Oxford, OXFORD, UNITED KINGDOM
Aner
Shalev
The Hebrew University of Jerusalem, JERUSALEM, ISRAEL
For a group $G$ and a positive real number $x$, define $d_G(x)$ to be the number of integers less than $x$ which are dimensions of irreducible complex representations of $G$. We study the asymptotics of $d_G(x)$ for algebraic groups, arithmetic groups and finitely generated linear groups. In particular we prove an "alternative" for finitely generated linear groups $G$ in characteristic zero, showing that either there exists $\alpha > 0$ such that $d_G(x)>x^\alpha$ for all large $x$, or $G$ is virtually abelian (in which case $d_G(x)$ is bounded).
Group theory and generalizations
General
1519
1537
10.4171/JEMS/339
http://www.ems-ph.org/doi/10.4171/JEMS/339
Geometric rigidity of $\times m$ invariant measures
Michael
Hochman
The Hebrew University of Jerusalem, JERUSALEM, ISRAEL
Measure rigidity, invariant measure, interval map, fractal geometry, geometric measure theory, scenery flow
Let $\mu$ be a probability measure on $[0,1]$ which is invariant and ergodic for $T_{a}(x)=ax\bmod1$, and $0
Dynamical systems and ergodic theory
Measure and integration
General
1539
1563
10.4171/JEMS/340
http://www.ems-ph.org/doi/10.4171/JEMS/340
Minkowski valuations intertwining the special linear group
Christoph
Haberl
Universität Salzburg, SALZBURG, AUSTRIA
Valuation, projection body, centroid body
All continuous Minkowski valuations which are compatible with the special linear group are completely classified. One consequence of these classifications is a new characterization of the projection body operator.
Differential geometry
General
1565
1597
10.4171/JEMS/341
http://www.ems-ph.org/doi/10.4171/JEMS/341
Unique Bernoulli g-measures
Anders
Johansson
University of Gävle, GÄVLE, SWEDEN
Anders
Öberg
Uppsala Universitet, UPPSALA, SWEDEN
Mark
Pollicott
University of Warwick, COVENTRY, UNITED KINGDOM
Bernoulli measure, g-measure, chains with complete connections
We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a $g$-measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique $g$-measures have Bernoulli natural extensions. We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the $g$-measure.
Dynamical systems and ergodic theory
Probability theory and stochastic processes
General
1599
1615
10.4171/JEMS/342
http://www.ems-ph.org/doi/10.4171/JEMS/342
Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric
Ngaiming
Mok
University of Hong Kong, HONG KONG, CHINA
Kähler manifold, holomorphic isometry, Bergman metric, bounded symmetric domain, holomorphic extension, total geodesy
We study the extension problem for germs of holomorphic isometries $f: (D;x_0) \to (\Omega;f(x_0))$ up to normalizing constants between bounded domains in Euclidean spaces equipped with Bergman metrics $ds_D^2$ on $D$ and $ds_\Omega^2$ on $\Omega$. Our main focus is on boundary extension for pairs of bounded domains $(D,\Omega)$ such that the Bergman kernel $K_D(z,w)$ extends meromorphically in $(z,\overline w)$ to a neighborhood of $\overline D \times D$, and such that the analogous statement holds true for the Bergman kernel $K_{\Omega}(\zeta,\xi)$ on $\Omega$. Assuming that $(D;ds_D^2)$ and $(\Omega;ds_\Omega^2)$ are complete K\"ahler manifolds, we prove that the germ of map $f$ extends to a proper holomorphic isometric embedding such that Graph$(f)$ extends to a complex-analytic subvariety on some neigborhood of $\overline D \times \overline{\Omega}$. In the event that the Bergman kernel $K_D(z,w)$ extends to a rational function in $(z;\overline w)$ and the analogue holds true for the Bergman kernel $K_{\Omega}(\zeta,\xi)$, we show that Graph$(f)$ extends to an affine-algebraic variety. Our results apply especially to pairs $(D,\Omega)$ of bounded symmetric domains in their Harish-Chandra realizations. When $D$ is the complex unit ball $B^n$ of dimension $n \ge 2$, we obtain a new rigidity result which guarantees the total geodesy of the map under certain conditions. On the other hand, we construct examples of holomorphic isometries of the unit disk into polydisks which are not totally geodesic, answering in the negative a conjecture of Clozel-Ullmo's.
Several complex variables and analytic spaces
General
1617
1656
10.4171/JEMS/343
http://www.ems-ph.org/doi/10.4171/JEMS/343
Rank gradient, cost of groups and the rank versus Heegaard genus problem
Miklós
Abért
Hungarian Academy of Sciences, BUDAPEST, HUNGARY
Nikolay
Nikolov
University of Oxford, OXFORD, UNITED KINGDOM
Rank gradient, cost, Heegaard genus, fixed price
We study the growth of the rank of subgroups of finite index in residually finite groups, by relating it to the notion of cost. As a by-product, we show that the ‘rank vs. Heegaard genus’ conjecture on hyperbolic 3-manifolds is incompatible with the ‘fixed price problem’ in topological dynamics.
Group theory and generalizations
Several complex variables and analytic spaces
Manifolds and cell complexes
General
1657
1677
10.4171/JEMS/344
http://www.ems-ph.org/doi/10.4171/JEMS/344