Fixed-point spectrum for group actions by affine isometries on Lp-spaces

The fixed-point spectrum of a locally compact second countable group G on lp is defined to be the set of real numbers p such that every action by affine isometries of G on lp admits a fixed-point. We show that this set is either empty, or is equal to a set of one of the following forms : [1,\pc[, [1,\pc[\{2} for some \pc<\infty or \pc=\infty, or [1,\pc], [1,\pc]\{2} for some pc<infty. This answers a question closely related to a conjecture of C. Drutu which asserts that the fixed-point spectrum is connected for isometric actions on Lp(0,1). We also study the topological properties of the fixed-point spectrum on Lp(X,\mu) for general measure spaces (X,\mu), and show partial results toward the conjecture for actions on Lp(0,1). In particular, we prove that the spectrum F_{L^{\infty}(X,\mu)(G,\pi) of actions with linear part \pi is either empty, or an interval of the form [1,\pc] or [1,\infty[, whenever \pi is an orthogonal representation associated to a measure-preserving ergodic action on a finite measure space (X,\mu).Comment: 25 page

On groups with Property (T_lp)

Let p be a real number with 1<p and different from 2. We study Property (T_lp) for a second countable locally compact group G. Property (T_lp) is a weak version of Kazhdan's Property (T), defined in terms of the orthogonal representations of G on the sequence space lp. We show that Property (T_lp) for a totally disconnected group G is characterized by an isolation property of the trivial representation from the quasi-regular representations associated to open subgroups of G. Groups with Property (T_lp) share some important properties with Kazhdan groups (compact generation, compact abelianization, ...). Simple algebraic groups over non-archimedean local fields as well as automorphism groups of regular trees have Property (T_lp). In the case of discrete groups, Property (T_lp) implies Lubotzky's Property tau and is implied by Property (F) of Glasner and Monod. We show that an irreducible lattice in a product of two locally compact groups G and H have Property (T_lp), whenever G has Property (T) and H is connected and minimally almost periodic.Comment: 17 page

Property (T) with respect to non-commutative Lp-spaces

International audienceWe show that a group with Kazhdan's property $(T)$ has property $(T_{B})$ for $B$ the Haagerup non-commutative $L_{p}(\mathcal{M})$-space associated with a von Neumann algebra $\mathcal{M}$, $

Algorithm Portfolios for Noisy Optimization

Noisy optimization is the optimization of objective functions corrupted by noise. A portfolio of solvers is a set of solvers equipped with an algorithm selection tool for distributing the computational power among them. Portfolios are widely and successfully used in combinatorial optimization. In this work, we study portfolios of noisy optimization solvers. We obtain mathematically proved performance (in the sense that the portfolio performs nearly as well as the best of its solvers) by an ad hoc portfolio algorithm dedicated to noisy optimization. A somehow surprising result is that it is better to compare solvers with some lag, i.e., propose the current recommendation of best solver based on their performance earlier in the run. An additional finding is a principled method for distributing the computational power among solvers in the portfolio.Comment: in Annals of Mathematics and Artificial Intelligence, Springer Verlag, 201

Emergence of macroscopic directed motion in populations of motile colloids

From the formation of animal flocks to the emergence of coordinate motion in bacterial swarms, at all scales populations of motile organisms display coherent collective motion. This consistent behavior strongly contrasts with the difference in communication abilities between the individuals. Guided by this universal feature, physicists have proposed that solely alignment rules at the individual level could account for the emergence of unidirectional motion at the group level. This hypothesis has been supported by agent-based simulations. However, more complex collective behaviors have been systematically found in experiments including the formation of vortices, fluctuating swarms, clustering and swirling. All these model systems predominantly rely on actual collisions to display collective motion. As a result, the potential local alignment rules are entangled with more complex, often unknown, interactions. The large-scale behavior of the populations therefore depends on these uncontrolled microscopic couplings. Here, we demonstrate a new phase of active matter. We reveal that dilute populations of millions of colloidal rollers self-organize to achieve coherent motion along a unique direction, with very few density and velocity fluctuations. Identifying the microscopic interactions between the rollers allows a theoretical description of this polar-liquid state. Comparison of the theory with experiment suggests that hydrodynamic interactions promote the emergence of collective motion either in the form of a single macroscopic flock at low densities, or in that of a homogenous polar phase at higher densities. Furthermore, hydrodynamics protects the polar-liquid state from the giant density fluctuations. Our experiments demonstrate that genuine physical interactions at the individual level are sufficient to set homogeneous active populations into stable directed motion

Universal behaviour of a wave chaos based electromagnetic reverberation chamber

In this article, we present a numerical investigation of three-dimensional electromagnetic Sinai-like cavities. We computed around 600 eigenmodes for two different geometries: a parallelepipedic cavity with one half- sphere on one wall and a parallelepipedic cavity with one half-sphere and two spherical caps on three adjacent walls. We show that the statistical requirements of a well operating reverberation chamber are better satisfied in the more complex geometry without a mechanical mode-stirrer/tuner. This is to the fact that our proposed cavities exhibit spatial and spectral statistical behaviours very close to those predicted by random matrix theory. More specifically, we show that in the range of frequency corresponding to the first few hundred modes, the suppression of non-generic modes (regarding their spatial statistics) can be achieved by reducing drastically the amount of parallel walls. Finally, we compare the influence of losses on the statistical complex response of the field inside a parallelepipedic and a chaotic cavity. We demonstrate that, in a chaotic cavity without any stirring process, the low frequency limit of a well operating reverberation chamber can be significantly reduced under the usual values obtained in mode-stirred reverberation chambers