Representations of algebraic groups over a 2-dimensional local field

We introduce a categorical framework for the study of representations of G(F), where G is a reductive group, and F is a 2-dimensional local field, i.e. F = K((t)), where K is a local field. Our main result says that the space of functions on G(F), which is an object of a suitable category of representations of G(F) with the respect to the action of G on itself by left translations, becomes a representation of a certain central extension of G(F), when we consider the action by right translations.     

Topological simplicity,commensurator super-rigidity and nonlinearities of Kac-Moody groups (appendix by P. Bonvin)

We provide new arguments to see topological Kac-Moody groups as generalized semisimple groups over local fields: they are products of topologically simple groups and their Iwahori subgroups are the normalizers of the pro-p Sylow subgroups. We use a dynamical characterization of parabolic subgroups to prove that some countable Kac-Moody groups with Fuchsian buildings are not linear. We show for this that the linearity of a countable Kac-Moody group implies the existence of a closed embedding of the corresponding topological group in a non-Archimedean simple Lie group, thanks to a commensurator super-rigidity theorem proved in the Appendix by P. Bonvin.     

On the existence of wave operators for some Dirac operators with square summable potential

We study the Dirac operators on the half-line. If the potential is square summable, we prove existence of the wave operators.     

A variational approach for compact homogeneous Einstein manifolds

((no abstract given))     

Absence of Positive Eigenvalues for the Linearized Elasticity System

In this paper, we prove that the linearized elasticity system has no eigenvalues in two geometric situations: the whole space and a local perturbation of the half-space. We consider the Lamé coefficients and the density varying in an unbounded part of the domain. For the whole space, we use the operations curl and div to reduce our system to a scalar problem and use a limiting absorption principle for the reduced scalar equation given by the partial Fourier transform. For the perturbed half-space, this decompositions being no longer valid, we give an other method based on a pseudo-decomposition using the operations div and curl in the horizontal direction. In contrast to the whole space case, the reduced problems depend strongly on the dual Fourier variable which do not enable us to use same techniques. To study these reduced problems, we use the analytic theory of linear operators.     

Gaussian images of surfaces and ellipticity of surface area functionals

((no abstract given))     

Going-down functors, the Künneth formula, and the Baum-Connes conjecture

We study the connection between the Baum-Connes conjecture for a locally compact group G with coeefficient A and the Künneth formula for the K-theory of tensor products by the corresponding crossed product . The main tool for this is obtained by an application of a general reduction procedure which allows us to analyze certain functors connected to the topological K-theory of a group in terms of their restrictions to compact subgroups. We also discuss several other interesting applications of this method, including a general extension result for the Baum-Connes conjecture.     

Embedding the diamond graph inLpand dimension reduction inL1

We show that any embedding of the level k diamond graph of Newman and Rabinovich [NR] into L_p, 1 < p 2, requires distortion at least . An immediate corollary is that there exist arbitrarily large n-point sets such that any D-embedding of X into requires . This gives a simple proof of a recent result of Brinkman and Charikar [BrC] which settles the long standing question of whether there is an L₁ analogue of the Johnson-Lindenstrauss dimension reduction lemma [JL].     

Volume,diameter and the minimal mass of a stationary 1-cycle

In this paper we present upper bounds on the minimal mass of a non-trivial stationary 1-cycle. The results that we obtain are valid for all closed Riemannian manifolds. The first result is that the minimal mass of a stationary 1-cycle on a closed n-dimensional Riemannian manifold Mⁿ is bounded from above by (n + 2)!d/4, where d is the diameter of a manifold Mⁿ. The second result is that the minimal mass of a stationary 1-cycle on a closed Riemannian manifold Mⁿ is bounded from above by where where is the filling radius of a manifold, and where is its volume.     

Friedmann cosmology and almost isotropy

In the Friedmann model of the universe, cosmologists assume that spacelike slices of the universe are Riemannian manifolds of constant sectional curvature. This assumption is justified via Schurs theorem by stating that the spacelike universe is locally isotropic. Here we define a Riemannian manifold as almost locally isotropic in a sense which allows both weak gravitational lensing in all directions and strong gravitational lensing in localized angular regions at most points. We then prove that such a manifold is Gromov-Hausdorff close to a length space Y which is a collection of space forms joined at discrete points. Within the paper we de.ne a concept we call an exponential length space and prove that if such a space is locally isotropic then it is a space form.     

Isoperimetric inequalities for nilpotent groups

We prove that every finitely generated nilpotent group of class c admits a polynomial isoperimetric function of degree c + 1 and a linear upper bound on its filling length function.     

Sharp rate of average decay of the Fourier transform of a bounded set

Estimates for the decay of Fourier transforms of measures have extensive applications in numerous problems in harmonic analysis and convexity including the distribution of lattice points in convex domains, irregularities of distribution, generalized Radon transforms and others. Here we prove that the spherical L ²-average decay rate of the Fourier transform of the Lebesgue measure on an arbitrary bounded convex set in $\mathbb{R}^{d}$ is $${\bigg(\int_{S^{d-1}}{\big|\widehat{\chi}_B(R\omega)\big|}^2d\omega \bigg)}^{{1}/{2}} \lesssim R^{-\frac{d+1}{2}}.\eqno(*)$$ This estimate is optimal for any convex body and in particular it agrees with the familiar estimate for the ball. The above estimate was proved in two dimensions by Podkorytov, and in all dimensions by Varchenko under additional smoothness assumptions. The main result of this paper proves (*) in all dimensions under the convexity hypothesis alone. We also prove that the same result holds if the boundary of is C^3/2.     

Property (T) and Kazhdan constants for discrete groups

Let be a group generated by a finite set S. We give a sufficient condition for to have Kazhdan's property (T). This condition is easy to check and gives Kazhdan constants. We give examples of groups to which this method applies. We prove that in some setting generic presentations define groups which satisfy this condition and thus have property (T). Moreover we prove that small changes in the presentation of a group satisfying this condition do not change the fact that the group has property (T).     

Explicit finite volume criteria for localization in continuous random media and applications

We give finite volume criteria for localization of quantum or classical waves in continuous random media. We provide explicit conditions, depending on the parameters of the model, for starting the bootstrap multiscale analysis. A simple application to Anderson Hamiltonians on the continuum yields localization at the bottom of the spectrum in an interval of size C for large , where stands for the disorder parameter. A more sophisticated application proves localization for two-dimensional random Schrödinger operators in a constant magnetic field (random Landau Hamiltonians) up to a distance from the Landau levels for large B, where B is the strength of the magnetic field.     

Cubulating small cancellation groups

We study the B(6) and B(4)-T(4) small cancellation groups. These classes include the usual C(1/6) and C(1/4)-T(4) metric small cancellation groups. We show that every finitely presented B(4)-T(4) or word-hyperbolic B(6) group acts properly discontinuously and cocompactly on a CAT(0) cube complex. We show that finitely generated infinite B(6) and B(4)-T(4) groups have codimension 1 subgroups and thus do not have property (T). We show that a finitely presented B(6) group is wordhyperbolic if and only if it contains no subgroup.     

Hausdorff dimension of limit sets of discrete subgroups of higher rank Lie groups

Let X be a globally symmetric space of noncompact type, and a discrete subgroup. Introducing an appropriate notion of Hausdorff measure on the geometric boundary of , we prove that for regular boundary points , the Hausdorff dimension of the radial limit set in is bounded above by the exponential growth rate of the number of orbit points close in direction to . Furthermore, for Zariski dense discrete groups we construct -invariant densities with support in every G-invariant subset of the limit set and study their properties. For a class of groups which generalises convex cocompact groups in the rank one setting, these densities allow to give a sharp estimate on the Hausdorff dimension of the radial limit set in each subset .     

Kazhdan's Property (T) for the unitary group of aseparable Hilbert space

We show that the unitary group of a separable Hilbert space has Kazhdan's Property (T), when it is equipped with the strong operator topology. More precisely, for every integer m 2, we give an explicit Kazhdan set consisting of m unitary operators and determine an optimal Kazhdan constant for this set. Moreover, we show that a locally compact group with Kazhdan's Property (T) has a finite Kazhdan set if and only if its Bohr compactification has a finite Kazhdan set. As a consequence, if a locally compact group with Property (T) is minimally almost periodic, then it has a finite Kazhdan set.     

A sharp bilinear restriction estimate for paraboloids

Recently Wolff [W3] obtained a sharp L² bilinear restriction theorem for bounded subsets of the cone in general dimension. Here we adapt the argument of Wolff to also handle subsets of elliptic surfaces such as paraboloids. Except for an endpoint, this answers a conjecture of Machedon and Klainerman, and also improves upon the known restriction theory for the paraboloid and sphere.