On the Bartnik extension problem for the static vacuum Einstein equations

We develop a framework for understanding the existence of asymptotically flat solutions to the static vacuum Einstein equations with prescribed boundary data consisting of the induced metric and mean curvature on a 2-sphere. A partial existence result is obtained, giving a partial resolution of a conjecture of Bartnik on such static vacuum extensions. The existence and uniqueness of such extensions is closely related to Bartnik's definition of quasi-local mass.Comment: 33 pages, 1 figure. Minor revision of v2. Final version, to appear in Class. Quantum Gravit

Bounds for moments of ℓ\ell-torsion in class groups

Fix a number field $k$, integers $\ell, n \geq 2$, and a prime $p$. For all $r \geq 1$, we prove strong unconditional upper bounds on the $r$-th moment of $\ell$-torsion in the ideal class groups of degree $p$ extensions of $k$ and of degree $n$ $S_n$-extensions of $k$, improving upon results of Ellenberg, Pierce and Wood as well as GRH-conditional results of Frei and Widmer. For large $r$, our results are comparable with work of Heath-Brown and Pierce for imaginary quadratic extensions of $\mathbb{Q}$. When $r=1$, our results are new even for the family of all quadratic extensions of $\mathbb{Q}$, leading to an improved upper bound for the count of degree $p$ $D_p$-extensions over $\mathbb{Q}$ (where $D_p$ is the dihedral group of order $2p$).Comment: 12 page

On contact numbers of totally separable unit sphere packings

Contact numbers are natural extensions of kissing numbers. In this paper we give estimates for the number of contacts in a totally separable packing of n unit balls in Euclidean d-space for all n>1 and d>1.Comment: 11 page

State-Insensitive Cooling and Trapping of Single Atoms in an Optical Cavity

Single Cesium atoms are cooled and trapped inside a small optical cavity by way of a novel far-off-resonance dipole-force trap (FORT), with observed lifetimes of 2 to 3 seconds. Trapped atoms are observed continuously via transmission of a strongly coupled probe beam, with individual events lasting ~ 1 s. The loss of successive atoms from the trap N = 3 -> 2 -> 1 -> 0 is thereby monitored in real time. Trapping, cooling, and interactions with strong coupling are enabled by the FORT potential, for which the center-of-mass motion is only weakly dependent on the atom's internal state.Comment: 5 pages, 4 figures Revised version to appear in Phys. Rev. Let

Teichm\"uller's problem in space

Quasiconformal homeomorphisms of the whole space Rn, onto itself normalized at one or two points are studied. In particular, the stability theory, the case when the maximal dilatation tends to 1, is in the focus. Our main result provides a spatial analogue of a classical result due to Teichm\"uller. Unlike Teichm\"uller's result, our bounds are explicit. Explicit bounds are based on two sharp well-known distortion results: the quasiconformal Schwarz lemma and the bound for linear dilatation. Moreover, Bernoulli type inequalities and asymptotically sharp bounds for special functions involving complete elliptic integrals are applied to simplify the computations. Finally, we discuss the behavior of the quasihyperbolic metric under quasiconformal maps and prove a sharp result for quasiconformal maps of R^n \ {0} onto itself.Comment: 25 pages, 2 figure