In this paper we prove that the defocusing, d-dimensional mass critical nonlinear Schrödinger initial value problem is globally well-posed and scattering for u 0 ∈ L 2 (R d ) and d ≥ 3. To do this, we will prove a frequency localized interaction Morawetz estimate similar to the estimate made in [10]. Since we are considering an L 2 -critical initial value problem we will localize to low frequencies.

In this paper we prove global well -posedness and scattering for the focusing, energy -critical nonlinear Schrödinger initial value problem in four dimensions. Previous work proved this in five dimensions and higher using the double Duhamel trick. In this paper, using long time Strichartz estimates we are able to overcome the logarithmic blowup in four dimensions.(1.2) is called energy -critical when p = 4 d−2 since a solution to (1.2) is invariant under the scaling(1.5) and (1.5) preserves the energy (1.4).There also exist the defocusing, energy -critical problems (F (u) = |u| 4 d−2 u), which are similar to the focusing problem in some ways, but also contain many important differences. The defocusing problem is now completely worked out.Proof: The proof of theorem 1.1 has involved contributions from a variety of authors.[11] proved theorem 1.1 for small data in both the focusing and defocusing problem.[11] also proved that (1.2) has a local solution for any initial data u 0 ∈Ḣ 1 (R d ), where the time of existence depends on the size and profile of u 0 .For large data, the seminal result was the work of [4] and [5], proving theorem 1.1 for radial data in dimensions d = 3, 4, and also that for more regular u 0 , this additional smoothness is preserved. See [21] for another proof of this last fact.[40] then extended theorem 1.1 to radial data in higher dimensions. Then [13] extended theorem 1.1 to general u 0 ∈Ḣ 1 when d = 3. Subsequently, [34] extended this to dimension d = 4, and [46], [47] extended theorem 1.1 to dimensions d ≥ 5. Remark: [48] and [30] reproved theorem 1.1 in dimensions three and four using the long time Strichartz estimates of [14]. We will use long time Strichartz estimates similar to the estimates of [30] in this paper as well.Returning to the focusing problem, we remark that theorem 1.1 does not hold for arbitrary data. In fact, by the virial identity (see for example [20])

Abstract. We revisit the scattering result of Holmer and Roudenko [5] on the radial focusing cubic NLS in three space dimensions. Using the radial Sobolev embedding and a virial/Morawetz estimate, we give a simple proof of scattering below the ground state that avoids the use of concentration compactness.

In this paper we prove that the defocusing, cubic nonlinear Schrödinger initial value problem is globally well-posed and scattering for u 0 ∈ L 2 (R 2 ). The proof uses the bilinear estimates of [67] and a frequency localized interaction Morawetz estimate similar to the high frequency estimate of [25] and especially the low frequency estimate of [28].

We present deep JHK photometry of the old and metal-rich open cluster NGC 6791. The photometry reaches below the main sequence turn-off to K = 16.5 mag. We combine our photometry with that from Stetson, Bruntt, & Grundahl (2003) to provide color-magnitude diagrams showing K vs. J-K, K vs. V-K, and V vs. V-K. We study the slope of the red giant branch in the infrared, but find that it is not a useful metallicity indicator for the cluster, nor any metal-rich cluster that lacks a well-populated red giant branch, because it is not linear, as has often been assumed, in K vs. J-K. The mean color of the red horizontal branch/red clump stars provide an estimate the cluster reddening, E(B-V) = 0.14 +/- 0.04 mag for [Fe/H] = +0.4 +/- 0.1. The mean magnitudes of these stars also provide a good distance estimate, (m-M)_0 = 13.07 +/- 0.04. Finally, we find that the isochrones of Yi, Kim, & Demarque (2003) provide optimal fits in V vs. B-V and V-K and K vs. J-K and V-K for such values if [Fe/H] lies between +0.3 and +0.5 (with a slight preference for +0.5) and ages between 9 Gyrs ([Fe/H] = +0.3) and 7.5 Gyrs ([Fe/H] = +0.5).Comment: Accepted for publication in AJ (Feb. 2005