Morse theory of the moment map for representations of quivers

The results of this paper concern the Morse theory of the norm-square of the moment map on the space of representations of a quiver. We show that the gradient flow of this function converges, and that the Morse stratification induced by the gradient flow co-incides with the Harder-Narasimhan stratification from algebraic geometry. Moreover, the limit of the gradient flow is isomorphic to the graded object of the Harder-Narasimhan-Jordan-H\"older filtration associated to the initial conditions for the flow. With a view towards applications to Nakajima quiver varieties we construct explicit local co-ordinates around the Morse strata and (under a technical hypothesis on the stability parameter) describe the negative normal space to the critical sets. Finally, we observe that the usual Kirwan surjectivity theorems in rational cohomology and integral K-theory carry over to this non-compact setting, and that these theorems generalize to certain equivariant contexts.Comment: 48 pages, small revisions from previous version based on referee's comments. To appear in Geometriae Dedicat

Surface currents and winds at the Delaware Bay mouth

Knowledge of the circulation of estuaries and adjacent shelf waters has relied on hydrographic measurements, moorings, and local wind observations usually removed from the region of interest. Although these observations are certainly sufficient to identify major characteristics, they lack both spatial resolution and temporal coverage. High resolution synoptic observations are required to identify important coastal processes at smaller scales. Long observation periods are needed to properly sample low-frequency processes that may also be important. The introduction of high-frequency (HF) radar measurements and regional wind models for coastal studies is changing this situation. Here we analyze synoptic, high-resolution surface winds and currents in the Delaware Bay mouth over an eight-month period (October 2007 through May 2008). The surface currents were measured by two high-frequency radars while the surface winds were extracted from a data-assimilating regional wind model. To illustrate the utility of these monitoring tools we focus on two 45-day periods which previously were shown to present contrasting pictures of the circulation. One, the low-outflow period is from 1 October through 14 November 2007; the other is the high-outflow period from 3 March through 16 April 2008. The large-scale characteristics noted by previous workers are clearly corroborated. Specifically the M2 tide dominates the surface currents, and the Delaware Bay outflow plume is clearly evident in the low frequency currents. Several new aspects of the surface circulation were also identified. These include a map of the spatial variability of the M2 tide (validating an earlier model study), persistent low-frequency cross-mouth flow, and a rapid response of the surface currents to a changing wind field. However, strong wind episodes did not persist long enough to set up a sustained Ekman response

Anticyclonic rings in the Gulf of Mexico

I Interntarional Symposium in Marine Sciences (ISMS07), Simposio GLOBER - IMBER España (2007), celebrado del 28 al 31 de marzo de 2007 en Valencia.-- 2 pages, 2 figuresThe Gulf of Mexico is a semi-closed sea that displays great variability in its circulation. The dynamics of this sea has been intensively studied not only by the scientific community but also by the oil industry, because of the presence of intense rings that may affect the offshore oils structures. A numerical model, the Colorado University Princeton Ocean Model (CUPOM), was develop and has been used during the last decades (Kantha and Clayson, 1994). Here we have used this model, with a 1/12 degree resolution in 24 sigma-layers, to obtain the temporal evolution of temperature and salinity vertical sections across an anticyclonic eddy that develops within the Gulf, and to analyze the behaviour of simulated parcel trajectories within this eddy. [...]Peer reviewe

Modelling the early evolution of Loop Current ring

Special issue Models and observations of Marine Systems.-- 12 pages, 13 figuresThe Colorado University Princeton Ocean Model (CUPOM) is used here to study the early stages in the life of Millennium, a mesoscale anticyclonic ring that detached from the Loop Current on April 2001 and lasted for more than 100 days. The numerical near-surface velocity field for the Gulf of Mexico is validated with the altimetry geostrophic velocities. The first 30 days of numerical data, before Millennium interacts with other mesoscalar features, are closely examined both from Eulerian and Lagrangian perspectives. During this time Millennium had a near-constant rotation period of 6.5 days, and particles do not leave the ring. Nevertheless, the distributions of temperature, salinity, and angular velocity confirm the existence of significant (possibly numerical) radial diffusion. Polar-coordinate phase plots for temperature–salinity anomalies and tangential–radial velocities, at several depths, illustrate the presence of an evolving oscillating pattern. Radial and tangential velocities change in phase, associated with vertical displacements of the isothermal and isohaline surfaces. A simple diffusion model with an effective diffusion coefficient of 200 m2 s−1 is appropriate to grossly simulate the temporal evolution of angular velocity within MillenniumThe first author wants to thank the Ministerio de Educación y Ciencia for funding her through an FPU grant (AP2003-3642). This work has been carried out as part of the CANOA project (CTM2005-00444/MAR) financed by the Spanish government through the Ministerio de Educación y CienciaPeer reviewe