A glimpse into the differential topology and geometry of optimal transport

This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It also establishes new connections --- some heuristic and others rigorous --- based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal.Comment: 27 page

Satellite sensor requirements for monitoring essential biodiversity variables of coastal ecosystems.

The biodiversity and high productivity of coastal terrestrial and aquatic habitats are the foundation for important benefits to human societies around the world. These globally distributed habitats need frequent and broad systematic assessments, but field surveys only cover a small fraction of these areas. Satellite-based sensors can repeatedly record the visible and near-infrared reflectance spectra that contain the absorption, scattering, and fluorescence signatures of functional phytoplankton groups, colored dissolved matter, and particulate matter near the surface ocean, and of biologically structured habitats (floating and emergent vegetation, benthic habitats like coral, seagrass, and algae). These measures can be incorporated into Essential Biodiversity Variables (EBVs), including the distribution, abundance, and traits of groups of species populations, and used to evaluate habitat fragmentation. However, current and planned satellites are not designed to observe the EBVs that change rapidly with extreme tides, salinity, temperatures, storms, pollution, or physical habitat destruction over scales relevant to human activity. Making these observations requires a new generation of satellite sensors able to sample with these combined characteristics: (1) spatial resolution on the order of 30 to 100-m pixels or smaller; (2) spectral resolution on the order of 5 nm in the visible and 10 nm in the short-wave infrared spectrum (or at least two or more bands at 1,030, 1,240, 1,630, 2,125, and/or 2,260 nm) for atmospheric correction and aquatic and vegetation assessments; (3) radiometric quality with signal to noise ratios (SNR) above 800 (relative to signal levels typical of the open ocean), 14-bit digitization, absolute radiometric calibration <2%, relative calibration of 0.2%, polarization sensitivity <1%, high radiometric stability and linearity, and operations designed to minimize sunglint; and (4) temporal resolution of hours to days. We refer to these combined specifications as H4 imaging. Enabling H4 imaging is vital for the conservation and management of global biodiversity and ecosystem services, including food provisioning and water security. An agile satellite in a 3-d repeat low-Earth orbit could sample 30-km swath images of several hundred coastal habitats daily. Nine H4 satellites would provide weekly coverage of global coastal zones. Such satellite constellations are now feasible and are used in various applications

Remote detection of invasive alien species

The spread of invasive alien species (IAS) is recognized as the most severe threat to biodiversity outside of climate change and anthropogenic habitat destruction. IAS negatively impact ecosystems, local economies, and residents. They are especially problematic because once established, they give rise to positive feedbacks, increasing the likelihood of further invasions and spread. The integration of remote sensing (RS) to the study of invasion, in addition to contributing to our understanding of invasion processes and impacts to biodiversity, has enabled managers to monitor invasions and predict the spread of IAS, thus supporting biodiversity conservation and management action. This chapter focuses on RS capabilities to detect and monitor invasive plant species across terrestrial, riparian, aquatic, and human-modified ecosystems. All of these environments have unique species assemblages and their own optimal methodology for effective detection and mapping, which we discuss in detail