ADE Spectral Networks

We introduce a new perspective and a generalization of spectral networks for 4d $\mathcal{N}=2$ theories of class $\mathcal{S}$ associated to Lie algebras $\mathfrak{g} = \textrm{A}_n$, $\textrm{D}_n$, $\textrm{E}_{6}$, and $\textrm{E}_{7}$. Spectral networks directly compute the BPS spectra of 2d theories on surface defects coupled to the 4d theories. A Lie algebraic interpretation of these spectra emerges naturally from our construction, leading to a new description of 2d-4d wall-crossing phenomena. Our construction also provides an efficient framework for the study of BPS spectra of the 4d theories. In addition, we consider novel types of surface defects associated with minuscule representations of $\mathfrak{g}$.Comment: 68 pages plus appendices; visit http://het-math2.physics.rutgers.edu/loom/ to use 'loom,' a program that generates spectral networks; v2: version published in JHEP plus minor correction

Parallel Mapper

The construction of Mapper has emerged in the last decade as a powerful and effective topological data analysis tool that approximates and generalizes other topological summaries, such as the Reeb graph, the contour tree, split, and joint trees. In this paper, we study the parallel analysis of the construction of Mapper. We give a provably correct parallel algorithm to execute Mapper on multiple processors and discuss the performance results that compare our approach to a reference sequential Mapper implementation. We report the performance experiments that demonstrate the efficiency of our method

Nested hierarchies in planar graphs

We construct a partial order relation which acts on the set of 3-cliques of a maximal planar graph G and defines a unique hierarchy. We demonstrate that G is the union of a set of special subgraphs, named `bubbles', that are themselves maximal planar graphs. The graph G is retrieved by connecting these bubbles in a tree structure where neighboring bubbles are joined together by a 3-clique. Bubbles naturally provide the subdivision of G into communities and the tree structure defines the hierarchical relations between these communities

Dimers and cluster integrable systems

We show that the dimer model on a bipartite graph on a torus gives rise to a quantum integrable system of special type - a cluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space of line bundles with connections on the graph. The sum of Hamiltonians is essentially the partition function of the dimer model. Any graph on a torus gives rise to a bipartite graph on the torus. We show that the phase space of the latter has a Lagrangian subvariety. We identify it with the space parametrizing resistor networks on the original graph.We construct several discrete quantum integrable systems.Comment: This is an updated version, 75 pages, which will appear in Ann. Sci. EN

Hybrid Satellite-Terrestrial Communication Networks for the Maritime Internet of Things: Key Technologies, Opportunities, and Challenges

With the rapid development of marine activities, there has been an increasing number of maritime mobile terminals, as well as a growing demand for high-speed and ultra-reliable maritime communications to keep them connected. Traditionally, the maritime Internet of Things (IoT) is enabled by maritime satellites. However, satellites are seriously restricted by their high latency and relatively low data rate. As an alternative, shore & island-based base stations (BSs) can be built to extend the coverage of terrestrial networks using fourth-generation (4G), fifth-generation (5G), and beyond 5G services. Unmanned aerial vehicles can also be exploited to serve as aerial maritime BSs. Despite of all these approaches, there are still open issues for an efficient maritime communication network (MCN). For example, due to the complicated electromagnetic propagation environment, the limited geometrically available BS sites, and rigorous service demands from mission-critical applications, conventional communication and networking theories and methods should be tailored for maritime scenarios. Towards this end, we provide a survey on the demand for maritime communications, the state-of-the-art MCNs, and key technologies for enhancing transmission efficiency, extending network coverage, and provisioning maritime-specific services. Future challenges in developing an environment-aware, service-driven, and integrated satellite-air-ground MCN to be smart enough to utilize external auxiliary information, e.g., sea state and atmosphere conditions, are also discussed

From rubber bands to rational maps: A research report

This research report outlines work, partially joint with Jeremy Kahn and Kevin Pilgrim, which gives parallel theories of elastic graphs and conformal surfaces with boundary. One one hand, this lets us tell when one rubber band network is looser than another, and on the other hand tell when one conformal surface embeds in another. We apply this to give a new characterization of hyperbolic critically finite rational maps among branched self-coverings of the sphere, by a positive criterion: a branched covering is equivalent to a hyperbolic rational map if and only if there is an elastic graph with a particular "self-embedding" property. This complements the earlier negative criterion of W. Thurston.Comment: 52 pages, numerous figures. v2: New example

Podcar City Washington, MTI Report S-13-03

On October 23-25, 2013, the Mineta Transportation Institute (MTI) co-presented a two-and-one-half day conference offering insights, strategies, and opportunities for cities to develop personal rapid transit (PRT) systems in their jurisdictions. These PRT systems are also known as automated guideway transit (AGT) or “podcars.” They are small driverless cars (4-6 passengers) that operate on an automated guideway system. Typically, they are used to transport passengers for first- and last- mile trips, such as between a transit station and an airport. However, they may be used for longer distances within or between neighborhoods or commerce centers. The International Institute of Sustainable Transportation (INIST) was the primary producer for the conference. Along with MTI, other co-producers included the U.S. Department of Transportation’s Federal Transit Administration (FTA), the Swedish Transportation Administration, George Mason University, Advanced Transit Association (ATRA), and Kompass. Sponsors included Encitra, Lea+Elliott, and Citizens for PRT