Singularities and Semistable Degenerations for Symplectic Topology

We overview our recent work defining and studying normal crossings varieties and subvarieties in symplectic topology. This work answers a question of Gromov on the feasibility of introducing singular (sub)varieties into symplectic topology in the case of normal crossings singularities. It also provides a necessary and sufficient condition for smoothing normal crossings symplectic varieties. In addition, we explain some connections with other areas of mathematics and discuss a few directions for further research.Comment: 18 pages, 2 figure

Counting genus zero real curves in symplectic manifolds

There are two types of $J$-holomorphic spheres in a symplectic manifold invariant under an anti-symplectic involution: those that have a fixed point locus and those that do not. The former are described by moduli spaces of $J$-holomorphic disks, which are well studied in the literature. In this paper, we first study moduli spaces describing the latter and then combine the two types of moduli spaces to get a well-defined theory of counting real curves of genus 0. We use equivariant localization to show that these invariants (unlike the disk invariants) are essentially the same for the two (standard) involutions on $\mathbb{P}^{4n-1}$.Comment: Publishing versio

UAV-enabled wireless-powered Iot wireless sensor networks

Future massive internet of thing (IoT) networks will enable the vision of smart cities, where it is anticipated that a massive number of sensor devices, in the order of tens of millions devices, ubiquitously deployed to monitor the environment. Main challenges in such a network are how to improve the network lifetime and design an e cient data aggregation process. To improve the lifetime, using low-power passive sensor devices have recently shown great potential. Ambient backscattering is a novel technology which provides low-power long-range wireless communication expanding the network lifetime signi cantly. On the other hand, in order to collect the sensed data from sensor devices deployed over a wide area, unmanned aerial vehicles (UAVs) has been considered as a promising technology, by leveraging the UAV's high mobility and line-of-sight (LOS) dominated air-ground channels. The UAV can act as data aggregator collecting sensed data from all sensors. In this thesis, we consider medium-access control (MAC) policies for two sensor data collection scenarios. First, the objective is to collect individual sensor data from the eld. The challenge in this case is to determine how a large number of sensors should access the medium so that data aggregation process performed in a fast and reliable fashion. Utilizing conventional orthogonal medium access schemes (e.g., time-division vi multiple access (TDMA) and frequency-division multiple access (FDMA)), is highly energy consuming and spectrally ine cient. Hence, we employ non-orthogonal multiple access (NOMA) which is envisaged as an essential enabling technology for 5G wireless networks especially for uncoordinated transmissions. In Chapter 2, we develop a framework where the UAV is used as a replacement to conventional terrestrial data collectors in order to increase the e ciency of collecting data from a eld of passive backscatter sensors, and simultaneously it acts as a mobile RF carrier emitter to activate backscatter sensors. In the MAC layer, we employ uplink power-domain NOMA scheme to e ectively serve a large number of passive backscatter sensors. Our objective is to optimize the path, altitude, and beamwidth of the UAV such that the network throughput is maximized. In Chapter 3, we consider the scenario where there are a separate data collector and RF carrier emitter such that the former is a gateway on the ground and the latter is a single UAV hovering over the eld of backscatter sensors. Secondly, we consider a case where only a function of sensed data is of interest rather than individual sensor values. A new challenge arises where the problem is to design a communication policy to improve the accuracy of the estimated function. Recently, over-the-air computation (AirComp) has emerged to be a promising solution to enable merging computation and communication by utilizing the superposition property of wireless channels, when a function of measurements are desired rather than individual in massive IoT sensor networks. One of the key challenges in AirComp is to compensate the e ects of channel. Motivated by this, in Chapter 4, we propose a UAV assisted communication framework to tackle this problem by a simple to implement sampling-then-mapping mechanism