The Emergence of Gravitational Wave Science: 100 Years of Development of Mathematical Theory, Detectors, Numerical Algorithms, and Data Analysis Tools

On September 14, 2015, the newly upgraded Laser Interferometer Gravitational-wave Observatory (LIGO) recorded a loud gravitational-wave (GW) signal, emitted a billion light-years away by a coalescing binary of two stellar-mass black holes. The detection was announced in February 2016, in time for the hundredth anniversary of Einstein's prediction of GWs within the theory of general relativity (GR). The signal represents the first direct detection of GWs, the first observation of a black-hole binary, and the first test of GR in its strong-field, high-velocity, nonlinear regime. In the remainder of its first observing run, LIGO observed two more signals from black-hole binaries, one moderately loud, another at the boundary of statistical significance. The detections mark the end of a decades-long quest, and the beginning of GW astronomy: finally, we are able to probe the unseen, electromagnetically dark Universe by listening to it. In this article, we present a short historical overview of GW science: this young discipline combines GR, arguably the crowning achievement of classical physics, with record-setting, ultra-low-noise laser interferometry, and with some of the most powerful developments in the theory of differential geometry, partial differential equations, high-performance computation, numerical analysis, signal processing, statistical inference, and data science. Our emphasis is on the synergy between these disciplines, and how mathematics, broadly understood, has historically played, and continues to play, a crucial role in the development of GW science. We focus on black holes, which are very pure mathematical solutions of Einstein's gravitational-field equations that are nevertheless realized in Nature, and that provided the first observed signals.Comment: 41 pages, 5 figures. To appear in Bulletin of the American Mathematical Societ

Computing distances and geodesics between manifold-valued curves in the SRV framework

This paper focuses on the study of open curves in a Riemannian manifold M, and proposes a reparametrization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. to define a Riemannian metric on the space of immersions M'=Imm([0,1],M) by pullback of a natural metric on the tangent bundle TM'. This induces a first-order Sobolev metric on M' and leads to a distance which takes into account the distance between the origins in M and the L2-distance between the SRV representations of the curves. The geodesic equations for this metric are given and exploited to define an exponential map on M'. The optimal deformation of one curve into another can then be constructed using geodesic shooting, which requires to characterize the Jacobi fields of M'. The particular case of curves lying in the hyperbolic half-plane is considered as an example, in the setting of radar signal processing

A 16 x 16 CMOS amperometric microelectrode array for simultaneous electrochemical measurements

There is a requirement for an electrochemical sensor technology capable of making multivariate measurements in environmental, healthcare, and manufacturing applications. Here, we present a new device that is highly parallelized with an excellent bandwidth. For the first time, electrochemical cross-talk for a chip-based sensor is defined and characterized. The new CMOS electrochemical sensor chip is capable of simultaneously taking multiple, independent electroanalytical measurements. The chip is structured as an electrochemical cell microarray, comprised of a microelectrode array connected to embedded self-contained potentiostats. Speed and sensitivity are essential in dynamic variable electrochemical systems. Owing to the parallel function of the system, rapid data collection is possible while maintaining an appropriately low-scan rate. By performing multiple, simultaneous cyclic voltammetry scans in each of the electrochemical cells on the chip surface, we are able to show (with a cell-to-cell pitch of 456 μm) that the signal cross-talk is only 12% between nearest neighbors in a ferrocene rich solution. The system opens up the possibility to use multiple independently controlled electrochemical sensors on a single chip for applications in DNA sensing, medical diagnostics, environmental sensing, the food industry, neuronal sensing, and drug discovery

3D mesh processing using GAMer 2 to enable reaction-diffusion simulations in realistic cellular geometries

Recent advances in electron microscopy have enabled the imaging of single cells in 3D at nanometer length scale resolutions. An uncharted frontier for in silico biology is the ability to simulate cellular processes using these observed geometries. Enabling such simulations requires watertight meshing of electron micrograph images into 3D volume meshes, which can then form the basis of computer simulations of such processes using numerical techniques such as the Finite Element Method. In this paper, we describe the use of our recently rewritten mesh processing software, GAMer 2, to bridge the gap between poorly conditioned meshes generated from segmented micrographs and boundary marked tetrahedral meshes which are compatible with simulation. We demonstrate the application of a workflow using GAMer 2 to a series of electron micrographs of neuronal dendrite morphology explored at three different length scales and show that the resulting meshes are suitable for finite element simulations. This work is an important step towards making physical simulations of biological processes in realistic geometries routine. Innovations in algorithms to reconstruct and simulate cellular length scale phenomena based on emerging structural data will enable realistic physical models and advance discovery at the interface of geometry and cellular processes. We posit that a new frontier at the intersection of computational technologies and single cell biology is now open.Comment: 39 pages, 14 figures. High resolution figures and supplemental movies available upon reques

Geometric deep learning: going beyond Euclidean data

Many scientific fields study data with an underlying structure that is a non-Euclidean space. Some examples include social networks in computational social sciences, sensor networks in communications, functional networks in brain imaging, regulatory networks in genetics, and meshed surfaces in computer graphics. In many applications, such geometric data are large and complex (in the case of social networks, on the scale of billions), and are natural targets for machine learning techniques. In particular, we would like to use deep neural networks, which have recently proven to be powerful tools for a broad range of problems from computer vision, natural language processing, and audio analysis. However, these tools have been most successful on data with an underlying Euclidean or grid-like structure, and in cases where the invariances of these structures are built into networks used to model them. Geometric deep learning is an umbrella term for emerging techniques attempting to generalize (structured) deep neural models to non-Euclidean domains such as graphs and manifolds. The purpose of this paper is to overview different examples of geometric deep learning problems and present available solutions, key difficulties, applications, and future research directions in this nascent field

CMOS design of chaotic oscillators using state variables: a monolithic Chua's circuit

This paper presents design considerations for monolithic implementation of piecewise-linear (PWL) dynamic systems in CMOS technology. Starting from a review of available CMOS circuit primitives and their respective merits and drawbacks, the paper proposes a synthesis approach for PWL dynamic systems, based on state-variable methods, and identifies the associated analog operators. The GmC approach, combining quasi-linear VCCS's, PWL VCCS's, and capacitors is then explored regarding the implementation of these operators. CMOS basic building blocks for the realization of the quasi-linear VCCS's and PWL VCCS's are presented and applied to design a Chua's circuit IC. The influence of GmC parasitics on the performance of dynamic PWL systems is illustrated through this example. Measured chaotic attractors from a Chua's circuit prototype are given. The prototype has been fabricated in a 2.4- mu m double-poly n-well CMOS technology, and occupies 0.35 mm/sup 2/, with a power consumption of 1.6 mW for a +or-2.5-V symmetric supply. Measurements show bifurcation toward a double-scroll Chua's attractor by changing a bias current

A new model for computing the evolution of the extracellular, innercellular and membrane potential simultaneously

Poster Presentation from Nineteenth Annual Computational Neuroscience Meeting: CNS*2010 San Antonio, TX, USA. 24-30 July 2010 In order to model extracellular potentials the Line-Source method provides [1] a very powerful and accurate approach. In this method transmembane fluxes are understood as sources for potential distributions which obey the Poission-equation with zero boundary conditions in the infinity. Its solutions reveal that the waveforms are proportional to local transmembrane net currents. The extracellular potentials are comparable small in amplitude and with the aid of their second special derivatives, it is possible to interpret them as additional fluxes to be included into the cable equation having an impact on the membrane potential of surrounding cells [2]. On this basis ephaptic interactions have been studied and have been considered to play a minor role in the network activity. This modeling study provides a new approach based on the first principle of the conservation of charges which leads to a generalized form of the cable equation taking into account the full three-dimensional detail of the cell’s geometry and the presence of the extracellular potential. So instead of coupling the compartment model and the model for extracellular potentials by means of the transmembrane currents, a non-linear system of partial differential equations is solved. Because the abstraction of deviding the cell’s geometry into compartments falls apart, it is possible to examine the contribution of the precise cell geometry to the signal processing while not neglecting the impact which could result from the extracellular potential. Some simulations of propagating action potentials on ramified geometries are going to be shown as well as the resulting distributions of extracellular action potentials