Publications

For smooth bounded domains in ℝn, we prove upper and lower L2 bounds on the boundary data of Neumann eigenfunctions, and we prove quasiorthogonality of this boundary data in a spectral window. The bounds are tight in the sense that both are independent of the eigenvalues; this is achieved by working with an appropriate norm for boundary functions, which includes a spectral weight, that is, a function of the boundary Laplacian. This spectral weight is chosen to cancel concentration at the boundary that can happen for whispering gallery-type eigenfunctions. These bounds are closely related to wave equation estimates due to Tataru. Using this, we bound the distance from an arbitrary Helmholtz parameter E>0 to the nearest Neumann eigenvalue in terms of boundary normal derivative data of a trial function u solving the Helmholtz equation (Δ−E)u=0. This inclusion bound improves over previously known bounds by a factor of E5/6, analogously to a recently improved inclusion bound in the Dirichlet case due to the first two authors. Finally, we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of Neumann eigenpairs on smooth planar domains. We show that the new inclusion bound improves the relative accuracy in a computed Neumann eigenvalue (around the 42000th) from nine to fourteen digits, with negligible extra numerical effort.

Pulsar timing array (PTA) collaborations in North America, Australia, and Europe, have been exploiting the exquisite timing precision of millisecond pulsars over decades of observations to search for correlated timing deviations induced by gravitational waves (GWs). The discovery space of this nanohertz band is potentially rich with populations of inspiraling supermassive black-holes binaries, decaying cosmic string networks, relic post-inflation GWs, and even non-GW imprints of axionic dark matter.
This article aims to provide an understanding of the exciting open science questions in cosmology, galaxy evolution, and fundamental physics that will be addressed by the detection and study of GWs at nanohertz frequencies. The focus of the article is on providing an understanding of the mechanisms by which PTAs can address specific questions in these fields, and to outline some of the subtleties and difficulties in each case. The material included is weighted most heavily towards the questions which we expect will be answered in the coming months to decades with PTAs; however, we have made efforts to include most currently anticipated applications of nanohertz GWs.

Matter evolved under influence of gravity from minuscule density fluctuations. Non-perturbative structure formed hierarchically over
all scales, and developed non-Gaussian features in the Universe, known as the Cosmic Web (1). To fully understand the structure formation of the Universe is one of the holy grails of modern astrophysics. Astrophysicists survey large volumes of the Universe (2–9) and employ a large ensemble of computer simulations to compare with the observed data in order to extract the full information of our own Universe. However, to evolve trillions of galaxies over billions of years even with the simplest physics is a daunting task. We build a deep neural network, the Deep Density Displacement Model (hereafter D3M), to predict the non-linear structure formation of the Universe from simple linear perturbation theory. Our extensive analysis, demonstrates that D3M outperforms the second order perturbation theory (hereafter 2LPT), the commonly used fast approximate simulation method, in point-wise comparison, 2-point correlation, and 3-point correlation. We also show that D3M is able to accurately extrapolate far beyond its training data, and predict structure formation for significantly different cosmological parameters. Our study proves, for the first time, that deep learning is a practical and accurate alternative to approximate simulations of the gravitational structure formation of the Universe.

We present a model for the interaction of the GD-1 stellar stream with a massive perturber that naturally explains many of the observed stream features, including a gap and an off-stream spur of stars. The model involves an impulse by a fast encounter, after which the stream grows a loop of stars at different orbital energies. At specific viewing angles, this loop appears offset from the stream track. The configuration-space observations are sensitive to the mass, age, impact parameter, and total velocity of the encounter, and future velocity observations will constrain the full velocity vector of the perturber. A quantitative comparison of the spur and gap features prefers models where the perturber is in the mass range of 106M⊙ to 108M⊙. Orbit integrations back in time show that the stream encounter could not have been caused by any known globular cluster or dwarf galaxy, and mass, size and impact-parameter arguments show that it could not have been caused by a molecular cloud in the Milky Way disk. The most plausible explanation for the gap-and-spur structure is an encounter with a dark matter substructure, like those predicted to populate galactic halos in ΛCDM cosmology. However, the expected densities of ΛCDM subhalos in this mass range and in this part of the Milky Way are 2−3σ lower than the inferred high density of the GD-1 perturber. This observation opens up the possibility that detailed observations of streams could measure the mass spectrum of dark-matter substructures and even identify individual substructures and their orbits in the Galactic halo.

Two fundamental difficulties are encountered in the numerical evaluation of time-dependent layer potentials. One is the quadratic cost of history dependence, which has been successfully addressed by splitting the potentials into two parts - a local part that contains the most recent contributions and a history part that contains the contributions from all earlier times. The history part is smooth, easily discretized using high-order quadratures, and straightforward to compute using a variety of fast algorithms. The local part, however, involves complicated singularities in the underlying Green's function. Existing methods, based on exchanging the order of integration in space and time, are able to achieve high order accuracy, but are limited to the case of stationary boundaries. Here, we present a new quadrature method that leaves the order of integration unchanged, making use of a change of variables that converts the singular integrals with respect to time into smooth ones. We have also derived asymptotic formulas for the local part that lead to fast and accurate hybrid schemes, extending earlier work for scalar heat potentials and applicable to moving boundaries. The performance of the overall scheme is demonstrated via numerical examples.

We produce light curves for all ~34,000 targets observed with K2 in Campaign 17 (C17), identifying 34 planet candidates, 184 eclipsing binaries, and 222 other periodic variables. The location of the C17 field means follow-up can begin immediately now that the campaign has concluded and interesting targets have been identified. The C17 field has a large overlap with C6, so this latest campaign also offers a rare opportunity to study a large number of targets already observed in a previous K2 campaign. The timing of the C17 data release, shortly before science operations begin with the Transiting Exoplanet Survey Satellite (TESS), also lets us exercise some of the tools and methods developed for identification and dissemination of planet candidates from TESS. We find excellent agreement between these results and those identified using only K2-based tools. Among our planet candidates are several planet candidates with sizes < 4 R_E and orbiting stars with KepMag < 10 (indicating good RV targets of the sort TESS hopes to find) and a Jupiter-sized single-transit event around a star already hosting a 6 d planet candidate.

In solving hard computational problems, semidefinite program (SDP) relaxations often play an
important role because they come with a guarantee of optimality. Here, we focus on a popular
semidefinite relaxation of K-means clustering which yields the same solution as the non-convex
original formulation for well segregated datasets. We report an unexpected finding: when data
contains (greater than zero-dimensional) manifolds, the SDP solution captures such geometrical
structures. Unlike traditional manifold embedding techniques, our approach does not rely on manually defining a kernel but rather enforces locality via a nonnegativity constraint. We thus call our
approach NOnnegative MAnifold Disentangling, or NOMAD. To build an intuitive understanding
of its manifold learning capabilities, we develop a theoretical analysis of NOMAD on idealized
datasets. While NOMAD is convex and the globally optimal solution can be found by generic SDP
solvers with polynomial time complexity, they are too slow for modern datasets. To address this
problem, we analyze a non-convex heuristic and present a new, convex and yet efficient, algorithm,
based on the conditional gradient method. Our results render NOMAD a versatile, understandable,
and powerful tool for manifold learning.

We test a method to reduce unwanted sample variance when predicting Lyman-α (lyα) forest power spectra from cosmological hydrodynamical simulations. Sample variance arises due to sparse sampling of modes on large scales and propagates to small scales through non-linear gravitational evolution. To tackle this, we generate initial conditions in which the density perturbation amplitudes are {\it fixed} to the ensemble average power spectrum -- and are generated in {\it pairs} with exactly opposite phases. We run 50 such simulations (25 pairs) and compare their performance against 50 standard simulations by measuring the lyα 1D and 3D power spectra at redshifts z=2, 3, and 4. Both ensembles use periodic boxes of 40 Mpc/h containing 5123 particles each of dark matter and gas. As a typical example of improvement, for wavenumbers k=0.25 h/Mpc at z=3, we find estimates of the 1D and 3D power spectra converge 34 and 12 times faster in a paired-fixed ensemble compared with a standard ensemble. We conclude that, by reducing the computational time required to achieve fixed accuracy on predicted power spectra, the method frees up resources for exploration of varying thermal and cosmological parameters -- ultimately allowing the improved precision and accuracy of statistical inference.

Pressure plays a key role in the study of quantum materials. Its application in angle resolved photoemission (ARPES) studies, however, has so far been limited. Here, we report the evolution of the k-space electronic structure of bulk Ca2RuO4, lightly doped with Pr, under uniaxial strain. Using ultrathin plate-like crystals, we achieve uniaxial strain levels up to −4.1%, sufficient to suppress the insulating Mott phase and access the previously unexplored electronic structure of the metallic state at low temperature. ARPES experiments performed while tuning the uniaxial strain reveal that metallicity emerges from a marked redistribution of charge within the Ru t2g shell, accompanied by a sudden collapse of the spectral weight in the lower Hubbard band and the emergence of a well-defined Fermi surface which is devoid of pseudogaps. Our results highlight the profound roles of lattice energetics and of the multiorbital nature of Ca2RuO4 in this archetypal Mott transition and open new perspectives for spectroscopic measurements.

The orbital properties of stars in the disk are signatures of their formation, but they are also expected to change over time due to the dynamical evolution of the Galaxy. Stellar orbits can be quantified by three dynamical actions, J_r, L_z, and J_z, which provide measures of the orbital eccentricity, guiding radius, and non-planarity, respectively. Changes in these dynamical actions over time reflect the strength and efficiency of the evolutionary processes that drive stellar redistributions. We examine how dynamical actions of stars are correlated with their age using two samples of stars with well-determined ages: 78 solar twin stars (with ages to ~5%) and 4376 stars from the APOKASC2 sample (~20%). We compute actions using spectroscopic radial velocities from previous surveys and parallax and proper motion measurements from Gaia DR2. We find weak gradients in all actions with stellar age, of (7.51 +/- 0.52, -29.0 +/- 1.83, 1.54 +/- 0.18) kpc km/s/Gyr for J_r, L_z, and J_z, respectively. There is, however, significant scatter in the action-age relation. We caution that our results will be affected by the restricted spatial extent of our sample, particularly in the case of J_z. Nevertheless, these action-age gradients and their associated variances provide strong constraints on the efficiency of the mechanisms that drive the redistribution of stellar orbits over time and demonstrate that actions are informative as to stellar age. The shallow action-age gradients combined with the large dispersion in each action at a given age, however, renders the prospect of age inference from orbits of individual stars bleak. Using the precision measurements of [Fe/H] and [α/Fe] for our stars we investigate the abundance-action relationship and find weak correlations. Similar to our stellar age results, dynamical actions afford little discriminating power between low- and high-α stars.