### 2017

Leaky oil droplets that are self-propelling due to their created concentration gradient form an ideal system for studying collective behavior. I will present a simple model that can be reduced to a system of non-Markov stochastic differential equations, allowing for analytical results that match the observed experimental system. The particles' interactive force is observed through their hovering above a bottom plate and their repelling nature. The model also displays a regime of super-diffusive scaling likely related to the mobility transition to a constant velocity solution (of the deterministic system). The single non-dimensional parameter in the model controls the history of interaction, allowing the system to go from having complete memory to behaving like interacting electrostatic potentials.

All numerical calculations will fail to provide a reliable answer unless the continuous problem under consideration is well posed. Well-posedness depends in most cases only on the choice of boundary conditions. In this paper we will highlight this fact, and exemplify by discussing well-posedness of a prototype problem: the time-dependent compressible Navier–Stokes equations. We do not deal with discontinuous problems, smooth solutions with smooth and compatible data are considered.

In particular, we will discuss how many boundary conditions are required, where to impose them and which form they should have in order to obtain a well posed problem. Once the boundary conditions are known, one issue remains; they can be imposed weakly or strongly. It is shown that the weak and strong boundary procedures produce similar continuous energy estimates. We conclude by relating the well-posedness results to energy-stability of a numerical approximation on summation-by-parts form. It is shown that the results obtained for weak boundary conditions in the well-posedness analysis lead directly to corresponding stability results for the discrete problem, if schemes on summation-by-parts form and weak boundary conditions are used.

The analysis in this paper is general and can without difficulty be extended to any coupled system of partial differential equations posed as an initial boundary value problem coupled with a numerical method on summation-by parts form with weak boundary conditions. Our ambition in this paper is to give a general roadmap for how to construct a well posed continuous problem and a stable numerical approximation, not to give exact answers to specific problems.

### 2016

There are numerous and diverse challenges associated with analyzing data collected from different fields of science and engineering. This talk consists of two parts. First, time-dependent oscillatory signals occur in a wide range of fields, including geophysics, biology, medicine, finance and social dynamics. Of great interest are techniques that decompose the time-dependent signals into multiple oscillatory components, with time-varying amplitudes and instantaneous frequencies. Such decompositions can help us better describe and quantify the underlying dynamics that govern the system. I will present a new advance in time-frequency representations whose effectiveness is justified by both numerical experiments and theoretical analysis. Second, the high-dimensionality of point cloud data makes investigating such data difficult. Fortunately, these data often locally concentrate along a low-dimensional subspace and this makes the problem more tractable. I will talk about utilizing low-dimensional structures for various data analysis objectives, ranging from recovering the underlying data in the presence of complex noise including Gaussian additive noise and large sparse corruptions, to recognizing subspace-based patterns in data, from robust algorithm design to theoretical analysis. The techniques for learning subspaces have broad applications: image processing, computer vision, bioinformatics, medicine, etc. At the end, I will talk about some future directions where both fields are involved.

Inspired by real-world networks consisting of layers that encode different types of connections, such as a social network at different instances in time, we study community structure in multilayer networks. We analyze fundamental limitations on the detectability of communities by developing random matrix theory for the dominant eigenvectors of modularity matrices that encode an aggregation of network layers. Aggregation is often beneficial when the layers are correlated, and it represents a crucial step for the discretization of time-varying network data, whereby layers are binned into time windows. We explore two methods for aggregation: summing the layers? adjacency matrices as well as thresholding this summation at some value. We develop theory for both large- and small-scale communities and analyze detectability phase transitions that are onset by varying either the density of within-community edges or community size. We identify layer-aggregation strategies that are optimal in that they minimize the detectability limit. Our results indicate good practices in the context of community detection for how to aggregate network layers, threshold pairwise-interaction data matrices, and discretize time-varying network data. We apply these results to synthetic and empirical networks, including a study of anomaly detection for the Enron email corpus.

LIGO’s detection of gravitational waves from a binary black hole merger inaugurates a completely new mode of observational astronomy and represents the culmination of a quest lasting half a century. After a brief review of gravitational waves in general relativity, I will discuss the detection itself. How do the LIGO instruments work? How do we know the signal was caused by a binary black hole merger? What does this detection tell us about binary black holes? Then I will focus on how this moment came to pass. The detection required many ingredients to be in place including (1) developments in theoretical relativity to allow proof that gravitational waves were not coordinate artifacts; (2) a bold vision to recognize that gravitational wave detection was not impossible; (3) technological developments of novel vacuum systems, lasers, optical coatings, active seismic isolation, etc.; (4) the successful conclusion of a 35 year effort to simulate binary black holes on the computer; (5) development of sophisticated, new data analysis methods to tease a waveform from noisy data; (5) the growth of the field of gravitational wave science from a handful of practitioners to the more than 1000 authors on the detection paper; and finally (6) the (nearly) unwavering support of the National Science Foundation. The first detection was followed by a second one in this first "science run" and soon another science run will begin. I will end with discussion of the future — more binary black holes, other sources of gravitational waves and what we might learn, instrument upgrades, new facilities — and other ways to detect gravitational waves — from space and from monitoring millisecond pulsars.

Wave breaking in deep oceans is a challenge that still defies complete scientific understanding. Sailors know that at wind speeds of approximately 5m/sec, the random looking windblown surface begins to develop patches of white foam ('whitecaps') near sharply angled wave crests. We idealize such a sea locally by a family of close to maximum amplitude Stokes waves and show, using highly accurate simulation algorithms based on a conformal map representation, that perturbed Stokes waves develop the university feature of an overturning plunging jet. We analyze both the cases when surface tension is absent and present. In the latter case, we show the pluning jet is regularized by capillary waves which rapidly become nonlinear Crapper waves in whose trough pockets whitecaps may be spawned.