# Applied Math Colloquium - Spring 2021

Colloquia are held on Fridays at 11:30 a.m. via Webex, unless otherwise noted. Webex information will be posted at a later date.

Colloquia are held on Fridays at 11:30 a.m. via Webex, unless otherwise noted. Webex information will be posted at a later date.

To join the Applied Mathematics Colloquium mailing list visit https://groups.google.com/a/njit.edu/forum/?hl=en#!forum/math-colloquium/join (Google Profile required). To join the mailing list without a Google Profile, submit the seminar request form.

**Jeff Calder (University of Minnesota)**

Random Walks and PDEs in Graph-Based Learning

I will discuss some applications of random walks and PDEs in graph-based learning, both for theoretical analysis and algorithm development. Graph-based learning is a field within machine learning that uses similarities between datapoints to create efficient representations of highdimensional data for tasks like semi-supervised classification, clustering and dimension reduction. There has been considerable interest recently in semi-supervised learning problems with very few

labeled examples (e.g., 1 label per class). The widely used Laplacian regularization is ill-posed at low label rates and gives very poor classification results. In the first part of the talk, we will use the random walk interpretation of the graph Laplacian to precisely characterize the lowest label rate at which Laplacian regularized semi-supervised learning is well-posed. At lower

label rates, where Laplace learning performs poorly, we will show how our random walk analysis leads to a new algorithm, called Poisson learning, that is probably more stable and informative than Laplace learning. We will conclude with some applications of Poisson learning to image classification and mesh segmentation of broken bone fragments of interest in anthropology.

**Alex Gittens (RPI)**

An Algorithm for Two-Cost Budgeted Matrix Completion

Matrix completion is a ubiquitous tool in machine learning and data analysis. Most work in this area has focused on the number of observations necessary to obtain an accurate lowrank approximation. In practice, however, the cost of observations is an important limiting factor, and experimentalists may have on hand multiple modes of observation with differing noise-vs-cost trade-offs. This work considers matrix completion

subject to such constraints: a budget is imposed and the experimentalist's goal is to allocate this budget between two sampling modalities in order to recover an accurate low-rank approximation. Specifically, we consider that it is possible to obtain low noise, high cost observations of individual entries or high noise, low cost observations of entire columns. We introduce a regression-based completion algorithm for this setting and experimentally verify the performance of our approach on both synthetic and real data

sets. When the budget is low, our algorithm outperforms standard completion algorithms. When the budget is high, our algorithm has comparable error to standard nuclear norm completion algorithms and requires much less computational effort

**Denis Silantyev (Courant Institute, NYU) **

Obtaining Stokes Wave with High-Precision Using Conformal Maps and Spectral Methods on Non-Uniform Grids

Two-dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth has a class of solutions called Stokes waves which is fully nonlinear periodic gravity waves propagating with the constant velocity. We developed a new highly efficient method for computation of Stokes waves. The convergence rate of the numerical approximation by a Fourier series is determined by the complex singularity of the travelling wave in the complex plane above the free surface. We study this singularity and use an auxiliary conformal mapping which moves it away from the free surface thus dramatically speeding up Fourier series convergence of the solution. Three options for the auxiliary conformal map are described with their advantages and disadvantages for numerics. Their efficiency is demonstrated for computing Stokes waves near the limiting Stokes wave (the wave of the greatest height) with 100-digit precision. Drastically improved convergence rate significantly expands the family of numerically accessible solutions and allows us to study the oscillatory approach of these solutions to the limiting wave in great detail.

**Haomin Zhou (Georgia Tech)**

Optimal Transport on Graphs with Some Applications

Optimal transport theory in continuous space has been extensively studied in the past few decades. In this talk, I will present the optimal transport theory on discrete spaces. Various recent developments related to free energy, FokkerPlanck equations, as well as Wasserstein distance on graphs will be presented. some of them are surprising in the discrete case. Applications in robotics as well as Schrodinger equation on graphs will be discussed briefly.

**Nigel Mottram (Glasgow)**

Active Nematic Liquid Crystals

Active nematic liquid crystals are fluids in which continuous internal energy generation, such as in bacterial suspensions and microtubule-forming suspensions, allows for spontaneous flow generation. In such fluids, the flow-generating agent usually possesses a shape anisotropy (defined by, for instance, the long axis of the bacterium or microtubule), with this symmetry giving rise to a liquid crystalline-like phase. Internally driven flows can lead to interesting effects such as self-organisation and non-equilibrium defect configurations. In this talk I will survey the main features of dense systems of such self-propelling agents and highlight some recent results from our work in the area. For instance, the director orientation and flow of an active nematic liquid crystal confined between two parallel glass plates shows a complex interaction between orientation, self-induced flow and the flow generated through director rotation (often called backflow). In two dimensions the behaviour is even more complicated, although in certain asymptotic limits (high or low Ericksen number) we discover the importance of corner regions in driving the bulk flow. Finally, I will consider some very recent experimental results and propose an area of future work which may lead to a deeper understanding of how bacteria use self-organisation to assist spreading of thin films.

**Horacio Rotstein (NJIT)**

Resonance-Based Mechanisms of Generation of Oscillations in Networks of Non-Oscillatory Neurons

Abstract: Several neuron types have been shown to exhibit (subthreshold) membrane potential resonance (MPR), defined as the occurrence of a peak in their voltage amplitude response to oscillatory input currents at a preferred (resonant) frequency. MPR has been investigated both experimentally and theoretically. However, whether MPR is simply an epiphenomenon or

it plays a functional role for the generation of neuronal network oscillations, and how the latent time scales present in individual, non-oscillatory cells affect the properties of the oscillatory networks in which they are embedded are open questions. We address these issues by investigating a minimal network model consisting of (i) a non-oscillatory linear resonator (bandpass

filter) with 2D dynamics, (ii) a passive cell (lowpass filter) with 1D linear dynamics, and (iii) nonlinear graded synaptic connections (excitatory or inhibitory) with instantaneous dynamics.We demonstrate that (i) the network oscillations crucially depend on the presence of MPR in the resonator, (ii) they are amplified by the network connectivity, (iii) they develop relaxation oscillations for high enough levels of mutual inhibition/excitation, and (iv) the network frequency monotonically depends on the resonator’s resonant frequency. We explain these phenomena using a reduced adapted version of the classical phase-plane analysis that helps uncovering the type of effective network nonlinearities that contribute to the generation of network oscillations. Our results have direct implications for network models of firing rate type and other biological oscillatory

**Chad Higdon-Topaz (Williams College)**

Quantitative Approaches to Social Justice

Civil rights leader, educator, and investigative journalist Ida B. Wells said that "the way to right wrongs is to shine the light of truth upon them." This talk will demonstrate how quantitative and computational approaches can shine a light on social injustices and help build solutions to remedy them. We will present quantitative social justice projects on topics ranging from diversity in art museums to equity in criminal sentencing to affirmative action, health care access, and other

fields. The tools engaged include data mining, crowdsourcing, data cleaning, clustering, hypothesis testing, statistical modeling, Markov chains, data visualization, and more.

**No Colloquium - Spring Recess**

**Robert Style (ETH Zurich)**

Phase Transformation in Soft Materials

I will talk about two seemingly unconnected topics, which actually have strong conceptual links. Firstly, I will discuss how growing ice into soft materials causes damage: for example in cryopreservation, or in the process of frost heave. It is commonly believed that this damage is due to the expansion of ice upon freezing, but this often plays only a minor role. I’ll show how we are measuring the stresses generated during freezing at the micron-scale, and describe how this is

caused by cryosuction — the major effect that underlies freezing-induced damage. Secondly, I will discuss how controlled phase separation can be used to generate vibrant structural colours, but how this is very hard to control for practical usage. We have shown that we can completely change this by performing the phase separation in soft, polymeric solids. Then the mechanical properties of the soft solid couple to the phase separation process, giving very different results.

**No Colloquium - Good Friday**

**David Anderson (University of Wisconsin)**

An Introduction to Stochastic Reaction Networks

Models of cellular processes are often represented with networks that describe the interactions between the constituent molecules. If the counts of the molecular “species" are low, then these systems are most often modeled stochastically using a continuous-time Markov chain. These stochastic reaction networks can be quite complex. However, hidden within the complexity there are sometimes underlying structures that, if properly quantified, give great insight into the dynamical or stationary behavior of the system. In this talk, I will begin with an introduction to the basic mathematical model and then provide a broad overview of research in this direction. I will detail some of the main results in the field, and I will present some open problems that people are actively working on. I plan to make this talk accessible to graduate students, though having some knowledge of continuous-time Markov chains would be helpful.

**Chris Rycroft (Harvard SEAS)**

Uncovering the Rules of Crumpling with a Data-Driven Approach

When a sheet of paper is crumpled, it spontaneously develops a network of creases. Despite the apparent disorder of this process, statistical properties of crumpled sheets exhibit striking reproducibility. Recent experiments have shown that when a sheet is repeatedly crumpled, the total crease length grows logarithmically [1]. This talk will offer insight into this surprising result by developing a correspondence between crumpling and fragmentation processes. We show how crumpling can be viewed as fragmenting the sheet into flat facets that are outlined by the creases, and we use this model to reproduce the characteristic logarithmic scaling of total crease length, thereby supplying a missing physical basis for the observed phenomenon [2].

This study was made possible by large-scale data analysis of crease networks from crumpling experiments. We will describe recent work to use the same data with machine learning methods to probe the physical rules governing crumpling. We will look at how augmenting experimental data with synthetically generated data can improve predictive power and provide physical insight [3].

[1] O. Gottesman et al., Commun. Phys. 1, 70 (2018).

[2] J. Andrejevic et al., Nat. Commun. 12, 1470 (2021).

[3] J. Hoffmann et al., Sci. Advances 5, eaau6792 (2019).

**Haimin Wang (New Jersey Institute of Technology)**

Data Intensive Study of Space Weather using Advanced Observations and Machine Learning

I will first introduce the importance of space weather research, and the challenges of data analysis in this subject. I will then summarize some discoveries using the 1.6m Goode Solar Telescope (GST) at Big Bear Solar Observatory (BBSO), with beautiful movies. Several of these works were published in Nature Journals: E.g. (1) The small scale magnetic structure evolution leading to solar eruptions that can affect earth—so called space weather. (2) With highest resolution observations, the photospheric magnetic structure changes can be tracked from flaring polarity inversion lines (PIL) propagating outwards. These include the sudden flare-induced rotation of sunspots and the increase of transverse magnetic fields near PIL. This provides some clear clues to the so-called “Dog vs. Tail” problem. (3) We recently found extremely strong (>5500G) fields in PIL of flaring sunspot in the famous solar active region of September 2017. Finally, I will introduce applications of machine learning and artificial intelligence in processing the “Big Data” from our observations. These examples will include automatic feature recognition and tracking, as well as the forecasting of solar eruptions. Overall, these discoveries have motivated us to collaborate with computer scientists and statisticians.

**Casey Diekman (NJIT)**

Data Assimilation and Dynamical Systems Analysis of Circadian Rhythmicity and Entrainment

Circadian rhythms are biological oscillations that align our physiology and behavior with the 24-hour environmental cycles conferred by the Earth’s rotation. In this talk, I will discuss two projects from my recent sabbatical that focus on circadian clock cells in the brain and the entrainment of circadian rhythms to the light-dark cycle. Most of what we know about the electrical activity of circadian clock neurons comes from studies of nocturnal (night-active) rodents, hindering the translation of this knowledge to diurnal (day-active) humans. In the first part of the talk, we use data assimilation and patch-clamp recordings from the diurnal rodent Rhabdomys pumilio to build the first mathematical models of the electrophysiology of circadian neurons in a day-active species. We find that the electrical activity of circadian neurons is similar overall between nocturnal and diurnal rodents but that there are some interesting differences in their responses to inhibition. In the second part of the talk, we use tools from dynamical systems theory to study the reentrainment of a model of the human circadian pacemaker following perturbations that simulate jet lag. We show that the reentrainment dynamics are organized by invariant manifolds of fixed points of a 24-hour stroboscopic map and use these manifolds to explain a rapid reentrainment phenomenon that occurs under certain jet lag.

April 26, 2021