2019 Faculty and Student Summer Talks
Talks begin at 2PM on Tuesdays (T) and Thursdays (R) in Cullimore Hall, room 611.
Talks begin at 2PM on Tuesdays (T) and Thursdays (R) in Cullimore Hall, room 611.
Date  Day  Speaker, Title, and Abstract 

June 6  T  Professor Travis Askham The Integral Equation Approach to the Numerical Solution of PDEs I will give a brief outline of the thinking and technology behind the use of integral equation representations to solve partial differential equations. Then, I'll present some recent work in this area on eigenvalue problems for Stokes flow and buckling analysis. 
June 6  R  Professor James MacLaurin Biological systems are able to demonstrate remarkable cohesive behavior in spite of considerable noise and variability. In this talk I outline techniques to understand how emergent phenomena such as cellular oscillations and spatiallyorganized neural activity can exist in spite of noise. In each case, I define a microscopic model that exhibits randomness and variability. I then analyze how this variability `scales up' to produce coherent macroscopic phenomena. In effect, symmetries at the microscopic level lead to preferred macroscopic behavior  since the macroscopic dynamics is typically attracting, it can persist for very long periods of time.

June 11  T  Professor Anand Oza This talk will describe systems in which active particles interact with their selfgenerated fluid flows. The first consists of oil droplets walking on the surface of a vibrating fluid bath, which has been shown to exhibit several features previously thought to be peculiar to the microscopic quantum realm. The second is a model of a "flock" of flapping swimmers interacting at high Reynolds number, which underscores how hydrodynamics might influence schooling behavior in animal collectives. The third consists of “surfers” that selfpropel on a vibrating fluid bath, a tunable prototype for interfacial active matter. In these systems, a complex dynamics arises from the fluidmediated interaction between particles and their histories. Andrew deStefan Optimal Sampling Paths for Autonomous Vehicles in Uncertain Ocean Flows

June 13  R  Professor Amitabha Bose Dr. Valeria Barra Efficient Representation of HighOrder Operators for the Numerical Solution of PDEs

June 25  T  Professor Enkeleida Lushi Will give an introduction into the world of motion without inertia, and with a focus on movement of microorganisms. Will explain the notions of Reynolds number and what swimming strategies can be employed in that regime. Will introduce the Stokes Equations, and the physical ideas like kinematic reversibility, quasistatic dynamics, drag anisotropy and stochastic trajectories. Mathematical tools such as the "Stokeslet" fundamental solution, boundary integral equations, multipole expansions in the far field, and slender body theory, will be introduced. Will also briefly discuss the method of images and Lorentz reflection theorem. Whenever needed, movies will illustrate the concepts.
Matthew Moye Data Assimilation Methods for Neuronal State and Parameter Estimation This work explores the use of data assimilation algorithms to estimate unobserved variables and unknown parameters of conductancebased neuronal models. Data assimilation (DA) is the optimal integration of noisy observations from a system with the output of a model describing that system in order to improve estimates of the system’s states and the model’s parameters. Modern DA techniques, such as the Unscented Kalman Filter (UKF) and 4DVariational methods (4DVar), are widely used in climate science and weather prediction, but have only recently begun to be applied in neuroscience. Sequential data assimilation techniques, such as the UKF, iteratively take in observational recordings to produce system estimates, whereas variational techniques attempt to minimize a cost function over a fixed time window. We demonstrate how to use DA to infer several parameters of the MorrisLecar model from a single voltage trace, and then extend our approach to more complex conductancebased models. These DA techniques have advantages and disadvantages. Namely, there is typically a gain in precision in exchange for computational time with the variational approach, but it has limitations when working with large time series data. We explore the utility of a twostage approach in which the fast timescale dynamics of a system are estimated via 4DVar, and slow timescale dynamics through the UKF. We apply this approach to currentclamp recordings of Drosophila clock neurons. 
June 27  R  Zhongcheng Lin In medical research, investigators usually face some informative censoring problems: that is, failure time and censoring time could be dependent with each other. Such a situation often occurs in clinical trials and survival analysis. Therefore, dependent censoring is an important issue to address. In this talk, I will first introduce some basic concepts regarding survival analysis such as rightcensored data, frailty models and archimedean copulas. These concepts are closely related to dependent censoring and several statistical models can be applied to model the dependence between timetoevent random variables. Moreover, the identifiability problem of dependent competing risk models is possible to be solved under some model assumptions. The dependence level between failure time and censoring time variable can be uniquely determined by the survival data. Also the marginal survival functions of the two variables are obtainable as the limits of copula graphic estimator. Finally, we will be developing new statistical methods that could be used to estimate the dependence level between failure time and censoring time in a nonparametric setting.
Circadian oscillators are found in a variety of species. The entrainment to a light dark cycle is one of the most important properties of a circadian system. A new tool, called the entrainment map, has been introduced to study the entrainment of a single oscillator. Here we generalize the map to study coupled circadian oscillators. The generalized map is a dimension reduced Poincar´e map on a Torus. By studying the geometry of the coupled system and parameterization, we reduce the dimension of the Poincar´e map from 4D to 2D. Studying the fixed points of the map, we are able to determine conditions for existence and stability of periodic orbits for the original system. In addition, by iterating the map and locating the stable and unstable manifold of the saddle points, we are able to understand the direction of entrainment and how the manifolds behave as a separatrix. Comparing with simulation, the map also provides a good approximation for the time to entrainment. 
July 2  T  Professor Enkeleida Lushi Will give a broad introduction to the area of active complex suspensions, which are fluids with moving/motile constituents in it. They are systems far from equilibrium and are often characterized by selforganization, collective motion, large number fluctuations. Wellknown examples include swarming bacteria colonies, actinmyosin beds, microtubulekinesin solutions (i.e. cell constituents), cell assemblies, and more recently externallydriven colloids. In low Re, the immersing fluid enables longrange interactions between individual active unit and as such affects the emerging selforganization. We'll discuss ways to model and compute the dynamics in these suspensions, whether in the continuum limit or tracing the individuals and their interactions. Linear stability for bacterial suspensions will be discussed, as well as their effective rheology, as examples of mathematical approaches to such problems.
Linwan Feng Numerical Methods for Dispersive Shallow Water Equations In the talk, we discuss numerical timestepping approaches for solving the dispersive shallow water wave equations. The equations take the form of nonlocal evolution equations where an elliptic operator is applied to one of the time derivatives. We examine two approaches for handling the nonlocal operator: (i) iterative methods that must be performed at each time step; and (ii) semiimplicit (ImExtype) time stepping methods that avoid inverting the full nonlocal operator (and also do not require subiterations). Guaranteeing stability for the semiimplicit approach is a nontrivial issue due to the fact that certain stiff terms in the equations are treated explicitly. We provide a stability theory which outlines how to choose the semiimplicit terms in such a way to guarantee numerical stability. We also discuss the situation when bottom topography is included in the system. 
July 9  T  Mahdi Bandegi In this talk, we discuss the global minimization of a large deviations rate function (the Helmholtz free energy functional) for the Boltzmann distribution, in large systems of interacting particles. Such systems are widely used as models in computational chemistry and molecular dynamics. Global minimizers of the rate function characterize the asymptotics of the partition function and thereby determine many important physical properties (such as selfasembly, or phase transitions). Finding and verifying local minima to the Helmholtz free energy functional is relatively straightforward. However, finding and verifying global minima is much more difficult since the Helmholtz energy is nonconvex and nonlocal. In our approach, instead of minimizing the original nonconvex functional, we find minimizers to a convex lower bound, which is related to the cone of copositive functionals. The socalled relaxed problems consists of a linear variational problem with an infinite number of Fourier constraints, leading to a variety of computational challenges. We then develop a fast solver based on matrixfree interior point algorithms by exploiting the Fourier structure in the problem.
Abstract TBA

July 11  R  Yixuan Sun In this talk I will talk about a recently introduced kernel technique, the weighted principal support vector machine (WPSVM), and its integration into a spatial point process framework. The WPSVM possesses an additional parameter, a weight parameter, besides the regularization parameter. Most statistical techniques, including WPSVM, have an inherent assumption of independence, which means the data points are not connected with each other in any manner. But spatial data violates this assumption. Correlation between two spatial data points increases as the distance between them decreases. However, through extensive simulations it has been shown that WPSVM performs better than other dimension reduction techniques. The main advantage of WPSVM comes from the fact that it can handle nonlinear relationships. I will also talk about an application to a real world data.
In this talk I will talk about a recently introduced kernel technique, the weighted principal support vector machine (WPSVM), and its integration into a spatial point process framework. The WPSVM possesses an additional parameter, a weight parameter, besides the regularization parameter. Most statistical techniques, including WPSVM, have an inherent assumption of independence, which means the data points are not connected with each other in any manner. But spatial data violates this assumption. Correlation between two spatial data points increases as the distance between them decreases. However, through extensive simulations it has been shown that WPSVM performs better than other dimension reduction techniques. The main advantage of WPSVM comes from the fact that it can handle nonlinear relationships. I will also talk about an application to a real world data.

July 16  T  Malik Chabane We consider resonant triad interactions of gravitycapillary waves and investigate in detail special resonant triads that exchange no energy during their interactions so that the wave amplitudes remain constant in time. After writing the resonance conditions in terms of two parameters (or two angles of wave propagation), we first identify a region in the twodimensional parameter space, where resonant triads can be always found, and then describe the variations of resonant wavenumbers and wave frequencies over the resonance region. Using the amplitude equations recovered from a Hamiltonian formulation for water waves, it is shown that any resonant triad inside the resonance region can interact without energy exchange if the initial wave amplitudes and relative phase satisfy the two conditions for fixed point solutions of the amplitude equations. Furthermore, it is shown that the symmetric resonant triad exchanging no energy forms a transversely modulated traveling wave field, which can be considered a twodimensional generalization of Wilton ripples. Optimal transport is a class of problems that was first examined by Gaspard Monge in the 18th century, but has only been recently tackled numerically due to theoretical and numerical difficulties. Producing provably convergent and efficient numerical schemes will allow the optimal transport problem to be utilized in a wide variety of fields such as data science, geometric optics, geoscience, inverse problems, meteorology, among others. The MongeAmpère PDE is derived from a simplification of the general mass transport problem with a cost function that satisfies some technical requirements. More general MongeAmpère type PDE, called prescribed Jacobian equations, arise in various applications including solving the optimal transport problem on the sphere, for example in geometric optics. The Wasserstein metric, defined as a distance metric between probability measures is another way to look at the optimal transport problem. The advantage of such a notion is in defining a metric space, which has ramifications in some applications such as gradient flows as well as image processing, seismic fullwaveform inversion and machine learning. We present a problem on solving the MongeAmpère equation on a sphere, and introduce one problem on gradient flows in Wasserstein space as well as hint at a possible numerical scheme for solving Wasserstein1. 
July 18  R  Professor Victor Matveev Neuroscience is a vast field that studies physical phenomena operating in a wide range of spatial and temporal scales, reflecting the complex multiscale nature of a functioning nervous system. The associated biological questions range in scale from single molecules such as ionic channel proteins, to the scale of the entire mammalian brain consisting of tens of billions of neurons. In this talk I will give an overview of some of the more fundamental concepts in neuroscience. In particular, I will introduce the socalled conductancebased HodgkinHuxley model of neuronal excitability, which is one of the most important building blocks of computational neuroscience. Understanding neuronal excitability, namely electric spike propagation along the neuronal axon, requires understanding the balance of concentrations of electrolytes such as sodium, potassium and calcium, as they move in and out of the cell across voltagesensitive transmembrane channel proteins. Interestingly, cell calcium in particular seems to control an even wider range of physiological cell processes, from gene expression to muscle contraction. I will give an overview of the most fundamental ingredients of cell calcium modeling, and list some interesting mathematical problems arising in the study of cell calcium dynamics. The emphasis in this talk will be made on simple explanations of the underlying biology, in order to provide a basic background for those that are interesting in delving further into this field of mathematical modeling.
Soheil Saghafi The nematode Caenorhabditis elegans produces undulatory locomotion as a result of alternating dorsal and ventral bends that propagate down the body of the worm. Although C. elegans locomotion involves just 66 neurons for which the anatomical location and connectivity has been fully mapped, the field still lacks an understanding of the neuronal properties that are critical for locomotion. We have developed a computational model of the motoneuron network and muscle cells underlying dorsal and ventral muscle activations. To gain insight into the aspects of the network that control locomotion, we are using Differential Evolution to find the connection strengths and intrinsic neuronal properties that optimize the match between the model output and data from real worms collected by Gal Haspel’s lab (Biological Sciences, NJIT).
Surfactants, also known as surfaceactive agents, change the interfacial properties between fluids by reducing surface tension. This characteristic of surfactants allows for many applications, which include detergents and soaps, the prevention of embolisms, and the stablization of emulsions. A “hybrid” numerical method has been developed to study the effects that surfactant solubility has on the dynamics of interfacial flow in the limit of a large bulk Peclet number, Pe. To accurately resolve the interface dynamics, the hybrid method uses a leading order asymptotic reduction of the governing equations in the limit as Pe tends to infinity. Until now, this method has been implemented to simulate the dynamics of drops and bubbles where the bulk concentration of soluble surfactant is solely located in the exterior fluid. Although this accuarately models some fluid systems, situations also arise where there is surfactant in the interior fluid. In this talk, I will introduce the hybrid method for interior flow, which will use complex variable techniques for 2D Stokes flow. 
July 23  T  Professor Michael Siegel We consider a model of a superhydrophobic cylinder rotating in a viscous liquid. The boundary of the cylinder is assumed to contain alternating noslip and noshear surfaces (of possibly different sizes), with transverse flow. Our main interest is in computing the hydrodynamic torque on the cylinder. An explicit solution to the flow problem is obtained using classical theory of complex variables, combined with asymptotic analysis. This work is joint with Ehud Yariv (Technion).
Yinbo Chen Many fundamental cell processes such as neurotransmitter and hormone release are activated by Ca2+ influx through transmembrane Ca2+ channels. Since optical Ca2+ imaging has very limited spatial and temporal resolution, computational modeling played an important role in the study of vesicle release and other cell processes activated by local Ca2+ signals. Computational studies revealed that localized Ca2+ elevations, termed nanodomains, form and collapse very rapidly in response to Ca2+ channel gating. This suggests that quasistationary solutions of reactiondiffusion equations describing Ca2+ ion movement and binding to intracellular Ca2+ molecules (termed Ca2+ buffers) capture the properties of Ca2+ nanodomains with sufficient accuracy. Previously developed approximations of stationary Ca2+ nanodomains such as the rapid buffering approximation (RBA), and the linear approximation proved very useful for understanding the properties of local Ca2+ signals. However, the accuracy of these approximants is restricted to specific regions in parameter space, and apart from RBA, they cannot be extended to more realistic biological buffers with multiple Ca2+ binding sites. Here we present several new approaches to approximate stationary singlechannel Ca2+ nanodomains with more accuracy in a wider range of model parameters. One of the new methods is based on matching the coefficients of shortrange Taylor series and longrange asymptotic series of the Ca2+ concentration distance dependence using simple interpolants. A second method is based on the variational approach, and involves a global minimization of a relevant functional with respect to parameters of a chosen approximation. Finally, we present a global error reduction method that provides an even better approximation accuracy over a wide range of distances and parameter values. Importantly, some of the presented approximants can be extended to more realistic buffers with two binding sites characterized by cooperative Ca2+ binding, such as calmodulin.
David Mazowieki Title TBA Abstract TBA

July 25  R 
Emel Khan Circadian rhythms are daily oscillations of activity which occur in liv ing organisms including animals, plants, fungi, and cyanobacteria. These rhythms have the defining properties of being persistent in constant con ditions, entrainable to external time cues, and sustainable over different temperatures. Our focus is on circadian rhythms in cyanobacteria, whose core clock is comprised of just three proteins, KaiA, KaiB and KaiC. When these proteins are mixed with ATP in a test tube, KaiC phospho rylates and dephosphorylates in an oscillatory manner. Here we analyze the mathematical model developed by Rust et al. to understand the cir cadian oscillations in cyanobacteria. Furthermore, we also address issues in the model and propose possible solutions. We are using simulations and dynamical systems theory to gain insight into the mechanisms of the core oscillator, as well as how the system entrains to lightdark cycles. Our longterm goal is to fully understand the timekeeping mechanism in cyanobacteria and how the properties of this clock may relate to circadian rhythms in mammals.
The aim of this talk is to solve the two dimensional time harmonic scattering problem via boundary integral equations of various type of layer representations. Stated differently, we solve the exterior Helmhotz equation of a bounded domain with an incoming plane wave. We also consider the problem of solving the scattering problem for single and multiple scatterers with varying smoothness. 
July 30  T  Beibei Li The false discovery rate (FDR) is the expected proportion of type I errors when conducting multiple comparisons. Controlling FDR is a powerful approach to multiple testing. In many large scale multiple testing applications, the hypotheses often have a graphical structure, such as gene ontology in gene expression data. However, it is challenging to control FDR when we incorporate the structure into developing testing procedures. In this talk, I will first introduce a new general approach for large scale multiple testing (GELS) which can simplify the problem of developing FDR controlling procedures as developing perfamily error rate (PFER) controlling procedures. Then, we have modified the GELS procedure and under this condition we want to develop a FDR controlling procedure under independence or positive regression dependence on subset(PRDS) for discrete data.
Twosample locationscale refers to a model that permits a pair of standardized random variables to have a common distribution. Functionbased hypothesis testing in these models refers to formal tests that would help decide whether or not two samples may have come from some locationscale family of distributions. For uncensored data, a comparison between two approaches of testing, one based on empirical characteristic functions (ECFs) and another on plugin empirical likelihood (PEL), is carried out. Sample means and standard deviations are used as plugins for both approaches. Results of numerical studies are reported.
We adapt the Vicsek model of selfpropelling particles to look at the collective motion for such swimmers in nontrivial 2D domains. We discuss the behavior in circular convex and nonconvex domains for a variety of densities, confinement sizes and alignment distances. The observed behavior of this model is compared with recent experimental results with bacteria. 
August 1  R  Diego Rios Using geometry, an initial travel path and time is found for a ray traveling through ocean water and sediment at a constant velocity profile. A discrete and continuous method is explored for a linear velocity profile once the ray refracts as it reaches the ocean sediment.
Rituparna Basak Application of Machine Learning to the StickSlip Dynamics of Particulate Media The stickslip transition of granular systems is related to earthquakes and avalanches, and therefore understanding the conditions leading to slip events is of general importance. Although stickslip behavior has been studied extensively, what triggers a slip event still remains unclear. We studied machine learning techniques to analysis of topological data such as force network and persistence diagrams (information of forces, Betti numbers and connectivity) evolving from discrete element simulations of granular system to understand the stickslip behavior. For predicting the next slip event, we are using some Machine learning algorithms such as Logistic Regression and Support vector machine on a large number of data. The results of the behavior of our model will be discussed.
Chao Cheng The stickslip transition of granular systems is related to earthquakes and avalanches, and therefore understanding the conditions leading to slip events is of general importance. Although stickslip behavior has been studied extensively, what triggers a slip event still remains unclear. The purpose of our study is to explore existence of precursors to slip events. For this purpose, we study a sheared system in stickslip regime via twodimensional discrete element simulations. Particular focus is on the evolution of force networks before and during slip events. We will show that some features of force network evolution could be used to gain insight into occurrence of a slip event. 
August 6  T  Kosuke Sugita We consider two dimensional Stokes flow problems with singularities in a viscous liquid. Such singularities appear depending on the geometries of the piecewise smooth boundary curves or multiple boundary conditions.
This talk presents an efficient robust numerical scheme which can be generally applied to a system of elliptic boundary value problems coupled at the boundary. The BVP models phenomena whose domain spans over multiple mediums, e.g. acoustic wave scattering through air and liquid. The algorithm, developed by Boubendir et. al., is a discretelevel adaptation to the domain decomposition method (DDM) for timeharmonic wave solutions. The new treatment crosspoints (special DDM nodes) is discussed in the context of solving a heterogeneousmedium scattering problem which involves the coupling of Finite Element Method and Boundary Element Method.
Brandon Behring The goal of this talk will be to introduce pointvortex motion and the integrable relative periodic "leapfrogging" solutions and the mysterious alebraic value for when the bifurcation of those orbits can occur. We will discuss how to both analyze and visualize the stability of periodic orbits in general using the "leapfrogging" solutions as an example. We will conclude with a novel semianalytic argument for the bifurcation value based on a a classical method due to G. W. Hill.

August 8  R  Ruqi Pei In this talk, we will give an overview of solving 2D elliptic PDEs using boundary integral equations. When the integrand has logarithmic singularities, classical Gaussian quadrature either cannot be implemented efficiently or it may simply fail to work. We will discuss the optimization procedure for the classic generalized Gaussian quadrature (GGQ) introduced by James Bremer using a panelbased method for the discretization of the boundary. Current work involves investigating a modification of GGQ to deal with some salient issues.
RamsauerTownsend (RT) resonance is a phenomenon in quantum mechanics (e.g. electron scattering) that allows waves to be fully transmitted. As a macroscopic example of RT resonance, the transmission of waves through chains of spherical granular particles has been studied (both experimentally and theoretically) to gain a better understanding of this quantum phenomenon. In this presentation, we will explore yet another possible analog of RT resonance: the transmission of waves through a chain of walking droplets interacting through wave fields.
Membrane filters have been widely used in industrial applications to remove contaminants and undesired impurities from the solvent. During the filtration process the membrane internal void area becomes fouled with impurities and as a consequence the filter performance deteriorates. In addition to the internal morphology of membrane filters, fouling mechanisms contribute to the complexity of the filtration process, in deterministic or stochastic ways. So far various mathematical models have been proposed to describe the membrane structure and stochasticity of particles flow individually but very few focus on both together. In this talk, we present an idealized mathematical model, in which a membrane consists of a series of bifurcating pores, which decrease in size as the membrane is traversed and particles are removed from the feed by adsorption within pores and sieving simultaneously. We focus on the modeling of the sieving process via a continuoustime Markov chain and show its explicit construction. Lastly, we discuss how filtration efficiency depends on the characteristics of the branching structure and show the coupling between the two fouling mechanisms.

Updated: August 15, 2019