2024 Faculty and Student Summer Talks
The talks will take place every Monday and Thursday from June 3, 2024, to August 1, 2024 at 2PM in Cullimore Hall Room 611.
Date  Day  Speaker, Title, and Abstract 

June 3  M 
Atul Anurag The Phase Space of the ThreeVortex Problem and its Application to VortexDipole Scattering The motion of three pointvortices in a 2D inviscid, incompressible fluid has been widely studied. Gröbli (1877) derived a closed system for the evolution of the side lengths of the triangle formed by the vortices. This system has been the basis of most studies of this problem. 
June 6  R 
Samantha Evans A Fast MeshFree Boundary Integral Method for Two Phase Flow with Soluble Surfactant and a Study of Electroconvective Flow The main focus of this talk is developing a fast and accurate boundary integral method to simulate the twophase flow of a viscous drop immersed in a surrounding viscous fluid which contains dissolved soluble surfactant. The drop is stretched by an imposed extensional flow. Surfactant is a surface active agent important in microfluidic applications that lowers surface tension and introduces a Marangoni force that typically opposes the imposed outer flow. However, it is difficult to adapt boundary integral methods to problems with soluble surfactant as the equation for the bulk surfactant does not have a Green’s function formulation. In addition, at large Peclet numbers that are characteristic of real physical systems, the concentration of soluble surfactant develops a transition or boundary layer adjacent to the drop interface which is difficult to resolve using traditional numerical methods. To address these difficulties, the hybrid numerical method that was first introduced in Booty and Siegel (2010) and extended to twophase flow with soluble surfactant in Xu, Booty, and Siegel (2013) is further developed. The hybrid method is based on an asymptotic reduction of the surfactant dynamics in the transition layer in the limit Pe→∞. This reduced advectiondiffusion equation for the concentration of bulk soluble surfactant has a Green's function formulation. A fast method for computing a timeconvolution integral that arises in the Green’s function formulation of the advectiondiffusion equation is introduced. The fast algorithm is based on a method originally introduced for Abel integral equations by Johannes Tausch. Results illustrating the speed and accuracy of the fast method will be presented. The secondary focus of this talk is conducting a study on electroconvective flow. The model considered is a symmetric binary electrolyte bounded by an ionselective surface on the bottom and a stationary reservoir on the top. The ionselective surface is impermeable to anions and permeable to cations. An external electric field is applied to drive the flow and transport of ions. The model is based on the electrostatically forced incompressible Stokes equations coupled to the Poisson and NernstPlanck equations. The streamfunction form of the Stokes equation is used, giving rise to the 2D force biharmonic equation. This fourth order boundary value problem results in high roundoff error when using ChebyshevFourier representation, as the roundoff error for the kth Chebyshev derivative grows like O(M^(2k) δ) where M is the number of discretization point in the vertical direction and δ is machine precision. To remedy this a numerical method to solve the system at higher accuracy using a spectrally accurate ultrasphericalFourier representation is developed. Results illustrating the electroconvective flow will be presented, including the expulsion of negative and positive free charge density into the bulk and the time and xaveraged current. 
June 10  M 
Jose Pabon Reduced Order Models of Hydrodynamically Interacting Flapping Wings Reducedorder modeling provides efficient means of elucidating the dynamical system characteristics of large, complex scenarios. Fish schools are examples of collective motion in which hydrodynamic interactions play a role in dynamical behavior and selforganization principles. However, the longtime evolution of hydrodynamically interacting collectives is challenging to investigate due to the persistent influence of longlived vortical structures, and the highresolution requirements of direct numerical simulation at large Reynolds numbers. Reducedorder models have therefore played an important role in theoretical investigations. We introduce new reducedorder models of swimmers that selfpropel by flapping, i.e., executing a prescribed periodic rigid body motion. The models are extensions of a discretetime dynamical system developed by Oza, Ristroph and Shelley in which flapping swimmers interact through periodically shed vortices. The models are used to investigate conditions under which hydrodynamic interactions lead to stable swimming configurations with an optimized swimming speed. Numerical results are presented. Additional exciting new avenues of research will be briefly discussed as well.

June 17  M 
Nicholas Harty Towards the Numerical Homogenization of TimeDomain Multiple Scattering by Resonators, and Finite Wires and Strips We will present some recent results relating to multiple scattering from objects in the plane, including preliminary results on the homogenization of materials.

June 17  M 
Patrick Grice Title and Abstract forthcoming 
June 20  R 
Kristian Nestor Model checks for censored twosample locationscale via estimated characteristic functions In this paper, we apply estimated characteristic functions for censored twosample locationscale model checks. The proposed test relies heavily on Stute and Wang's KaplanMeier integrals for the characteristic function as well as the plugins for means and variances. As would be expected with plugins, however, the test is not asymptotically distribution free. Bootstrap is employed to obtain critical values needed for the testing. Numerical studies validate the proposed testing method.

June 20  R 
Tareq Aldirawi Conformal Risk Control and applications on Multilabel Classification Conformal prediction is a popular, modern technique for providing valid predictive inference for arbitrary machine learning models. By measuring the model's performance on a calibration dataset of featureresponse pairs and then postprocessing the model, we construct prediction sets that bound the miscoverage. We will also focus on the extension of conformal prediction where the natural notion of error is not simply miscoverage. Finally, we use conformal prediction framework to address the uncertainty quantification of multilabel classification problems along with the tools of multiple hypothesis testing. 
June 27  R 
Mark Fassano On Modeling a Thin Oil Film Covering Rigid and Deformable Objects In this talk, we present a derivation to describe the dynamics of a thin oil film covering a quasistatic water drop both under the excitement of a MHzfrequency SAW propagating through the underlying substrate. It has been shown in experiments that if the displacements caused by such a wave are in a given range (low to moderate values), then the oil film (on the order of hundreds of micrometers) will be translated along the direction of propagation while the water drop will slightly deform, however, remain stationary (with fixed contact lines). Beginning from the assumption that both phases individually satisfy the Stokes flow equations, we are able to follow the general outline to derive the thinfilm equation to get a modified version of that for both phases. In this way, we reduce the problem from solving for velocities and pressures of both phases to just solving fourthorder PDEs for the height profiles of the two phases. The talk concludes by looking at numerical results of the case of a rigid obstacle with a brief comparison to experiments.

June 27  M 
Philip Zaleski Iterated Function Systems and Stochastic Gradient Descent We introduce iterated function systems (IFS) by constructing the Barnsley Fern and the Collage theorem. We formulate stochastic gradient descent (SGD) as an iterated function system and present a one dimensional theory for convergence and uniqueness of invariant measures. Lastly, we apply our Theorem to a double well potential and compare the exact dynamics to the diffusion approximation. 
July 1  M 
Austin Juhl Certifying Stability in RungeKutta Schemes: Algebraic Conditions and Semidefinite Programming Numerical stability is a critical property for a timeintegration scheme. In the context of RungeKutta methods applied to stiff differential equations, Astability is one of the most basic and practically important notions of stability. Dating back to the work of Dahlquist, it has been known that Astability is equivalent to the RungeKutta stability function satisfying a particular convex feasibility problem. Specifically, up to a transformation, the stability function lies in the convex cone of \emph{positive functions}. In recent years, sumofsquares optimization and semidefinite programming have become valuable tools in developing rigorous certificates of stability in dynamical systems. Therefore, it is natural to employ these convex optimization tools for the purpose of rigorously certifying A and A(α)stability in RungeKutta methods. Two distinct convex feasibility problems defined by linear matrix inequalities are introduced. The first approach employs sumofsquares programming applied to the RungeKutta $E$polynomial, making it applicable to both A and A(α)stability. The second approach refines the algebraic conditions for Astability, as developed by Cooper, Scherer, Türke, and Wendler (CSTW), to incorporate the RungeKutta order conditions. The theoretical enhancement of the algebraic conditions facilitates the practical application of the refined conditions for certifying Astability within a computational framework. The Epolynomial and CSTW methodologies are utilized to obtain rigorous stability certificates for several implicit RungeKutta schemes proposed in the literature. Specific attention is given to certifying the implicit RungeKutta schemes utilized in the SUite of Nonlinear and DIfferential/ALgebraic equation Solvers (SUNDIALS). 
July 8  M 
Matthew Illingworth On Correlating Topology and Performance of Membrane Filter Pore Networks Membrane filtration is an important and ubiquitous process in industrial applications, and there is a growing body of mathematical models that capture this complex process. Previous theoretical work models the internal structure of membrane filters as a network of cylindrical pores whose radii are drawn from a uniform distribution, with fouling modeled as an adsorption process; i.e. the gradual accretion of fouling particles on the inner walls of the pores. Simulationbased approaches are used to measure membrane filter performance, using metrics such as total throughput and accumulated foulant concentration. In the present work, we investigate the correlation between the performance of these networks and their topological properties, in order to discover optimal pore topologies for membrane filter design. We use persistent homology as our principal tool for quantifying topological features, where the radii of a network’s pores are represented by a collection of twodimensional points known as a persistence diagram. The data encoded in these persistence diagrams are then statistically correlated with the performance metrics, particularly with total throughput. Favorably strong correlation between total throughput and filter topology will be presented.

July 8  M 
Andrew White Role of Calcium Buffers in Synaptic Neurotransmitter Release and its shortterm Modulation Neurotransmitter release is caused by the entry of calcium ions into the cell in response to a depolarizing pulse ("action potential"), and their subsequent binding to calcium sensor proteins that fuse neurotransmittercontaining vesicle with the synaptic membrane. The probability (efficiency) of such fusion may transiently change from pulse to pulse, either increasing (shortterm facilitation) or decreasing (shortterm depression). Facilitation and depression are collectively termed shortterm synaptic plasticity (STSP or STP). In this project we investigate STP by modeling calcium dynamics inside the synapse, focusing on the role of calcium buffers in STP phenomenon. This involves numerical solution of the corresponding reactiondiffusion PDE system describing the evolution of calcium and buffer concentrations. It had been shown previously that facilitation may result from the depletion of buffer inside the synaptic terminal (“facilitation by buffer saturation”), or from the dislocation of buffer from the membrane upon calcium binding (“facilitation by buffer translocation”). However, we find other, previously unreported dynamic behaviors in the case of calcium buffers with more than one calcium binding site, such as calmodulin or calretinin. Our goal is to fully explore the dependence of shortterm calcium dynamics on calcium buffering properties, focusing on buffers characterized by cooperative calcium binding. We will also discuss possible extensions of this work, including extensions to other fundamental calciumdependent cell mechanism. 
July 11  R 
Bryan Currie On the maximum value of the stairs2 index Measures of tree balance play an important role in different research areas such as mathematical phylogenetics or theoretical computer science. The balance of a tree is usually quantified in a single number, called a balance or imbalance index, and several such indices exist in the literature. Here, we focus on the stairs2 balance index for rooted binary trees, which was first introduced in the context of viral phylogenetics but has not been fully analyzed from a mathematical viewpoint yet. While it is known that the caterpillar tree uniquely minimizes the stairs2 index for all leaf numbers and the fully balanced tree uniquely maximizes the stairs2 index for leaf numbers that are powers of two, understanding the maximum value and maximal trees for arbitrary leaf numbers has been an open problem in the literature. We fill this gap by showing that for all leaf numbers, there is a unique rooted binary tree maximizing the stairs2 index. Additionally, we obtain recursive and closed expressions for the maximum value of the stairs2 index of a rooted binary tree with n leaves 
July 11  R 
Joseph D'Addesa Stability and Spreading of Sessile Drops on Vertically Vibrating Substrates In this talk we examine the stability and spreading of a water drop under the influence of vertical oscillations. We start by introducing the thin film model which includes inertial effects and briefly go through the derivation. We will then discuss some simulation results and how this dynamic system can be described by a surrogate static system modeled by the YoungLaplace Equation. Finally we will investigate the different forces acting on the drop in order to gain some insight into the drop’s spreading dynamics. This can be useful in pattern formation and coating of surfaces where control over the spreading of fluids is important. 
July 15  M 
Prianka Bose Modeling NonIsochronous Cyclic Rhythms: Relating Taal and Clave In this talk, we will explore the rhythmic patterns in music, focusing on Taal and Clave, which represent isochronous and nonisochronous cyclic rhythms, respectively. We will discuss the significance of these rhythmic structures and their properties, including beat generation and synchronization continuation. The primary objective of this study is to develop a hybrid model that combines biophysical and artificial neural network (ANN) techniques to analyze and replicate these rhythmic patterns. Our approach involves enhancing the Beat Generator (BG) model, originally based on the HodgkinHuxley framework, to handle both isochronous and nonisochronous synchronization. This enhancement is achieved by incorporating the Tapping Neuron (TN), a Morris Lecar model, which uses frequency information from the BG and counting values from the ANN. The model determines the cycle length, including taps and rests, with the rests learned and classified by the ANN. We will present methods showing how the hybrid model generates voltagetime course traces of nonisochronous and cyclic rhythms. The BG model's ability to replicate isochronous sound files and perform beat tracking will be shown, highlighting its alignment with external stimuli. Further, we will illustrate the mapping of Clave to Taal counting, the classification of Clave types by the ANN, and the hybrid model's replication of nonisochronous cyclic rhythms. Results on Son Clave forecasting, peak location, and the differentiation between 4/4 and 12/8 Clave will also be discussed. 
July 18  R 
Connor Greene Optimal Polynomial Bases on the Square and Cube Polynomial interpolation is a ubiquitous tool in numerical analysis. We present a method for deriving collocation points that attain faster convergence in integration and interpolation errors than standard tensorproduct Chebyshev points in 2 and 3 dimensions, with ready generalization to higher dimensions. The natural polynomial basis associated to the points is efficient in "Euclidean" degree, which is the relevant measure for interpolating or integrating a generic analytic function. In addition to this efficiency advantage, there are natural algorithms based on the fast Fourier transform for working with the points; the conversion of point values to polynomial basis coefficients (and vice versa) can be performed in O(n log n) time. We compare the results of interpolating and integrating with these points to other "reduced" sets of points, including the "Padua" points. 
Jul 22  M 
Matthew Cassini Volume Integral Method for Electromagnetic Equations Understanding electromagnetic wave behavior has many applications such as detecting defects in manufacturing metamaterials and nanophotonic crystals. In order to study this, we will discuss the direct scattering problem of a locally perturbed infinite layer in which Maxwell’s Equations reduces to the Helmholtz Equation. We will introduce the FloquetBloch transform to convert the system of equations into a quasiperiodic system in two spatial variables before applying a volume integral equation and FFT to solve this problem. Ongoing work will include expanding to Maxwell’s Equations and solving the inverse problem. 
July 22  M 
Zhiwen (Esther) Wang Deep Learning Methods in Spatial Intensity Estimation In spatial point processes intensity estimation, traditional methods like kernel estimators and point process models (PPM) have been effective in capturing linear correlations between points. However, they fall short when dealing with nonlinear correlations. Deep learning models, such as Neural Networks, Autoencoders, and Variational Autoencoders (VAE), offer a promising alternative to address these limitations due to their inherent properties and settings. These models are widely used and acknowledged for their flexibility and capability to handle complex, nonlinear relationships and allows it to maintain the flexibility of traditional models, such as bandwidth selection in kernel estimators and feature selection in PPM. Block Bootstrap in Phylogenetic Tree Estimation In phylogeny estimation, Felsenstein’s bootstrap support values are the most common approach to assess the reliability and robustness of phylogenies estimated from sequence data. Bootstrap support values are based on resampling with replacement the sites of a reference alignment until obtaining a bootstrap alignment 
July 22  M 
Nataran Rezaei Instability between the twolayer Poiseuille Flow with the VOF Method In this talk, I will present the numerical simulation of the instability in a twolayer flow. We will examine the wave evolution using the VOF (Volume Of Fluid) method. Furthermore, the interface instability results from the Chebyshev spectral collocation method and finite difference method are presented to validate our outcomes. Finally, we will investigate the effect of stabilizing and destabilizing parameters on the growth rate. 
July 25  R 
David Mazowiecki Numerical Simulation of Multiple Particles Undergoing Quincke Rotation We study several dielectric particles suspended in a dielectric fluid in stokes flow under a uniform applied electric field. Depending on physical parameters and field strength, a single particle can have either a stable or unstable solution. In the unstable case, the solution bifurcates into a class of solutions where the polarization is antiparallel to the applied field, and the particles rotate about an axis perpendicular to the applied field. We present the relevant boundary integral equations, and compare our results with those found in the literature. 
July 29  M 
Michael Storm Spectral Stability of Pulse Solutions of the SwiftHohenberg Equation with Computer Assisted Proofs The SwiftHohenberg equation is a PDE noted for its use in modeling pattern formation. We consider the problem of determining the spectral stability of pulse solutions. This can be challenging since there are infinitely many eigenvalues, so we count a related object called conjugate points, of which there are the same number of unstable eigenvalues. We discuss the numerics involved in finding conjugate points and how computer assisted proofs can be applied. 
July 29  M 
Elizabeth Tootchen Title and Abstract forthcoming 
July 29  M 
Souaad Lazergui Title and Abstract forthcoming 
Updated: July 24, 2024