2025 Faculty and Student Summer Talks

June 2

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Dr. Roy Goodman

A MATLAB disaster (or, why not to blindly trust your numerics)

Prof. Goodman will give an example of a short talk that succeeds in its objectives (in his opinion). He will then discuss some pitfalls to avoid when speaking about your research. Time permitting, he may also discuss another important issue: managing numerical experiments.

June 9

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Michael Storm

Spectral Stability of the Swift-Hohenberg Equation with Computer Assisted Proofs

The Swift-Hohenberg equation is a partial differential equation noted for its use in modeling pattern formation. In this talk we consider the problem of determining the spectral stability of standing pulse solutions. A difficulty in determining spectral stability is that the point spectrum can be challenging to compute. To deal with this, we count a related object called conjugate points, which have a 1-1 correspondence with unstable eigenvalues. We discuss the numerics involved in counting conjugate points and how computer assisted proofs can be applied to give a rigorous proof of the spectral stability/instability of a pulse solution.

June 12

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Zheng Zhang

Transfer learning under stochastic block models

In this talk, we study the transfer learning problem under stochastic block models(SBM), which aim to improve the fit on target data by borrowing information from useful source data. We propose a transfer learning algorithm that automatically leverages the commonality in the presence of unknown discrepancy, by optimizing an estimated bias-variance decomposition. Numerical studies confirm the effectiveness of the algorithm

June 16

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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. Simulation-based 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 two-dimensional 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. A purely topological effect on total throughput will be presented, along with the temporal evolution of filter topology over the lifetime of the filter.

June 19

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Souaad Lazergui

Analysis of Multiple Scattering In High Frequency Regime for Dirichlet and Neumann Problems

June 23

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Mark Fasano

Modeling Drops Driven by Surface Acoustic Waves (SAWs)

We present a theoretical study, supported by simulations and simple experiments, on the spreading of a silicone oil drop under MHz-frequency SAW excitation in the underlying solid substrate. Our time-dependent theoretical model uses the long wave approach and considers interactions between fluid dynamics and acoustic driving. For the macroscopic silicon oil drops in this study, acoustic forcing arises from Renolds stress variations in the liquid due to changes in the intensity of the acoustic field leaking from the SAW beneath the drop and the viscous dissipation of the leaked wave. Both experiments and simulations show that after an initial phase where the oil drop deforms to accommodate acoustic stress, it accelerates, achieving nearly constant speed over time, leaving a thin wetting layer. Our model indicates that the steady speed of the drop results from the quasi-steady shape of its body. The drop’s shape and speed are further clarified by a simplified traveling wave-type model that highlights various physical effects.  

In the second half of the talk, we discuss an extension of this model that describes the dynamics of SAW-driven drops covering solid obstacles. The solid obstacle contributes to both the gravitational and capillary terms in the evolution equation for the drop height consistent with the literature. Numerical results show that for both bump and ramp shaped obstacles qualitative trends with respect to the SAW amplitude are captured by the simulations consistent with experiments.

June 26

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Atul Anurag

The Global Phase Space of the Three-vortex Interaction System and its Application to Vortex-Dipole Scattering

The main focus of this dissertation is the development a fast and accurate boundary integral method to simulate the two-phase 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 surface 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 two-phase 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 P e → ∞. This reduced advection-diffusion equation for the concentration of bulk soluble surfactant has a Green’s function formulation. A fast method for computing a time-convolution integral that arises in the Green’s function formulation of the advection-diffusion 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 second focus of this dissertation is a study of electroconvective flow. The model considered is a symmetric binary electrolyte bounded by an ion-selective surface on the bottom and a stationary reservoir on the top, periodic in the horizontal direction and wall-bounded in the vertical direction. The ion-selective 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 Nernst-Planck equations. The streamfunction form of the Stokes equation is used, giving rise to the 2D forced biharmonic equation. This fourth order boundary value problem results in severe amplification of round-off error when using a Chebyshev-Fourier representation, as the round-off error for the kth Chebyshev derivative grows like O(N^(2k)δ) where N is the number of discretization points 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 ultraspherical-Fourier 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 space-averaged current.

June 30

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Joseph D'Addesa

Stability and Spreading of water drops on Vertically Vibrating Substrates

We investigate experimentally and theoretically the spreading and deformation of fluid puddles on bounded substrates under vertical surface vibrations. For a fluid volume with a characteristic height on the order of the capillary length, a finite-radius puddle remains stable in the absence of forcing. Upon applying vibrations, the puddle front expands progressively, with the extent of spreading increasing with vibration strength. We find that the equilibrium puddle radius increases with both liquid volume and excitation amplitude. This behavior is captured by an effective capillary length framework, where vibrational forcing reduces the apparent surface tension. These findings offer new insight into how dynamic surface forces influence the shape and stability of vibrated fluid volumes.

July 3

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Andrew White

Modeling Calcium-Induced-Calcium-Release: The Effects of Buffers on Calcium Wave Propagation

This project explores calcium-induced calcium release (CICR) and the resultant propagation of calcium waves. CICR is a biological phenomenon whereby the influx of calcium ions into a cell triggers the release of yet more calcium from intracellular stores called the endoplasmic and sarcoplasmic reticulum (ER/SR). This positive feedback is caused by the opening of calcium- sensitive calcium channels on the surface of the ER/SR membrane. This feedback mechanism is crucial for muscle cell contraction, endocrine hormone release, oocyte fertilization, and other fundamental cell functions. The calcium channels responsible for CICR are the inositol triphosphate receptors (IP₃R) and ryanodine receptors (RyR). In many cells, such receptors are distributed in clusters along the ER/SR membrane, leading to calcium waves as the activation of one cluster triggers the activation (firing) of a nearby channel cluster.

To model this behavior, we utilize what is known as the “Fire-Diffuse-Fire” model (FDFM). Our goal is to explore how intracellular substances called calcium buffers affect the propagation and speed of calcium waves by binding to free calcium and affecting the resultant calcium dynamics. Previous work utilized a homogenized model which exhibits continuous waves to study the effect of calcium buffers. These buffers can bind one or two calcium ions and, depending on the binding dynamics, were found to prevent or empower the propagation of continuous Ca²⁺ waves. Our goal is to extend this work to the case of the more biophysical FDFM model, which describes saltatory wave propagation between neighboring channel clusters. Preliminary results describing the effect of simple buffers with one calcium binding site on 1D wave propagation will be presented.

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Bryan Currie

Extremal Trees and Values of the Maximum Increase in Widths Index

Mathematical phylogenists are interested in improving models for evolution, and current models tend to yield trees that seem more "balanced" than trees inferred from genetic data. However, since tree balance is ambiguous, there are several tree balance indices which capture different notions of balance. One less-understood tree balance index is maximum increase in widths, which in particular lacks understanding when limited to binary trees. We fill a gap in the literature by describing the maximum value of the index over all leaf numbers as well as counting the maximizing trees for some special leaf numbers.

July 7

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Patrick Grice

The Inverse Elasto-Acoustic Obstacle Problem

The inverse elasto-acoustic problem consists of shape reconstruction and classification of an elastic obstacle immersed in an acoustic medium. The motivation for our work is to improve the stability and efficiency of acoustic imaging methods in layered media which occur frequently in applications.

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Justin Maruthanal

Hopf Bifurcations in the Stoichiometric Model for Centrosome Motion in the C. elegans embryo

The Caenorhabditis elegans, C. elegans, is a model organism used for various studies. Its embryo has been observed to undergo an asymmetric cell division. During cell division, the two centrosomes align themselves along the anterior-posterior axis of the cell. During this alignment process, they go closer to the posterior end and are noted to undergo transverse oscillations. I am studying a stoichiometric model to study this phenomenon. 
In this model, microtubules nucleating and growing from the centrosomes can bind to cortical force generators, CFGs, on the surface. In a stoichometric framework, the CFGs can bind to one MT at a time. A Hopf BIfurcation is observed in this model as the number of CFGs increase. Using the method of multiple scales, I hope to study the nature and size of this bifurcation.

July 10

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Nan Zhou

Point Vortex Dynamics: Three-Pair Vortex Model and Continuation

This talk studies periodic motion in a symmetric three–pair point vortex system, motivated by classical vortex “leapfrogging” dynamics in two–dimensional incompressible flow. While exact solutions are known for two vortex pairs, adding a third pair destroys integrability and requires numerical methods. We use continuation techniques to track families of periodic orbits as the distance between vortex pairs varies, and compare different numerical formulations. The results reveal how symmetry-breaking and deformation of vortex trajectories emerge as interactions between vortex pairs become stronger.

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Elizabeth Tootchen

Modeling Atonia in a REM Sleep Network

One of the characteristics of REM sleep is atonia, or muscle paralysis due to the inhibition of motoneurons in the spinal cord. A role of atonia is to prevent one from acting out dreams during REM sleep, which could cause inadvertent harm to the individual or those around them. REM active neurons, which are part of the sleep-wake circuitry, project via two pathways to the spinal cord motoneurons which control atonia. Damage to the ventral sublaterodorsal nucleus (vSLD) or ventral medial medulla (VMM), which are key elements of these pathways, may result in an individual experiencing REM sleep without atonia. We construct a firing rate model that combines the sleep-wake cycle and atonia pathways to discern the roles of vSLD and VMM on network dynamics. By functionally disabling key pathways in the atonia circuit, using phase plane analysis and numerical simulations, we suggest various explanations that may underlie REM sleep without atonia.

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Nastaran Rezaei

Instability of two-layer flows

July 14

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Matthew Cassini

Volume Integral Method for Electromagnetic Equations

Scattering problems have a wide range of applications, from military to electronics to medicine and from science to production. Our work is motivated by applications in optics and material sciences, such as non-destructive testing of defects in periodic structures such as nanophotonic crystals.  These materials are part of high-tech sensor devices widely used in, for example, aviation, space, and medicine. In practice, the manufacturing of these types of materials may contain defects.  The use of wave scattering provides an efficient way to non-destructively test the performance of the production of these materials. The scattered wave by the medium under consideration allows us to predict the existence of a defect.  We are particularly interested in considering a sample of materials on the nanoscale, which contains many periodic cells.  By sending electromagnetic waves (such as light),  the wave scattered by the domain is governed by Maxwell's equations defined in the domain given by the characterization of the sample of material. The main objective of our work is to numerically solve Maxwell's equations under the assumption that the sample contains a defect locally included in a period cell.  Compared to existing works on numerical methods for Maxwell's equations, we have to deal with a large domain consisting of many periodic cells while the existence of defects breaks the periodicity, which causes challenges in solving the problem.  The method that we proposed is based on the use of the Floquet-Bloch transform technique in \cite{haddar_nguyen2017} to reformulate the equations into a system for which the discretized version gives a system of quasi-periodic equations. Such equations can be solved in a smaller domain (a single period cell).  We then write the quasi-periodic equation under the volume integral formulation in \cite{NGUYEN201559} and combine it with FFT to efficiently discretize the equation and then iteratively solve with GMRES. We perform numerical experiments with varying scattering obstacles and discuss the convergence. This work will be followed by ongoing mathematically rigorous proof.

July 17

R

Philip Zaleski

Convergence of Markov Chains for Stochastic Gradient Descent with Separable Functions

Stochastic gradient descent (SGD) is a popular algorithm for minimizing objective functions that arise in machine learning. For constant step-size SGD, the iterates form a Markov chain on a general state space. Focusing on a class of nonconvex objective functions, we establish a “Doeblin-type decomposition,” in that the state space decomposes into a uniformly transient set and a disjoint union of absorbing sets. Each absorbing set contains a unique invariant measure, with the set of all invariant measures being the convex hull. Moreover, the set of invariant measures is shown to be a global attractor of the Markov chain with a geometric convergence rate. The theory is highlighted with examples that show (1) the failure of the diffusion approximation to characterize the long-time dynamics of SGD; (2) the global minimum of an objective function may lie outside the support of the invariant measures (i.e., even if initialized at the global minimum, SGD iterates will leave); and (3) bifurcations may enable the SGD iterates to transition between local minima.

July 21

M

Luc Brancheau

Neural Field Models

Hodgkin-Huxley models are well-known for their success in capturing the behavior of individual neurons. However, these types of models can become unwieldy as the number of neurons increases. This can make them inappropriate for modeling the emergent properties of large clusters of neurons. Instead, we consider a neural field approach, considering the average behavior of neurons to keep the dynamics tractable.

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Gabriel Masarwa

Dance of the Spinning Micro-Particles Above a Wall

Modeling systems of microswimmers in free space and half free space using Rotlet singularities.

July 24

R

José Pabón

Reduced Order Models of Hydrodynamically Interacting Flapping Wings

Fish schools exhibit a collective behavior and self-organization that is mediated by hydrodynamic interactions between individual fish. However, the long-time evolution of hydrodynamically interacting collectives is challenging to investigate due to the persistent influence of long-lived vortical structures, and the high-resolution requirements of direct numerical simulation at large Reynolds numbers. Reduced-order models have therefore played an important role in theoretical investigations of collectives of swimming bodies. The main results detailed herein are several new reduced-order models of swimmers that self-propel by flapping, i.e., by executing a prescribed periodic rigid body motion. The models are extensions of a discrete-time dynamical system developed previously in the literature in which flapping swimmers interact through periodically shed vortices. The extensions include allowing a variable separation distance between swimmers, and more faithful treatment of wing-wing interactions by modeling neighboring wings as point vortices. The models are used to investigate conditions under which hydrodynamic interactions lead to stable swimming configurations. Analytical results include closed form expressions for the velocity potential and total forces on the wings for an arbitrary number of swimmers. Numerical results illustrate steady state solution branches and their stability. Taken together, the analytical and numerical results exhibit favorable agreement with laboratory experiments on flapping wings in a water tank. and elucidate how hydrodynamic interactions can influence the structure and stability of collectives of flapping bodies.

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Liuming (Jack) Wang

Bootstrap Nonparametric Inference Under Data Integration

Integrating data from multiple sources under distribution shifts challenges nonparametric inference, especially with differing marginal distributions but shared conditional structures. This paper presents a framework for estimating and quantifying uncertainty in the target mean function without requiring dataset similarity. Using Reproducing Kernel Hilbert Spaces (RKHS) and kernel ridge regression, we estimate source deviations and integrate them via weighted estimation.

A multiplier bootstrap method constructs confidence intervals, with proofs of local and global consistency under mild conditions on sample sizes and regularization. Unlike transfer or distributed learning, it handles arbitrary shifts, achieving asymptotic normality. The simulations demonstrate robustness to source-target differences, and application to Alzheimer's data (hippocampal volume vs. cognitive scores in age groups) shows tighter intervals than target-only analysis. Future work extends to survival analysis in cardiovascular datasets.

July 28

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Joseph Canavatchel

Well Posedness of 3D Vortex Sheets with Surface Tension and Density Jumps

We start by dividing $\mathbb{R}^3$ into two regions $\Omega_+$ and $\Omega_-$ and are separated by a surface S. The regions each represent an irrotational and inviscid fluid with velocity $u_i$ and density $\rho_i$, and the surface will evolve in time as PDE of the velocity and density. This will be called the two fluid interface problem, and the goal for this paper is to establish that the two fluid problem with two different densities is locally well posed in $\mathbb{R}^3$. To do this we will use the method of energy estimates, by defining an energy and showing it is bounded for some time.

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Ellison O'Grady

Spirometry in the Closed-loop Neural Model of Breathing Control

Spirometry is a common pulmonary function test often used as a diagnostic tool in identifying breathing problems. Broadly, the major pathologies we are considering can be either restrictive or obstructive, representing different mechanical problems in the lung. We examine the current closed-loop lung model and spirograms produced from publicly available datasets in order to identify key elements of the flow-volume curve and determine how to bridge the gap between the model and the clinical data, namely via the implementation of bang-bang control and the inclusion of novel anatomical elements to the model. 

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Tamanna

Models of Bursting Pacemaker Neurons in the Pre-Bötzinger Complex

In this work, we reproduce results from the conductance-based bursting pacemaker neuron models of Butera, Rinzel, and Smith (1999).

The pre-Bötzinger complex generates the rhythmic neural activity in mammalian respiration. We study two models: the first model generates bursting via fast activation and slow inactivation of a persistent sodium current, while in the second model bursting arises via a fast-activating persistent sodium current and slow activation of a potassium current. We compare membrane potential dynamics and burst characteristics across the two models.

July 31

R

Xiaotian Mu

From Bulk to Single-Cell: Functional Deconvolution for Precise Cell Subpopulation Identification

We present a functional deconvolution approach to integrate bulk RNA-seq data with single-cell RNA-seq data for identifying cell subpopulations related to patient outcomes. Bulk RNA-seq has large sample sizes and strong statistical power, but its signals are mixed across different cell types. In contrast, single-cell RNA-seq provides cell-level resolution but usually includes only a small number of patients and suffers from sparsity. To combine the strengths of both data types, we extend existing cell-selection methods by modeling outcome-associated effects as a continuous surface over the low-dimensional structure of single-cell data, rather than selecting cells in a binary way. Using downsampling experiments, we show that the identified outcome-related cell signals are stable when single-cell data are reduced, but become weaker when bulk sample size is small. Simulation results further demonstrate that the proposed method can recover both sharp and smooth changes of effects across cell states. This framework provides a more flexible way to link bulk outcomes with single-cell populations.

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Hong Xiao

Analyzing point patterns on networks

Traditional spatial analysis deals with open spaces, without a network. However, real life problems, like traffic accidents, can only occur on roads. So a correct spatial analysis of traffic accident locations must take account of the layout of the road network. This motivates us to do spatial statistics on a linear network, where we analyze spatial patterns and processes that occur on linear structures, like rivers, roads, pipelines, or nerve fibres. The analysis of linear networks could be very complicated due to the geometrical complexities of the network. Here we propose three projects to deal with linear network problems.

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Zhiwen (Esther) Wang

Bandwidth Selection of Density Estimators over Treespaces

A kernel density estimator (KDE) is one of the most popular nonparametric density estimators. In this paper we focus on a best bandwidth selection method for use in an analogue of a classical KDE using the tropical symmetric distance, known as a tropical KDE, for use over the space of phylogenetic trees. We propose the likelihood cross validation (LCV) for selecting the bandwidth parameter for the KDE over the space of phylogenetic trees. In this paper, first, we show the explicit optimal solution of the best-fit bandwidth parameter via the LCV for tropical KDE over the space of phylogenetic trees.  

Then, computational experiments with simulated datasets generated under the multispecies coalescent (MSC) model show that a tropical KDE with the best-fit bandwidth parameter via the LCV perform better than a tropical KDE with an estimated best-fit bandwidth parameter via nearest neighbors in terms of accuracy and computational time.   Lastly, we apply our method to empirical data from the Apicomplexa genome.