# Applied Math Colloquium - Fall 2019

Colloquia are held on Fridays at 11:30 a.m. in Cullimore Lecture Hall II, unless noted otherwise. Refreshments are served at 11:30 a.m.

Colloquia are held on Fridays at 11:30 a.m. in Cullimore Lecture Hall II, unless noted otherwise. Refreshments are served at 11:30 a.m.

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.

**Yuan-nan Young**, New Jersey Institute of Technology

Hosted by: David Shirokoff

**From Self-Assembly of Nanoscale Lipid Molecules to Micronscale Vesicle Hydrodynamics Under Adhesion**

In this talk I will present some of the work completed during my sabbatical. The first subject is the self-assembly of lipid molecules in a solvent. The second subject is on the effects of adhesion on vesicle hydrodynamics in various flows. In the first half I will demonstrate how interactions between amphiphilic particles (such as lipid macromolecules) in a solvent can lead to formation of structures such as micelles and bilayer membranes. Using a hybrid description for coarse-grained lipid molecules that interact via a continuum action field we convert the problem to a boundary integral equation with moving boundaries. From the simulations we are able to demonstrate that such formulation is adequate to capture the Helfrich free energy for an elastic membrane. In the second half I will present how adhesion between two vesicles (enclosed lipid bilayer membranes) affect their hydrodynamics under various flowing conditions. These works are summarized in manuscripts submitted to SIAM Journal of Multiscale modeling and Physical Review Fluids, respectively.

**Marsha J. Berger**, Courant Institute, NYU

Hosted by: Christina Frederick

**Progress in Modeling of Asteroid-Generated Tsunamis**

Could an asteroid that explodes over the ocean generate a tsunami threatening coastal populations far away? We show simulations of tsunami propagation from asteroid-generated airbursts. We then present a 1D model with an explicit solution to understand the unexpected results. The model is then extended to explore the effects of dispersion and compressibility. We end with a discussion of appropriate tools to study the more serious case of an asteroid that impacts the water.

**Sean Lawley**, University of Utah

Hosted by: James MacLaurin

**New Questions in Diffusion Theory Prompted by Biology**

Modern biology is posing new questions in the mathematical theory of diffusion. In the first part of this talk, I will outline several problems regarding diffusion in a stochastic environment. These problems stem from diverse biological applications, including chemoreception, neural communication, and insect respiration. Mathematically, the models take the form of partial differential equations with stochastic boundary conditions. In the second part of the talk, I will focus on extreme first passage times (FPTs) of diffusion. While FPTs are often used to estimate timescales in biology, the overwhelming majority of studies focus on the time it takes a single diffusing searcher to reach a target. However, in many scenarios the more relevant timescale is the FPT of the first searcher to reach a target from a large group of searchers. This fastest FPT depends on extremely rare events and its mean can be orders of magnitude faster than the mean FPT of a given single searcher. I will (a) show how rigorous probabilistic methods can determine all the moments of this fastest FPT in a general setting and (b) discuss the implications of this theory.

**Andrej Košmrlj**, Princeton University

Hosted by: Enkeleida Lushi

**Phase Separation in Multicomponent Liquid Mixtures**

Multicomponent systems are ubiquitous in nature and industry. While the physics of binary and ternary liquid mixtures is well-understood, the thermodynamic and kinetic properties of N-component mixtures with N>3 have remained relatively unexplored. Inspired by recent examples of intracellular phase separation, which have important cellular functions, we investigate equilibrium phase behavior and morphology of N-component liquid mixtures within the Flory-Huggins theory of regular solutions. In order to determine the number of coexisting phases and their compositions, we developed a new algorithm for constructing complete phase diagrams, based on numerical convexification of the discretized free energy landscape. Together with a Cahn-Hilliard approach for kinetics, we employ this method to study mixtures with N=4 and 5 components. In this talk I will discuss both the coarsening behavior of such systems, as well as the resulting morphologies in 3D. The topology of phase separated mixtures can be described in terms of graphs, which enabled us to reverse engineer the interaction parameters of components in order to obtain all topologically distinct morphologies and to achieve a range of desired packing structures, such as nested "Russian dolls" and encapsulated emulsions droplets.

**Tamer Zaki**, Johns Hopkins University

Hosted by: Yuan-Nan Young

**Vorticity Dynamics in Viscoelastic Turbulence**

The response of Couette flow to streamline curvature is a canonical problem that models the flow dynamics in curved shear layers when there is a large separation of scales, near small-scale roughness, and near two-fluid interfaces. In Newtonian flows, a vorticity perturbation is injected at the wavy surface, and diffuses into the bulk of the fluid. Two dimensionality precludes vorticity tilting and stretching, and the maximum enstrophy is at the wall. This understanding of the behavior of vorticity is challenged when the fluid has even small levels of elasticity. A vorticity maximum can emerge in the bulk of the fluid, localized at a wall-normal distance above the surface undulation. The vorticity equation alone is not sufficient to understand this effect, and additional evolution equations for the polymer stress are required to fully describe the dynamics. In the case of an Oldroyd-B fluid, the influence of the polymer stress can be expressed in terms of an evolution equation for the spanwise polymer torque, which is the curl of the polymer force. Together with the spanwise vorticity equation, they provide a complete description of the problem. Asymptotic analyses demonstrate that the flow response depends on two lengthscales, normalized by the wavelength of the surface undulation: (i) the depth of the channel and (ii) the depth of the critical layer of a shear-wave. The latter emerges because vorticity can travel as a wave along the tensioned streamlines in viscoelastic fluids, and is central to the amplification of vorticity in the bulk of the fluid. A phase diagram is constructed based on the two parameters, and each regime is explained in terms of the dynamics of vorticity and polymer torque. Finally, experimental measurements that confirm the theoretical predictions will be presented.

**Nathan Glatt-Holtz**, Tulane University

Hosted by: David Shirokoff

**A Bayesian Approach to Quantifying Uncertainty in Divergence Free Flows**

In this talk, we treat the inverse problem of estimating a divergence free flow field u from the partial and noisy observation of a passive scalar θ. Such a problem arises in a range of engineering and geological fields involving fluid flows. The problem is in general ill-posed, however becomes well-posed with a statistical regularization. Our solution is a Bayesian posterior distribution, that is a probability measure µ which precisely quantifies uncertainties in u once one specifies models for measurement error and a prior knowledge for u. We present some of our recent work which analyzes µ both analytically and numerically. In particular we discuss a posterior contraction (consistency) result as well as some Markov Chain Monte Carlo (MCMC) algorithms which we have developed and refined and rigorously analyzed to effectively sample from µ.

*This is joint work with Jeff Borggaard, Justin Krometis and Cecilia Mondaini.*

**Ian Tice**, Carnegie Mellon University

Hosted by: Michael Siegel

**Analysis of Free Boundary Problems in Fluid Mechanics**

A free boundary problem in fluid mechanics is one in which the fluid domain is not specified a priori and evolves in time with the fluid. Such problems are ubiquitous in nature and occur at a huge range of scales, from dew drops, to waves on the ocean, to the surface of a star. In this talk I will review the basic features of free boundary problems and how we incorporate interesting interfacial physics effects. I will also survey recent work on various viscous free boundary problems

**Pak-Wing Fok**, University of Delaware

Hosted by: Anand Oza

**Simulating the Vulnerable Atherosclerotic Plaque through Morphoelasticity**

Atherosclerosis is a manifestation of cardiovascular disease consisting of the buildup of inflamed arterial plaques. Because most heart attacks are caused by the rupture of unstable "vulnerable" plaque, the characterization of plaques and their vulnerability remains an outstanding problem in medicine.

Morphoelasticity is a mathematical framework commonly employed to describe tissue growth. Its central premise is the decomposition of the deformation gradient into the product of an elastic tensor and a growth tensor. In this talk, I will present some recent efforts to simulate atherosclerosis using morphoelasticity theory. I will discuss idealized axisymmetric arteries as well as more general geometries that are solved using the finite element method. Results are discussed in the context of arterial remodeling and enhanced ultrasound images of advanced plaques.

**Darlayne Addabbo**, University of Notre Dame

Hosted by: Denis Blackmore

**Discrete and Ultra-discrete Integrable Systems and Representation Theory**

The KdV equation is an integrable equation which models shallow water waves. It appears in both pure and applied mathematics and is part of an infinite hierarchy of differential equations that can be solved exactly. The solutions, called tau-functions, to this infinite hierarchy of equations correspond to matrix elements for the action of a group. In this talk, we will discuss hierarchies of difference equations rather than differential equations, but will emphasize the analogy with the KP and KdV hierarchies.

Ultra-discretization is a limiting procedure for obtaining integrable cellular automata from discrete and continuous integrable systems. A famous example of this is the Box-Ball System which was first introduced by Takahashi and Satsuma in 1990 and was shown to be obtained from the KdV equation via ultra-discretization by Matsukidaira, Satsuma, Takahashi, and Tohikiro in 1996. We will discuss this Box-Ball System and some of its interesting properties as well as current and future research in this direction. No prior knowledge of these areas will be assumed.

Some of this work is joint with Maarten Bergvelt.

**Rachel Ward**, ICES, University of Texas (Austin)

Hosted by: Christina Frederick

**Stochastic Gradient Descent, from Linear Systems to Neural Networks**

Stochastic Gradient Descent the key algorithm behind powerful neural network optimization algorithms today. There is a wide gap between the setting where the behavior of SGD is understood theoretically (linear systems), and the setting where it is most useful in practice (nonlinear neural network functions). We discuss recent theoretical advances in closing this gap, as well as remaining mysteries and questions.

**Antoine Cerfon**, Courant Institute, NYU

Hosted by: Travis Askham

**Efficient Integral Equation Based Solvers for Optimized Magnetic Fusion Devices **

The stellarator is an unusual magnetic fusion concept, with no obvious symmetries in the magnetic field used to confine the hot fuel in the plasma state. This peculiarity translates into additional degrees of freedom for appropriately tailoring the confining magnetic field, which make the stellarator uniquely well suited for numerical design and optimization, and may open a cost-effective path to commercial fusion energy. In this talk, I will introduce the stellarator concept, and describe the mathematical and computational challenges which arise in the current framework for reactor optimization. I will then present new approaches based on integral equation formulations we developed to address some of these challenges, and discuss key open problems which must be tackled if one desires to provide fast and robust design optimization codes to physicists and engineers.

**Catalin Turc**, New Jersey Institute of Technology

Hosted by: David Shirokoff

**Planewave Density Interpolation Methods for 3D Helmholtz and Maxwell Boundary Integral Equations**

I will present my recent work completed during my Sabbatical at PUC Chile on density interpolation methods leading to universal quadratures for fast, high-order evaluations of 3D potentials and corresponding Boundary Integral Operators for Helmholtz and Maxwell equations. The density interpolation method regularizes the singularities of the layer potentials Green functions associated with a linear constant coefficient PDE via certain surface interpolation problems whose bases functions consist of families of smooth solutions of that PDE. The ensuing quadratures are universal in the sense that they can handle at once singular and nearly-singular layer potentials, as well as all types of singularities (e.g. integrable, Cauchy principal value, and Hadamard finite parts). The density interpolation method is applicable to both collocation as well as Galerkin discretizations of 3D BIE. We showed that this method is capable of producing high-order solutions of 3D BIE in complex multi-scale geometries associated with technological applications of interest. This is joint work with Carlos Perez-Arancibia (PUC Chile), Luiz Faria (INRIA Paris), and Constantine Sideris (EE USC).

**No Colloquium - Thanksgiving Break**

**David Bader**, New Jersey Institute of Technology

Hosted by: Michael Siegel*On Twitter*

**(Title and Abstract TBA)**

*Updated: November 11, 2019*