Applied Math Colloquium

Paulo Arratia, University of Pennsylvania
Life in complex fluids 

Many microorganisms (e.g. bacteria, algae, sperm cells) move in fluids or liquids that contain (bio)-polymers and/or solids. Examples include human cervical mucus, intestinal fluid, wet soil, and tissues. These so-called complex fluids often exhibit non-Newtonian rheological behavior due to the non-trivial interaction between the fluid microstructure and the applied stresses. In this talk, I will show how the presence of particles and polymers in the fluid medium can strongly affect the motility (i.e. swimming) behavior of microorganisms such as the bacterium E. coli. For bacteria moving in particle suspensions of different (particle) sizes, we find a regime in which larger (passive) particles can diffuse faster than smaller particles: the particle long-time effective diffusivity exhibits a peak in particle size, which is a deviation from classical thermal diffusion. A minimal model qualitatively explains the existence of the effective diffusivity peak and its dependence on bacterial concentration. These results have broad implications on characterizing active fluids using concepts drawn from classical thermodynamics. For swimmers (E. coli and C. reinhardtii) moving in polymeric liquids, we find that fluid elasticity can significantly affect the run-and-tumble mechanism characteristic of E. coli, for example, as well as the swimming speed and kinematics of both pushers and pullers. These results demonstrate the intimate link between swimming kinematics and fluid rheology and that one can control the spreading and motility of microorganisms by tuning fluid properties.

Uwe Beuscher, W. L. Gore & Associates
Investigation of the correlation between gas/liquid porometry and particle filtration using simple network models

In order to better understand the performance of microporous membranes for particle filtration, it is important to determine and measure the appropriate membrane structural properties that influence filtration. One such property is the pore size distribution and a common technique for measuring pore sizes or porous structures is gas/liquid capillary porometry. In this method, the pore size is determined by measuring the capillary pressure distribution of the sample, which is correlated to a pore size distribution using a very simple model structure of parallel capillaries. A numerical study using a simple network model for the porous material, however, has shown that the morphological structure strongly influences the porometry results. Model structures with different morphology and pore size information lead to varying porometry results at similar filtration behavior and vice versa. An attempt is made to correlate numerical porometry and filtration performance as is often done in experimental studies. Finally, the limitations of porometry in predicting filtration behavior are illustrated.

Aleksandar Donev, New York University
Large Scale Brownian Dynamics of Confined Suspensions of Rigid Particles

We introduce new numerical methods for simulating the dynamics of passive and active Brownian colloidal suspensions of particles of arbitrary shape sedimented near a bottom wall. The methods also apply for periodic (bulk) suspensions. Our methods scale linearly in the number of particles, and enable previously unprecedented simulations of tens to hundreds of thousands of particles. We demonstrate the accuracy and efficiency of our methods on a suspension of boomerang-shaped colloids. We also model recent experiments on active dynamics of uniform suspensions of spherical microrollers.

Sue Ann Campbell, University of Waterloo
Mean Field Analysis and the Dynamics of Large Networks of Neurons

We use mean field analysis to study emergent behaviour in networks of all-to-all coupled, pulse-coupled neurons. The individual neurons are represented using a class of two-dimensional integrate and fire model. The mean field model is derived using a population density approach, moment closure assumptions and a quasi-steady state approximation. The resulting model is a system of switching ordinary differential equations, which exhibits a variety of smooth and nonsmooth bifurcations. We show that the results of the mean field analysis are a reasonable prediction of the behaviour seen in numerical simulations of large networks and discuss how the presence of parameter heterogeneity and noise affects the results. This is joint work with Wilten Nicola.

Gene Wayne, Boston University
Dynamical Systems and the Two-Dimensional Navier-Stokes Equations

Two-dimensional fluid flows exhibit a variety of coherent structures such as vortices and dipoles which can often serve as organizing centers for the flow. These coherent structures can, in turn, sometimes be associated with the existence of special geometrical structures in the phase space of the equations and in these cases the evolution of the flow can be studied with the aid of dynamical systems theory.

Jerome Darbon, Brown University
On Convex Finite-Dimensional Variational Methods in Imaging Sciences, and Hamilton-Jacobi Equations

We consider standard finite-dimensional variational models used in signal/image processing that consist in minimizing an energy involving a data fidelity term and a regularization term. We propose new remarks from a theoretical perspective which give a precise description on how the solutions of the optimization problem depend on the amount of smoothing effects and the data itself. The dependence of the minimal values of the energy is shown to be ruled by Hamilton-Jacobi equations, while the minimizers u(x,t) for the observed images x and smoothing parameters t are given by u(x,t)=x - grad H( grad E(x,t)) where E(x,t) is the minimal value of the energy and H is a Hamiltonian related to the data fidelity term. Various vanishing smoothing parameter results are derived illustrating the role played by the prior in such limits. Finally, we briefly present an efficient numerical numerical method for solving certain Hamilton-Jacobi equations in high dimension and some applications in optimal control.

Sanjoy Mahajan, Olin College (secondary affiliation: Massachusetts Institute of Technology)
Street-Fighting Mathematics for Better Teaching and Thinking

With traditional science and mathematics teaching, students struggle with fundamental concepts. For example, they cannot reason with graphs and have no feel for physical magnitudes. Their instincts remain Aristotelian: In their gut, they believe that force is proportional to velocity. With such handicaps in intuition and reasoning, students can learn only by rote. I'll describe these difficulties using mathematical and physical examples, and illustrate how street-fighting mathematics and science---the art of insight and approximation---can improve our thinking and teaching, the better to handle the complexity of the world.

Jonathan Luke

(co-hosted by CSLA, Dept. of Physics, and the NJIT Institute for Teaching Excellence)

Becca Thomases, University of California, Davis
Microorganism Locomotion in Viscoelastic Fluids

Many microorganisms and cells function in complex (non-Newtonian) fluids, which are mixtures of different materials and exhibit both viscous and elastic stresses. For example, mammalian sperm swim through cervical mucus on their journey through the female reproductive tract, and they must penetrate the viscoelastic gel outside the ovum to fertilize. In micro-scale swimming the dynamics emerge from the coupled interactions between the complex rheology of the surrounding media and the passive and active body dynamics of the swimmer. We use computational models of swimmers in viscoelastic fluids to investigate and provide mechanistic explanations for emergent swimming behaviors. I will discuss how flexible filaments (such as flagella) can store energy from a viscoelastic fluid to gain stroke boosts from fluid elasticity. I will also describe a 3D simulation of the model organism C. Reinhardtii that we used to separate stroke effects and fluid effects and explore why strokes that are adapted to Newtonian fluid environments might not do well in viscoelastic environments.

Michael Shelley, New York University / Flatiron Institute
Active Mechanics in the Cell

Many fundamental phenomena in eukaryotic cells -- nuclear migration, spindle positioning, chromosome segregation -- involve the interaction of often transitory structures with boundaries and fluids. I will discuss the interaction of theory and simulation with experimental measurements of active processes within the cell. This includes understanding the force transduction mechanisms underlying nuclear migration, spindle positioning and oscillations, as well as how active displacement domains of chromatin might be forming in the interphase nucleus.

Maxim Olshanskii, University of Houston
Finite Element Methods for PDEs Posed on Surfaces

Partial differential equations posed on surfaces arise in mathematical models for many natural phenomena: diffusion along grain boundaries, lipid interactions in biomembranes, pattern formation, and transport of surfactants on multiphase flow interfaces to mention a few. Numerical methods for solving PDEs posed on manifolds recently received considerable attention. In this talk we review recent developments in this field and focus on an Eulerian finite element method for the discretization of elliptic and parabolic partial differential equations on surfaces which may evolve in time. The method uses traces of finite element space functions on a surface to discretize equations posed on the surface. The talk explains the approach, reviews analysis and demonstrates results of numerical experiments. The problems addressed include diffusion along time-dependent surfaces and fluid systems on manifolds.

Yoichiro Mori, University of Minnesota
Stability of Planar Fronts of the Bidomain Equation

The bidomain model is the standard model describing electrical activity of the heart. In this talk, I will discuss the stability of planar front solutions of the bidomain equation with a bistable nonlinearity (the bidomain AllenCahn equation) in two spatial dimensions. In the bidomain Allen-Cahn equation a Fourier multiplier operator whose symbol is a positive homogeneous rational function of degree two (the bidomain operator) takes the place of the Laplacian in the classical Allen-Cahn equation. Stability of the planar front may depend on the direction of propagation given the anisotropic nature of the bidomain operator. Our analysis reveals that planar fronts can be unstable in the bidomain Allen-Cahn equation in striking contrast to the classical or anisotropic Allen-Cahn equations. We identify two types of instabilities, one with respect to long-wavelength perturbations, the other with respect to medium-wavelength perturbations. Interestingly, whether the front is stable or unstable under long-wavelength perturbations does not depend on the bistable nonlinearity and is fully determined by the convexity properties of a suitably defined Frank diagram. On the other hand, stability under intermediate-wavelength perturbations does depend on the choice of bistable nonlinearity. Intermediate-wavelength instabilities can occur even when the Frank diagram is convex, so long as the bidomain operator does not reduce to the Laplacian. We shall also give a remarkable example in which the planar front is unstable in all directions.
This is joint work with Hiroshi Matano of the University of Tokyo and Mitsunori Nara of Iwate University.