Applied Math Colloquium
Spring 2018
Spring 2018
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. For questions about the seminar schedule, please contact David Shirokoff.
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Date  Speaker, Affiliation, and Title  Host 

February 2 
Paulo Arratia, University of Pennsylvania

Lou Kondic 
February 9 
Uwe Beuscher, W. L. Gore & Associates

Linda Cummings 
February 16 
Aleksandar Donev, New York University

Lou Kondic 
February 23 
Sue Ann Campbell, University of Waterloo We use mean field analysis to study emergent behaviour in networks of alltoall coupled, pulsecoupled neurons. The individual neurons are represented using a class of twodimensional integrate and fire model. The mean field model is derived using a population density approach, moment closure assumptions and a quasisteady 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. 
Casey Diekman 
March 2 
Gene Wayne, Boston University Twodimensional 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. 
Roy Goodman 
March 9 
Jerome Darbon, Brown University We consider standard finitedimensional 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 HamiltonJacobi 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 HamiltonJacobi equations in high dimension and some applications in optimal control. 
David Shirokoff 
March 23 
Sanjoy Mahajan, Olin College (secondary affiliation: Massachusetts Institute of Technology) 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 streetfighting mathematics and sciencethe art of insight and approximationcan improve our thinking and teaching, the better to handle the complexity of the world. 
Jonathan Luke (cohosted by CSLA, Dept. of Physics, and the NJIT Institute for Teaching Excellence) 
April 6 
Becca Thomases, University of California, Davis Many microorganisms and cells function in complex (nonNewtonian) 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 microscale 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. 
YuanNan Young 
April 13 
Michael Shelley, New York University / Flatiron Institute 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. 
Anand Oza 
April 20 
Maxim Olshanskii, University of Houston 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 timedependent surfaces and fluid systems on manifolds. 
Shahriar Afkhami 
April 27 
Yoichiro Mori, University of Minnesota 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 AllenCahn 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 AllenCahn 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 AllenCahn equation in striking contrast to the classical or anisotropic AllenCahn equations. We identify two types of instabilities, one with respect to longwavelength perturbations, the other with respect to mediumwavelength perturbations. Interestingly, whether the front is stable or unstable under longwavelength 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 intermediatewavelength perturbations does depend on the choice of bistable nonlinearity. Intermediatewavelength 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. 
YuanNan Young 
Updated: April 18, 2018