Taylor dispersion, first described by G.I. Taylor in 1953, occurs when the diffusion of a solute in a pipe or channel is enhanced by a background flow, in that the solute asymptotically approaches a form that solves a different diffusion equation (with larger diffusion coefficient, in an appropriate moving frame). By introducing scaling variables, I'll show how one can obtain this asymptotic form in a very natural way using a center manifold reduction. This is joint work with Margaret Beck and Gene Wayne.

Abstract: We present recent results of a numerical bifurcation study of multidimensional
planar viscous shock waves in Magnetohydrodynamics using the Evans function.
We describe the numerical methods used and the bifurcation properties observed.

In this talk we introduce the Maslov index as a tool to analyze stability of steady state solutions to a reaction-diffusion equation in one spatial dimension. We show that the path of unstable subspaces associated to this equation is governed by a matrix Riccati equation whose solution $S$ develops singularities when changes in the Maslov index occur. We then prove that at these singularities the change in Maslov index equals the number of eigenvalues of $S$ that increase to $+\infty$ minus the number of eigenvalues that decrease to $-\infty$.

Zebrafish (Danio rerio) is a small fish with distinctive black and yellow stripes that form due to the interaction of different pigment cells. Working closely with the biological data, we present an agent-based model for these stripes that accounts for the migration, differentiation, and death of three types of pigment cells. The development of both wild-type and mutated patterns will be discussed, as well as the non-local continuum limit associated with the model.
(work is with Bjorn Sandstede)

A problem of great interest regarding conservation laws is the propagation of certain special initial data, and in particular, the behavior and evolution of the solution. For example, if we start with initial data in the form of a Markov process (in $x$), will this property continue to be preserved in time? What about other more general classes of initial data? I aim to address this and other questions, as well as provide some insight into the so-called $n$-point equations, the relationships between positions and velocities at certain points.

Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and algorithms to solve and analyze problems that involve fluid flows governed by classical Navier-Stokes (NS) equations.

I will outline the proof that steady, incompressible Navier-Stokes flows posed over the moving boundary, $y = 0$, can be decomposed into Euler and Prandtl flows globally in the tangential variable, assuming a sufficiently small velocity mismatch. The main obstacles in the analysis center around obtaining sharp decay rates for the linearized profiles and the remainders. The remainders are controlled via a high-order energy method, supplemented with appropriate embedding theorems, which I will present.

Recent numerical studies have revealed certain families of functions play a crucial role in the long time behaviour of the 2D Navier-Stokes equation with periodic boundary conditions. These functions, called bar and dipole states, exist as quasi-stationary solutions and attract trajectories of all other initial conditions exponentially fast. If the domain is the symmetric torus, then the dipole states dominate, while on the asymmetric torus the bar states dominate.

An agent based model has been developed by B. Sandstede and A. Volkening to better understand the pattern formation on zebrafish in early development. This model tracks the migration, birth, death, and form changes of 5 different pigment cells. These interactions are dependent on short- and long-range length scales and several other parameters, where different parameter regimes correspond to different types of cell mutations. While this model has provided new insight on the development of zebrafish stripes, the robustness of the model has yet to be explored.

In this talk I first recall some mathematical results in the study of classical Prandtl boundary layer theory. And then, I will focus on the related well-posedness and convergence theories for the characteristic MHD boundary layer. Under the assumption that the initial tangential magnetic field is not zero, we establish the local-in-time existence and uniqueness of solutions to the nonlinear MHD boundary layer equations.

In this talk, we will discuss the dynamics of rogue waves in nonlinear Schrodinger (NLS) equations and discrete variants thereof. Initially, we will consider NLS equations with variable coefficients which can be converted into their integrable siblings by utilizing suitable transformations. Then, the Peregrine soliton will be fed to the transformation employed.

Planar spiral waves have been observed in many natural systems and also as solutions of reaction-diffusion equation.

Recent advances in X-ray tomography and high performance computing enable modeling of the spatio-temporal dynamics of signaling pathways within a realistic 3D mammalian cell. We study the process of a chemical signal that is activated at random locations on cell membrane diffuses across cytosol and into nucleus.

Obtaining predictive dynamical equations from data lies at the heart of
science and engineering modeling, and is the linchpin of our technology.
In mathematical modeling one typically progresses from observations of
the world (and some serious thinking!) first to equations for a model,
and then to the analysis of the model to make predictions. Good mathematical
models give good predictions (and inaccurate ones do not) - but the computational
tools for analyzing them are the same: algorithms that are typically based
on closed form equations.

We construct stationary soliton states in a one-component, two dimensional nonlinear Schroedinger equation with a ring-shaped trap and repulsive interatomic interactions.

We consider the plasma confined in a general axisymmetric spatial domain with perfect conducting boundary which reflects particles specularly, and look at a certain class of equilibria, assuming axisymmetry in the problem. We prove a sharp criterion of spectral stability under these settings. Moreover, we provide several explicit families of stable/unstable equilibria using this criterion.

Spiral wave patterns observed in models of cardiac arrhythmias and chemical oscillations develop alternans and stationary line defects, both of which can be thought of as period-doubling instabilities. These instabilities are observed on bounded domains, and may be influenced by the spiral core, far-field asymptotics, and boundary conditions. In this talk, we introduce a methodology to disentangle the impacts of each region on the instabilities by analyzing spectral properties of spiral waves and boundary sinks on bounded domains with appropriate boundary conditions.

Neural oscillations, including rhythms in the beta1 band (12-20 Hz), are important in various cognitive functions. Often neural networks receive rhythmic input at frequencies different than their natural frequency, but very little is known about how such input affects the network's behavior. We use a simplified, yet biophysical, model of a beta1 rhythm that occurs in the parietal cortex, in order to study its response to oscillatory inputs.

Molecules of a nematic liquid crystal tend to align along preferred directions, producing partially-ordered material. This results in peculiar physical properties that render them useful in a range of applications, including their widespread use in optical displays. When confined to thin planar cells or channels, the fluid flow extends the applications of nematics further to optofluidic devices and guided micro-cargo transport. Our goal is to examine the intrinsic coupling between the fluid motion and the orientation of nematic molecules within a reduced Beris-Edwards framework.

In this talk, we present our recent results on the density fluctuation field for the stirring process with births and deaths at two sites at the boundary which tend to impose a current into the system. Assuming correlation estimates, we derive an equation for the limiting kernel of the variance of the fluctuation field.