Welcome to my personal portal on the Information Superhighway! Thanks for stopping by.
My research interests are infinite dimensional dynamical systems and partial differential equations.
More recently, I've been focusing on generating informative data visualizations of other problems which have spatial or high-dimensional aspects. One example is available if you click on this map:
It's an interactive map of NY State Senate District 18, colored in a way useful to campaign efforts.
Another example is this map of Baltimore, colored in by frequency of 911 calls for assault:
The horizontal and vertical axes represeint longitudinal and latitudinal spatial coordinates. A neighborhood colored in red indicates a relatively high frequency of assault calls, while yellow indicates
a relatively low frequency of assault calls.
A similar visual is the following map of New York City, colored in by frequency of 911 calls for harrassment:
Again, the horizontal and vertical axes represeint longitudinal and latitudinal spatial coordinates. In this map, a neighborhood colored in yellow indicates a relatively high frequency of harrassment calls, while purple indicates
a relatively low frequency of harrassment calls. The brightest yellow neighborhood can be seen in Brooklyn and is Crown Heights. The second-brightest yellow neighborhood is Midtown.
My mathematics research has focused on applying Center Manifold
theory to explain Taylor Dispersion, a phenomenon occuring in pipe flows. Taylor dispersion concerns the long-time behavior of how solutes or other passive scalars diffuse in longitudinally extended fluid domains.
Dye moving through a pipe.
The same dye, in a moving frame.
Here is a list of publications and preprints:
"Rigorous Justification of Taylor Dispersion via center manifolds and hypocoercivity," with
M. Beck and C.E. Wayne. Submitted. [PDF]
"Analysis of enhanced diffusion in Taylor dispersion via a model problem," with M. Beck and C.E. Wayne. In the special volume of the Fields Institute Communications entitled "Hamiltonian PDEs and Applications", Springer-Verlag, New York (2015)
"The Stubborn Roots of Metabolic Cycles," with E. Reznik and A. Watson.
Journal of the Royal Society Interface (2013), volume 10 no. 83. [Interface].
Confused about "point", "continuous", "residual", and "essential" spectrum? Here's a writeup that may help.
A Mathematica notebook containing some demonstrations you may find useful (possibly after some modification) for your teaching;
right-click, shift-click, or two-finger click to save the .nb file.
A numerical simulation of the FPU "alpha" model (quadratic nonlinearity) on a finite lattice (64 interior oscillators) with Dirichlet boundary conditions. "Recurrence" occurs around 65 seconds, near the end of the video. Mathematica notebook available