SCHOLARLY INTERESTS
I am primarily interested in numerical analysis and applied mathematics. The title of the volume my first published paper appears in says it best: Math Is For Solving Problems (Editors L. Pamela Cook and Victor Roytburd, SIAM, 1996).
My thesis work (under Julian D. Cole and Don W. Schwendeman) was in theoretical aerodynamics/ scientific computing. There I showed that it was possible to compute solutions of the Transonic Small Disturbance equations that in the case of flow around a slender body of revolution without fore-aft symmetry possessed vanishingly small shocks. This was a very tricky calculation because it involves trying to obtain the solution accurately as near as one can get to a rather nasty singularity. In 1998, I was invited to present these results at the 2nd Theoretical Fluid Mechanics meeting at the annual American Institute of Aeronautics and Astronautics conference. Along the way we stumbled upon an interesting way to discretize the radial derivatives in our partial differential equation. I later discovered that Professor Ron Mickens of Clark Atlanta University had been writing about these same kinds of "nonstandard finite difference" schemes for years.
I have always been interested in applying mathematics to unusual phenomena so when I had a sabbatical coming up in Spring 2000 I suggested to my colleague David Edwards that we come up with a new mathematical model for "how movies make money." I call this area "cinematic box-office dynamics" and have written two papers on the topic (Edwards and Buckmire, 2001; Edwards, Buckmire and Ortega-Gingrich, 2013) and conducted a lot of research (which sadly has not resulted in publication, yet).
Most of my mathematical work has been
investigating nonstandard finite difference schemes, particularly Mickens
discretizations of the radial operators in the cylindrical and spherical
Laplacian operators.
A famous old problem, called the Bratu problem, is a nonlinear version of
Helmholtz equation where the right hand side of
Laplacian u = - f is exponential. In cylindrical and planar
coordinates there are known exact solutions to the Bratu problem, making it
ideal for numerical benchmarking of my nonstandard finite difference scheme(s).
More recently I have expanded my work to venture into the area of data science. Specifically, applying machine learning teachniques to produce mathematical models of the commitment decisions of students who are admitted to colleges.
AUTOMATHOGRAPHY
DATA SCIENCE & SOCIAL JUSTICE
DATA SCIENCE / MACHINE LEARNING
NONSTANDARD FINITE DIFFERENCES
MATHEMATICAL MODELLING
CLASSROOM VOTING / PEER INSTRUCTION
DIVERSITY, EQUITY & INCLUSION IN MATHEMATICS
COMPUTATIONAL AERODYNAMICS
GENDER, SEXUALITY and THE LAW
VISUALIZATION OF CONCEPT MAPS
October 2001: I have become interested in doing mathematical education research on how students visualize Concept Maps in Linear Algebra. Here are some of the Concept Maps that I have produced so far.
UNDERREPRESENTATION IN MATHEMATICS
Last modified: January 25, 2023