Professionally, I'm a faculty member at the University of Arizona. I joined the Arizona faculty in 1988, and I've put down roots in the years since. My research extends over applied mathematics, physics, and computer science. In my spare time, I'm a software engineer: I write and distribute free software.
I'm a broad-spectrum computational and applied mathematician, probably the broadest on the Arizona faculty. My central research interest is asymptotic methods, in many contexts. That includes random processes and their associated `rare events', random matrix theory, and singular perturbation theory applied to the partial differential equations of theoretical physics. I'm also interested in areas of applied probability not related to asymptotic methods (e.g., the algebraic theory of Markov chains), in statistics, and in theoretical computer science. I'm now becoming interested in bioinformatics: discrete mathematics and probability applied to computational biology.
Papers on Special FunctionsAs an offshoot of the probabilistic work described below, I've recently produced two preprints on classical special function theory, i.e., the theory of ordinary differential equations in the complex domain. "Transforming the Heun equation to the hypergeometric equation, Part I" works out transformations analogous to the classical hypergeometric transformations of Kummer and Goursat, which apply to solutions of the Heun equation rather than to hypergeometric functions. (PDF, PS; also available as arXiv: math.CA/0203264.) "Algebraic solutions of the Lamé equation, revisited" studies the special case when the (algebraic-form) Lamé equation, which is a second-order differential equation on an elliptic curve, has only algebraic solutions. (PDF, PS; also available as arXiv: math.CA/0206285.)
A recent paper of mine that's halfway between applied mathematics and theoretical physics is "How an anomalous cusp bifurcates in a weak-noise system" [Physical Review Letters 85 (2000), 1358]. (PDF, PS; also available as arXiv: cond-mat/0006119.) It extends classical catastrophe theory by studying the bifurcation of a `nongeneric cusp', which has a nonpolynomial normal form. A few more details are in a Stochaos '99 proceedings article. (PDF, PS.) A paper that's closer to theoretical physics is "Noise-activated escape from a sloshing potential well" [Physical Review Letters, 86 (2001), 3942]. (PDF, PS; also available as arXiv: cond-mat/0006120.) It extends the Kramers theory of noise-induced escape to the case when the confined particle is periodically driven. This work has applications in nonequilibrium physics and chemistry, and elsewhere in stochastic modeling. My collaborator in much of it has been Dan Stein of the University of Arizona Physics Department.
A much longer paper, which appeared in the applied mathematics literature, is "Limiting exit location distributions in the stochastic exit problem" [SIAM Journal of Applied Mathematics 57 (1997) 752]. (PDF, PS.) It's a starting point for learning about my approach to the stochastic exit problem, which is the problem of determining how a drift-diffusion process exits from a specified domain. The key concept is the concept of an `optimal trajectory': the most probable sample path extending from an attractor to a specified endpoint, in the limit of weak volatility. My approach goes beyond the `large deviation principles' that others have used. A recent preprint, "Droplet nucleation and domain wall motion in a bounded interval" [Physical Review Letters 87 (2001), 270601] analyses the phenomenon of stochastic exit in a spatially extended one-dimensional system: a micromagnetic model of noise-induced magnetization reversal. (PDF, PS; also available as arXiv: cond-mat/0108217.)
In Fall 2002, I'm teaching my new graduate courses Math/CSc 535 ("The Mathematics of Computer Graphics") and Math 577-1 ("Software Tools for Biotechnology") In Spring 2001, I last taught my basic graduate course Math/CSc 589 ("Software Tools for Computational Science and Engineering").
I frequently teach courses in statistics, such as Math 263 ("Introduction to Statistics and Biostatistics"), and discrete mathematics, such as Math/CSc 243 ("Discrete Mathematics in Computer Science"), and Math/CSc 543 ("The Theory of Graphs and Networks"). Also, I've several times taught Ph.D.-level complex analysis, real analysis, and probability.
Mike Peralta (Arizona Ph.D., Physics, 1999), whose dissertation was "Statistical Simulation of Complex Correlated Semiconductor Devices". Mike works at Texas Instruments' Burr-Brown division. Graduate students interested in working in any of the areas mentioned above, or on mathematical problems relevant to technology, should contact me: I have funding available.
In my spare time, I write mathematically-oriented free software, or open source software as it's called
these days. I'm the primary author of the GNU Plotting Utilities
package, which is being distributed by the Free Software Foundation. It includes a
powerful C/C++ library, GNU libplot, for exporting 2-D vector
graphics in many file formats. A manual is available on-line. (PDF, PS.)
I've also integrated and enhanced GNU libxmi,
which is a scan-conversion library based on X11 code. A manual for
libxmi is available on-line. (PDF, PS.)
I have an extensive background in Unix system administration and network administration. It's been many years since I wrote large amounts of network code, but I used to amuse myself by adding support for non-TCP/IP protocols to Unix workstations, via kernel hacking. Oldtimers may remember the MIT Chaosnet protocol; that was one of them.
I maintain a local software page showing what software packages have been developed in the Arizona Mathematics Department, besides my own.
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