Dr. Philip J. Holmes
Department of Mechanical and Aerospace Engineering
and Program in Applied and Computational Mathematics
Princeton, NJ 08544-5263
Philip Holmes was born in North Lincolnshire, UK, in 1945 and studied engineering at the Universities of Oxford and Southampton, where he fell among mathematicians and began working on nonlinear dynamics and chaos. In 1977 he emigrated to the US. He taught at Cornell University until 1994, when he moved to Princeton, where he is now Professor of Mechanics and Applied Mathematics. Most of his current work is in biology and neuroscience, addressing the question "How are neural spikes turned into behavior?" Collaborating with biologists, neuroscientists and psychologists, he approaches this through studies of running cockroaches, swimming lampreys, and thinking humans. He has co-authored over 200 papers and four books, including "Celestial Encounters" with Florin Diacu: an historical account of the origins of chaos theory. He is a Fellow of the American Academy of Arts and Sciences and a Foreign Member of the Hungarian Academy of Sciences. He has also published four collections of poems; the most recent, "Lighting the Steps," appeared from Anvil Press (London, UK) in 2002.
33 Years of Nonlinear Dynamics: Less is More and More is Different
In the early 1970's dynamical systems theory was just reapproaching earth after a 70-year sojourn in the stratosphere of pure mathematics. Catastrophe theory was hot (if controversial), complexity was yet to come, and some prominent engineers and applied mathematicians told me that chaos didn't exist, or would be irrelevant if it did. I will review some of the successes and failures of nonlinear dynamics since that time, traveling back to its origins in the work of Poincare, and returning to current frontiers in infinite-dimensional evolution equations, hybrid and piecewise-smooth systems, and stochastic models.
A non-technical, historical survey, but it helps if you like differential equations, even from a distance. Needs laptop projector.
An earlier version of this talk was given as the opening plenary lecture at the 5th EUROMECH Nonlinear Dynamics Conference in August 2005.
Does Math Matter to Gray Matter? Optimal Decisions from Stochastic Dynamics.
The sequential probability ratio test (SPRT) is optimal in that it allows one to accept or reject hypotheses, based on noisy incoming evidence, with the minimum number of observations for a given level of accuracy. There is increasing neural and behavioral evidence that primate and human brains employ a continuum analogue of SPRT: the drift-diffusion (DD) process. I will review this and also describe how a biophysical model of a pool of spiking neurons can be simplified to a phase oscillator and analysed to yield spike rates in response to stimuli. These spike rates tune DD parameters via neurotransmitter release. This study is a small step toward the construction of a series of models, at different time and space scales, that link neural spikes to human decisions. And yes, of course math matters*.
A little more technical, uses calculus and elementary differential equations, introduces necessary background from cognitive psychology and neuroscience. Needs laptop projector.
*The Institute of Mathematics and Its Applications (Minnesota) sponsors a public lecture series called "Math Matters," in which an earlier version of this talk was given. I am indebted to IMA for this happy phrase.
How Do Cockroaches Run So Fast Without Thinking About It?
I will discuss joint work with John Schmitt, Raffaele Ghigliazza, Justin Seipel, Raghavendra Kukillaya, Bob Full and Dan Koditschek, in which nonlinear mechanics and hybrid dynamical systems meet biology. Motivated by Full's experimental studies of running insects at UC Berkeley, we propose a hierarchy of models for the dynamics of legged locomotion. We start with energetically conservative bipedal models (each leg corresponding to the front/rear/opposite-middle stance tripod used by many insect species), move on to activated hexapedal models, and end by describing a central pattern generator of bursting neurons linked via simplified muscles to more realistic leg geometries. I believe that massive, detail-packed simulations do not necessarily confer understanding, and in reviewing our work, I shall stress the relevance of simple models.
More technical, uses linear algebra and differential equations, introduces necessary background from biomechanics and neuroscience. Needs laptop projector.
What Do Poems and Differential Equations Share? Some Thoughts on Metaphors and Models.
I will describe some similarities and differences between the worlds of applied mathematics and poetry. Scientific examples are drawn from celestial mechanics and dynamical systems theory, and literary examples from the verse of Gerard Manley Hopkins and others.
"Without poetry our science will appear incomplete ... "
Matthew Arnold, `The Study of Poetry'
Completely non-technical, but it helps if you like differential equations and poetry. If there's a sympathetic Creative Writing Program or Department of English, it might be fun to combine this with a short reading of my own poetry. Currently needs only overhead projector and eager ears.
See article, "Our Models are our Metaphors," SIAM News 35 (5), June 2002, for an interview with Dr. Holmes.