This summer I've been hanging out in Cambridge Massachusetts, working on quantum gravity and also having some fun. Not so long ago I gave a talk on cellular automata at Boston University, thanks to a kind invitation from Bruce Boghosian, who is using cellular automata to model cool stuff like emulsions:
1) Florian W. J. Weig, Peter V. Coveney, and Bruce M. Boghosian, Lattice- gas simulations of minority-phase domain growth in binary immiscible and ternary amphiphilic fluid, preprint available as cond-mat/9705248.
As you add more and more of an amphiphilic molecule (e.g. soap) to a binary immiscible fluid (e.g. oil and water), the boundary layer likes to grow in area. This is why you wash your hands with soap. There are various phases depending on the concentrations of the three substances - a "spongy" phase, a "droplet phase", and so on - and it is very hard to figure out what is going on quantitatively using analytical methods.
Luckily, one can simulate this stuff using a cellular automaton! Standard numerical methods for solving the Navier-Stokes equation tend to outrun cellular automata when it comes to plain old hydrodynamics, but with these fancy "ternary amphiphilic fluids", cellular automata really seem to be the most practical way to study things - apart from experiments, of course. This is very heartwarming to me, since like many people I've been fond of cellular automata ever after learning of John Conway's game of Life, and I've always hoped they could serve some practical purpose.
I spoke about the thesis of my student James Gilliam and a paper we wrote together:
2) James Gilliam, Lagrangian and Symplectic Techniques in Discrete Mechanics, Ph.D. thesis, Department of Mathematics, University of Riverside, 1996.
John Baez and James Gilliam, An algebraic approach to discrete mechanics, Lett. Math. Phys. 31 (1994), 205-212. Also available in LaTeX form as http://math.ucr.edu./home/baez/ca.tex
Here the idea was to set up as much as possible of the machinery of classical mechanics in a purely discrete context, where time proceeds in integer steps and the space of states is also discrete. The most famous examples of this "discrete mechanics" are cellular automata, which are the discrete analogs of classical field theories, but there are also simpler systems more reminiscent of elementary classical mechanics, like a particle moving on a line - where in this case the "line" is the integers rather than the real numbers. It turns out that with a little skullduggery one can apply the techniques of calculus to some of these situations, and do all sorts of stuff like prove a discrete version of Noether's theorem - the famous theorem which gives conserved quantities from symmetries.
After giving this talk, I visited my friend Robert Kotiuga in the Functorial Electromagnetic Analysis Lab in the Photonics Building at Boston University. "Photonics" is the currently fashionable term for certain aspects of optics, particularly quantum optics. As befits its flashy name, the Photonics Building is brand new and full of gadgets like a device that displays Maxwell's equations in moving lights when you speak the words "Maxwell's equations" into an inconspicuous microphone. (It also knows other tricks.) Robert told me about what he's been doing lately with topology and finite-element methods for solving magnetostatics problems - this blend of higbrow math and practical engineering being the reason for the somewhat tongue-in-cheek name of his office, inscribed soberly on a plaque outside the door.
Like the topologist Raoul Bott, Kotiuga started in electrical engineering at McGill University, and gradually realized how much topology there is lurking in electrical circuit theory and Maxwell's equations. Apparently a paper of his was the first to cite Witten's famous work on Chern-Simons theory - though presumably this is merely a testament to the superiority of engineers over mathematicians and physicists when it comes to rapid publication. In fluid dynamics, the integral of the following quantity
v . curl(v)
(where v is the velocity vector field) is known as the "helicity functional". Kotiuga been studying applications of the same mathematical object in the context of magnetostatics, namely
A . curl(A)
where A is the magnetic vector potential. It shows up in impedance tomography, for example. But in quantum field theory, a generalization of this quantity to other forces is known as the "Chern-Simons functional", and Witten's work on the 3-dimensional field theory having this as its Lagrangian turned out to revolutionize knot theory. Personally, I'm mainly interested in the applications to quantum gravity - see "week56" for a bit about this. Here are some papers Kotiuga has written on the helicity functional, or what we mathematicians would call "U(1) Chern-Simons theory":
3) P. R. Kotiuga, Metric dependent aspects of inverse problems and functionals based helicity, Journal of Applied Physics, 70 (1993), 5437-5439.
Analysis of finite element matrices arising from discretizations of helicity functionals, Journal of Applied Physics, 67 (1990), 5815-5817.
Helicity functionals and metric invariance in three dimensions, IEEE Transactions on Magnetics, MAG-25 (1989), 2813-2815.
Variational principles for three-dimensional magnetostatics based on helicity, Journal of Applied Physics, 63 (1988), 3360-3362.
Later Jon Doyle, a computer scientist at M.I.T. who had been to my talk, invited me to a seminar at M.I.T. where I met Gerald Sussman, who with Jack Wisdom has run the best long-term simulations of the solar system, trying to settle the old question of whether the darn thing is stable! It turns out that the system is afflicted with chaos and can only be predicted with any certainty for about 4 million years... though their simulation went out to 100 million.
Here are some fun facts: 1) They need to take general relativity into account even for the orbit of Jupiter, which precesses about one radian per billion years. 2) They take the asteroid belt into account only as modification of the sun's quadrupole moment (which they also use to model its oblateness). 3) The most worrisome thing about the whole simulation - the most complicated and unpredictable aspect of the whole solar system in terms of its gravitational effects on everything else - is the Earth-Moon system, with its big tidal effects. 4) The sun loses one Earth mass per 100 million years due to radiation, and another quarter Earth mass due to solar wind. 5) The first planet to go is Mercury! In their simulations, it eventually picks up energy through a resonance and drifts away.
For more, try:
4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system, Science, 257, 3 July 1992.
Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion of Pluto is chaotic, Science, 241, 22 July 1988.
James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom, The outer solar system for 200 million years, Astronomical Journal, 92, pp 176-194, July 1986, reprinted in Lecture Notes in Physics #267 -- Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986.
James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay Sussman, A digital orrery, in IEEE Transactions on Computers, C-34, No. 9, pp. 822-831, September 1985, reprinted in Lecture Notes in Physics #267, Springer Verlag, 1986.
Meanwhile, I've also been trying to keep up with recent developments in n-category theory. Some readers of "This Week's Finds" have expressed frustration with how I keep tantalizing all of you with the concept of n-category without ever quite defining it. The reason is that it's a lot of work to write a nice exposition of this concept!
However, I eventually got around to taking a shot at it, so now you can read this:
5) John Baez, Introduction to n-categories, to appear in 7th Conference on Category Theory and Computer Science, eds. E. Moggi and G. Rosolini, Springer Lecture Notes in Computer Science vol. 1290, Springer, Berlin. Preprint available as q-alg/9705009 or at http://math.ucr.edu/home/baez/ncat.ps
There are different definitions of "weak n-category" out there now and it will take a while of sorting through them to show a bunch are equivalent and get the whole machinery running smoothly. In the above paper I mainly talk about the definition that James Dolan and I came up with. Here are some other new papers on this sort of thing... I'll just list them with abstracts.
6) Claudio Hermida, Michael Makkai and John Power, On weak higher dimensional categories, 104 pages, preprint available at http://hypatia.dcs.qmw.ac.uk/authors/M/MakkaiM/papers/multitopicsets/
Inspired by the concept of opetopic set introduced in a recent paper by John C. Baez and James Dolan, we give a modified notion called multitopic set. The name reflects the fact that, whereas the Baez/Dolan concept is based on operads, the one in this paper is based on multicategories. The concept of multicategory used here is a mild generalization of the same-named notion introduced by Joachim Lambek in 1969. Opetopic sets and multitopic sets are both intended as vehicles for concepts of weak higher dimensional category. Baez and Dolan define weak n-categories as (n+1)-dimensional opetopic sets satisfying certain properties. The version intended here, multitopic n-category, is similarly related to multitopic sets. Multitopic n-categories are not described in the present paper; they are to follow in a sequel. The present paper gives complete details of the definitions and basic properties of the concepts involved with multitopic sets. The category of multitopes, analogs of the opetopes of Baez and Dolan, is presented in full, and it is shown that the category of multitopic sets is equivalent to the category of set- valued functors on the category of multitopes.
7) Michael Batanin, Finitary monads on globular sets and notions of computad they generate, available as postscript files at http://www-math.mpce.mq.edu.au/~mbatanin/papers.html
Consider a finitary monad on the category of globular sets. We prove that the category of its algebras is isomorphic to the category of algebras of an appropriate monad on the special category (of computads) constructed from the data of the initial monad. In the case of the free n-category monad this definition coincides with R. Street's definition of n-computad. In the case of a monad generated by a higher operad this allows us to define a pasting operation in a weak n-category. It may be also considered as the first step toward the proof of equivalence of the different definitions of weak n-categories.
7) Carlos Simpson, Limits in n-categories, approximately 90 pages, preprint available as alg-geom/9708010.
We define notions of direct and inverse limits in an n-category. We prove that the (n+1)-category nCAT' of fibrant n-categories admits direct and inverse limits. At the end we speculate (without proofs) on some applications of the notion of limit, including homotopy fiber product and homotopy coproduct for n-categories, the notion of n-stack, representable functors, and finally on a somewhat different note, a notion of relative Malcev completion of the higher homotopy at a representation of the fundamental group.
8) Sjoerd Crans, Generalized centers of braided and sylleptic monoidal 2-categories, Adv. Math. 136 (1998), 183-223.
Recent developments in higher-dimensional algebra due to Kapranov and Voevodsky, Day and Street, and Baez and Neuchl include definitions of braided, sylleptic and symmetric monoidal 2-categories, and a center construction for monoidal 2-categories which gives a braided monoidal 2-category. I give generalized center constructions for braided and sylleptic monoidal 2-categories which give sylleptic and symmetric monoidal 2-categories respectively, and I correct some errors in the original center construction for monoidal 2-categories.
© 1997 John Baez