Archive for the ‘Math’ Category

More on category theory

June 30, 2006

While reading this mildly entertaining discussion (among mathematicians) on the merits and misdemeanours of category theory, i’ve stumbled upon this sci.physics.research thread (also available here) with the subject How does category theory help?. The thread starts with a skeptic tone, as exemplified by this fun quote:

is almost impossible for me to read contemporary mathematicians who, instead of saying Petya washed his hands,” write simply: There is a t1 < 0 such that the image of t1 under the natural mapping t1 -> Petya(t1) belongs to the set of dirty hands, and a t2, t1 < t2 <= 0, such that the image of t2 under the above-mentioned mapping belongs to the complement of the set defined in the preceding sentence.”

but of course there’s many a post defending CT, as well as a lot of (i’m actually tempted to say, every single one you’ll ever need) references to papers, books and everything categorical [1]. You’ll find in there all the usual suspects (Baez, Weiss, …) engaged in a very civilized and informative exchange about virtually every single topic related to CT (including lambda calculus and its application to physics!) and then more, like comments on Arnol’d’s geometrical way of thinking (to name a random issue that caught my eye).

As for me, the jury’s still out, although the more time i spend far from (theoretical) computer science, the less i find CT appealing in physics. On the other hand, when i’m just trying to play some maths, CT is really fun in its own sake and i have the nagging suspicion that i like it because it is sometimes posed as the unified theory of math. Whatever.

[1] If you just want a quick link to references to get started, follow this one to a post by Chris Hillman (author of his own Categorical Primer).


Parenthetical geometry

June 25, 2006

sicmIf i were asked for my preferred field in physics, i’d have a really hard time choosing just one, but it would surely relate in one way or the other to differential geometry. Picking up a programming language, on the other hand, would be far easier, for i feel at home among Scheme’s lots of infuriating, silly parenthesis. Jack Wisdom and Gerry Sussman have managed to bring together the best of both worlds in their book Structure and Interpretation of Classical Mechanics, freely available on-line. The book, aptly dedicated to the Principle of Least Action, is an amazing journey through modern classical mechanics, using two apparently different languages: differential geometry and scheme. There you’ll find all the expected topics: Lagrangian and Hamiltonian formulations, the rigid body, phase space structure, canonical transformations and a very complete treatment of perturbation theory in non-linear systems. What makes this book different (and extremely fun) is what one may call its computational stance: in the authors’ words,

Computational algorithms are used to communicate precisely some of the methods used in the analysis of dynamical phenomena. Expressing the methods of variational mechanics in a computer language forces them to be unambiguous and computationally effective. Computation requires us to be precise about the representation of mechanical and geometric notions as computational objects and permits us to represent explicitly the algorithms for manipulating these objects. Also, once formalized as a procedure, a mathematical idea becomes a tool that can be used directly to compute results.[…]
Our requirement that our mathematical notations be explicit and precise enough that they can be interpreted automatically, as by a computer, is very effective in uncovering puns and flaws in reasoning.

As i mentioned, the computer language chosen is Scheme, for pretty good reasons (besides the obvious one of Sussman being one of the language’s inventors). To begin with, Scheme’s syntax is so simple that one can learn it on the go, although that simplicity does not preclude in any way powerful abstraction means and natural expression of symbolic computations. As a matter of fact, it favours it, as testified by the book’s accompanying library, scmutils. Thanks to it,
expressing mathematical equations in a way understandable by a computer is often a natural exercise. On the other hand, Scheme is an interpreted language, which means that you have at your disposal an interactive environment to play with. The convenience of it for exploratory purposes is hard to overstate.

Wisdom and Sussman have been using SICM for teaching classical mechanics at MIT during several years, and you can find additional material in the course’s website. But the fun does not end there. The 2005 booklet Functional Differential Geometry is an unconventional introduction to differential geometry using SICM’s schemy approach, which is also being followed and extended to Lie Groups by Will Farr, who has started an effort to port (part of) scmutils to PowerPC architectures [1]. Finally, there’s a SICM reading group in sore need of contributions and discussion (hint, hint), but with some potentially useful tips.

Happy hacking!

[1] Those of you who enjoyed my swimming post will probably be interested in this article by Wisdom that i found in Will’s blog,

The cyclist team

May 29, 2006

 Fieldtheory Figs FeynmancovGeorgia Tehch’s Pedrag Cvitanovic’ and friends write physics and maths books under the nome de guerre of The cyclist team. These books are interesting in many ways. First of all, they are comprehensive and of excellent quality, although, fortunately, these are not extremely rare in the field. What is not so common is the dose of humour and engaging wittiness distilled in their pages. And, besides, they’re being written over a span of many years in a totally public manner: you can view and download them in PDF of Postscript and are frequently updated. As they explain in their webbook rationale, they don’t even plan to ever publish them:

The relevant parts of a good text will be printed and perused, no less than a good electronic preprint. A bad text should be junked anyway. If a student in Buenos Aires or Salamanca reads a chapter and is wiser for it, that is all it takes to make us happy. The webbook has done something to further little piece of wisdom that we know and love.

The oldest book is Pedrag’s (Quantum) Field Theory, and its companion (and more modern) lecture notes: Quantum Field Theory, a cyclist tour. In Pedrag’s own words:

Relax by reading Classics Illustrated, diagrammatic, Predragian vision of field theory. The exposition assumes no prior knowledge of anything (other than Taylor expansion of an exponential, taking derivatives, and inate knack for doodling). The techniques covered apply to QFT, Stat Mech and stochastic processes.

As is a norm, the book’s site contains many other bits of additional information, including the delicious fable of Quefithe.

Next comes the Group Theory Book, which, under the subtitle of Birdtracks, Lie’s and Exceptional Groups and spanning almost 300 pages, will tell you all you’ll ever need about Lie Groups and Algebras. This nice PDF presentation makes for a good summary of its contents, or, as Pedrag says, of “most of the Webbook at a cyclist pace, in 50 overheads” (see also here for more short intros). In case you’re wondering, birdtracks are to Lie Groups what Feynman’s diagrams to QFT, and then more. As you can see, Pedrag loves diagrams and pictures, in a way that reminds me of Penrose’s fondness for geometrical descriptions (actually, birdtracks have many a point in common with Penrose’s diagrammatic tensor notation, who even wrote a letter claiming his precedence on it). And, again, don’t miss the book’s site for lots of additional goodies.

Finally, there’s the Chaos Book, probably my favourite. Again, the authors introduce it far better than i would:

Quite a few excellent mathematics monographs on nonlinear dynamics and ergodic theories have been published in last three decades. On the whole, they are unreadable for non-mathematicians, and they give no hint that the theory is applicable to problems of physics, chemistry and other sciences.
By now, there are also many physics textbooks on “chaos”. Most lack depth, and many of them are plain bad, emphasizing pictorial and computer-graphics aspects of dynamics and short changing the student on the theory. That’s a pity, as the subject in its beauty and intellectual depth ranks alongside statistical mechanics and quantum field theory, with which it shares many fundamental techniques. The book represents authors’ attempt to formulate the subject as one of the basic cornerstones of the advanced graduate physics curriculum of future.

The amount of additional information for this book is almost overwhelming, including computer programs, additional exercises (the book itself contains many) and a long list of projects written by students. I won’t try to summarize the wide range of themes covered by the book (here you have the table of contents of its three volumes–classical chaos, quantum chaos and appendices), but a very good way of getting a glimpse of its scope and fun style is reading its Overture (PS). An amazing way to become acquainted with an amazing subject!

The cyclist team

Categorical spacetime

May 17, 2006

John Baez is preparing a colloquium at the Perimeter Institute where the application of Category Theory (CT) to quantum spacetime will be discussed. This page contains links to (a few) introductory and (many) advanced papers on how the ideas of CT (and its generalization to n-categories) can be used to analyze the category of Hilbert spaces.

I’ve developed a lively interest in CT during my computer science wanderings , and even wrote an elementary introduction (geared to programmers) that you may find useful (if you’re new to the field and don’t mind digressions into computer programming).

Categories are deceptively simple: they abstract the notion of sets and their morphisms, or, if you prefer, of objects and their transformations. Thus, a category is defined as a set of objects related by arrows that are composable (in the functional sense). Composition is associative and there’s an identity arrow for each object in the category. As you can easily see, virtually any mathematical (and, by extension, physical) structure can be modeled as a category. For instance, your objects can be groups, with arrows representing group morphisms. Or you could model any discrete physical system’s evolution, by taking as objects its states and as arrows its transitions. With the latter example, we begin to see how CT relates to physics: think of arrows as processes or object transformations. CT somehow captures their essence: one moves from the basic category definition to an exploration of morphisms (called functors) between categories, and from here to constructing categories whose objects are categories, with functors as arrows. Iterate and you’re soon talking about processes consisting of processes consisting of processes consisting of… But of course that’s not all: one also studies morphisms between functors (called natural transformations), and obtains a very precise statement of how any object is equivalent to the set its transformations: objects just disappear on behalf of their relationships! This result, called Yoneda’s Lemma, is beautifully presented in Barry Mazur’s When is a thing equal to some other thing? and further explored in Brown and Porter’s Category Theory: an abstract setting for analogy and comparison (both of them being also an excellent introduction to CT). Although i’m not privy with the applications of CT to LQG and other ‘relationist’ approaches, i think that this blurring of objects in favor of their transformations are at the heart of CT’s appeal to some physicists.

Even discounting its (highly tentative) applications to physics, CT raises deep issues in pure mathematics and philosophy: for instance, this very interesting entry of the Stanford Encyclopedia of Philosophy ends its introduction with these words:

Category theory is both an interesting object of philosophical study, and a potentially powerful formal tool for philosophical investigations of concepts such as space, system, and even truth. It can be applied to the study of logical systems in which case category theory is called “categorical doctrines” at the syntactic, proof-theoretic, and semantic levels. Category theory is an alternative to set theory as a foundation for mathematics. As such, it raises many issues about mathematical ontology and epistemology. Category theory thus affords philosophers and logicians much to use and reflect upon.

All that said, i’m a bit skeptical on the usefulness of CT as a means of synthesizing new laws. For what i’ve seen, it is very powerful for analyzing and discovering properties of already known structures, but i have yet to find a convincing case where it is used to reveal brand new laws. For instance, CT is used in Axiomatic QFT to formalize the theory axioms (see, e.g., Buchholz’s Algebraic quantum field theory: a status report or any other of the review papers on Local Quantum Physics), but it’s not clear to me whether CT is, in this context, more than a convenient mathematical language. Let me, however, rush to say that i’m by no means an expert, and that reading Baez’s writings above may well change my mind!

At any rate, i do like Category Theory, if only for its mathematical beauty, and i think it’s very worthwhile spending a little time learning a bit about it if you are interested in mathematical physics (see here for more of my recommended readings).

Update: Over at A Neighborhood of Infinity, there’s a discussion about “two kinds of maths” (namely, structural and content-providing) which somehow coincides with my half-baked feelings about CT (and, as a bonus, provides a link to yet another proof of Yoneda’s lemma):

Most branches of mathematics have a mixture of the two types of theorem. Typically the structure theorems are used as a tool to discover content. In some sense the content is the end and the structure is the means. But category theory seems different in this regard – it seems to be mainly about structure. Every time I read category theory I see ever more ingenious and abstract tools but I don’t really see what I think of as content. What’s hard in category theory is getting your head around the definitions and often the theorems are trivial consequences of these. For example consider the Yoneda lemma as described here. This is a pretty challenging result for beginning category theorists. And yet these notes say: “once you have thoroughly understood the statement, you should find the proof straightforward”. This exactly fits with what I’m getting at: the theorem is an immediate consequence of the definitions and the proof essentially shows that the machinery does what you’d expect it to do.

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