Algebraic Topology No Longer Ineffective
April 16th, 2007 by WaltIn a comment thread at n-category cafe, John Baez has linked to the electronic version of a big bowl of ice cream. Investigating the link in his comment, I came across the web page of Francis Sergeraert, who has linked to his papers and talks.
Sergeraert and his collaborators have pioneered a program of computational algebraic topology, and it is amazing what they have already acheived. For example, they have developed effective versions of the Serre and Eilenberg-Moore spectral sequences.
These kinds of algorithms exert a powerful hold on my imagination. When I first tried to learn commutative algebra, I found much of the subject impenetrable. Then later when I learned about Gröbner bases, I suddenly found everything I found hard to understand became easy to understand. Now the Gröbner basis algorithm is too slow to implement by hand other than toy examples, but having effectively computable toy examples was enough for me. Commutative algebra textbooks are full of toy examples, but my suspicious unconscious mind was sure that they were tricking me, and that the toy examples were not to be trusted. Learning how to compute new examples allowed me to shut my unconscious up.
April 16th, 2007 at 10:10 pm
Your link to Francis Sergeraert’s web page has incorrect HTML. Somehow a <br> got into its midst.
April 17th, 2007 at 7:27 am
Thanks! I blame society.
April 17th, 2007 at 12:29 pm
I agree: it would be almost impossible to overstate the value of simple but nontrivial examples in illustrating the power and limitations of any theory, and in many cases, sufficiently generic nontrivial examples may be out of the reach of pencil and paper. The historical significance of the rise of “computational x theory”, where x = “group”, “algebraic geometry”, “algebraic topology”, etc., should not be underestimated.
April 17th, 2007 at 4:52 pm
And Haskell gives a nice concrete non-trivial representation of a category for you to play with. (Though deciding exactly which category can be a bit tricky.)
April 17th, 2007 at 5:47 pm
I once brought up the idea of the “new” subject of computational algebraic topology, and all the other grad students laughed at me. Then again, maybe I was joking…maybe it was “with”.
April 18th, 2007 at 12:55 pm
Hi, sigfpe,
I seem to be trying to instantly grok months of posts at four blogs. I am mulling trying to take up my expository student paper “Categorical Primer” again, if only to add Terry Tao’s “sizing pullbacks via Cauchy-Schwarz” thing as an exercise. One of the lacks I sorely felt in “Primer” was the dearth of simple but nontrivial applications to CS (other than remarking that the “join” operation in a relational database is the kind of fibered square or pullback Terry Tao mentioned). Anyway, have you tried to write about this in true TWF style? I confess I feel some internal resistance to the prospect of trying to learn Haskell just so I can read your blog!
David Corfield mentioned in N-category cafe the influence of Atiyah on Gower’s thought as described in the essay he cited. “Simple but nontrivial” is one of my favorite phrases, so I thought I’d mention that I got this from Atiyah’s address to the LMS, “The Unity of Mathematics”, which I highly recommend to everyone!
April 20th, 2007 at 10:41 am
> the dearth of simple but nontrivial applications to CS
This book has a bunch of examples. For example F-algebras and F-coalgebras are very pretty and unify a whole bunch of disparate looking operations, especially the folds and unfolds. No Haskell in that book but it also misses out on monads.
Reynolds parametricity is really cool. The most best intro I know is Theorems for Free!, but that particular paper isn’t all that categorical. In a suitable version of polymorphic lambda calculus, natural transforms are trivial enough to make them extremely easily grasped and yet non-trivial enough that statements about them are actually useful in programming. (The “Theorems for Free” being such examples.) This gives what I think is the easiest way to grasp things like the Yoneda lemma.
And the highly non-trivial applications of monads are ubiquitous. It’d be a pity not to mention those in a primer!
> true TWF style?
I’m not sure exactly how you’d define that. I guess it would be in a style written to be understood by mathematicians without a background in programming. Hmmm…
April 21st, 2007 at 11:46 am
The bulk of “pure Mathematics” as well as my Mathematiucal Biology, Mathematical Physics, and Mathematical Economics publications are not merely “Simple but nontrivial” but “Simple, elementary, but novel and nontrivial.” This does not, however, seem to lead to tenure.
April 21st, 2007 at 12:48 pm
JVP: seconded. I’ll raise you that it doesn’t lead to getting a job in the first place either.
April 22nd, 2007 at 9:54 am
John: seems true. Off the top of my head, there are several possible reasons:
(1) Schools are in the Prestige game, and “Simple, elementary” looks to them less likely to increase prestige, whereas someone doing categorical quantum knot theory of Ricci flows will add luster.
(2) Schools (I mean mostly colleges and universities) are torn between their teaching mandate and their research mandate. “Simple, elementary” look to them like teaching content, not research, and they are very unclear on how the two mandates interact.
(3) Schools are businesses. “Simple, elementary” turns them off in the cover letter, before they get to the part about how I’ve won contracts from Army, Navy, Air Force, NASA, Department of Energy, and others. I want them clear on the concept that I can be a “cash cow” or “rainmaker” — but they are addicted to thinking of full-time faculty as (literally) million-dollar compensation committments, which is why more than half of all college courses in the USA are taught by temps, adjuncts, instructors, and other part-timers without benefits.
(4) Science popularizers (in general, including Math) have less respect than Scientists, in general.
(5) Schools are unclear on how cheap computers have shifted the line between trivial and nontrivial, simple and non-simple; nor do most know about “experimental mathematics.”
(6) When I say that I have (as of yesterday) 1,555 entries in the Online Encyclopedia of Integer Sequences, and 200 entries at Prime Curios, they don’t believe the numbers, or know that they mean. My 19 entries at MathWorld can be explained as proof that I am likely to be a good teacher, if my work appears in the #1 online Math encyclopedia, at various levels.
(7) Academe is biased, strongly, against Generalists, and towards Specialists. An oddball such as myself, with degrees in 3 different subjects, who has taught a dozen other subjects, appears both unfocused, and (if believed) an emotional threat to narrow specialists.
I am actually looking at entry-level jobs where I leave out all my advanced degrees in a dumbed-down resume. I was found not qualified for a $20/hour Japanese automobile assembly line, because I could not produce a letter of recommendation from my high school shop teacher. Given that I wentr to high school in the early-to-mid 1960s, that’s not a surprise. My guess is that it’s illegal to say “you’re too old” so they found a sneaky way to weed out the overexperienced.
My son will collect his double B.S. in Math and Computer Science this summer, and go to a top-10 Law School — almost surely the youngest such student in the USA this year. Mathematically gifted (and published) he has read the handwriting on the wall. If we mathematicians are so smart, why aren’t we rich? The question almost answers itself.
April 25th, 2007 at 2:02 pm
Two significant footnotes to my oversimplification above: “If we mathematicians are so smart, why aren’t we rich?”
(1) April 24, 2007
A Wealth of Smarts Does Not Guarantee Actual Wealth
http://sciam.com/article.cfm?articleID=258C68D0-E7F2-99DF-3FE5895C3D920B22&chanID=sa003
A new analysis of data from a long-term study shows that you don’t have to be smart to be wealthy
By David Biello
“… A detailed study of 7,000-plus Americans followed since their teen years in the late 1970s reveals that intelligence provides more earning power but not necessarily more accumulated wealth. “The smarter you are, the more income you have,” explains economist Jay Zagorsky of Ohio State University, who analyzed the data. “For wealth, there is no relationship.”…
(2) “Make Less than $240 Million? You’re Off Top Hedge Fund List”, by Jenny Anderson and Julie Creswell, The New York Times, Tuesday 24 April 2007, pp.A1 and C6.
“James Simons, a 69-year-old publicity-shy former math professor, uses complex mathematical models to make bets on stocks, bonds and commodities, among other things. His earnings last year were $1.7 billion…”
Correct. $1.7 x 10^9
That raises the average for all Mathematcians. Alas, none of it trickles down to me. How about you?
April 26th, 2007 at 4:29 pm
I would appreciate your opinion on this geometry.thank you in advance.