The n-Category Café

For some inimitable Arnoldian exposition on Hilbert’s thirteenth problem see the first pages of the lecture From Hilbert’s Superposition Problem to Dynamical Systems.

Do I seriously understand right that this list is being used as the basis for a DARPA grant program (DARPA Mathematical Challenges, BAA 07-68)? I find this shocking, because I think the list of problems is abysmal.

It’s silly to request proposals for “major mathematical breakthroughs” on the Riemann hypothesis, but at least it’s a worthy goal. By contrast, some of the problem statements appear to me to be gibberish. For example, in Problem 7, is there any meaning to the question “As data collection increases can we ‘do more with less’ by finding lower bounds for sensing complexity in systems?” I have no idea why we are trying to do more with less when data collection is increasing, what lower bounds have to do with actually accomplishing anything, or what it means to sense complexity in systems. The problem title (“Occam’s Razor in Many Dimensions”) and second sentence (“This is related to questions about entropy maximization algoriths.”) help only a little in clarifying.

By contrast, Problem 6 (“Computational Duality: Duality in mathematics has been a profound tool for theoretical understanding. Can it be extended to develop principled computational techniques where duality and geometry are the basis for novel algorithms?”) is already solved. Certainly the statement has been achieved, although one could always develop these ideas further.

Many of the other problems are poorly phrased or extremely speculative. I’m seriously offended by the form of this solicitation for proposals. I imagine it won’t be a big problem, since what will end up happening is that proposers add a brief description of why their proposal is relevant to these problems and DARPA funds the same stuff it would have anyway. However, it’s ridiculous. PIs shouldn’t have to try to guess what the program manager is talking about, and the government shouldn’t waste money trying to flesh out the program manager’s wild ideas.

My thoughts exactly anonymous.

What i find even more offending, is the “between the lines” comparison of this guy himself to Hilbert in his motivation. Let alone the quoting of Kelvin and Einstein to give weight to this nonsense, (He should have read Baez’s crackpot story)

My only hope is that this gibberish will eventually lead to a decent set of problems, how general they may be, to be tackled by the 21st century mathematicians..

I’m pretty underwhelmed by this list of problems. Some are quite vague; others seem like random stabs in the dark.

For one of the latter sort consider, problem 23. It first urges that we “SETTLE THE SMOOTH POINCARE CONJECTURE IN DIMENSION 4”. That’s a worthy goal: it would be very nice to know if every smooth compact manifold homotopy equivalent to the 4-sphere is not just homeomorphic but diffeomorphic to $S^4$ with its usual smooth structure.

But then, it asks “What are the implications for space-time and cosmology?” It’s not clear there are any such implications. There might be, if the answer uses a lot of gauge theory. But there might not. People used gauge theory to construct exotic smooth structures on $\mathbb{R}^4$, but nobody really knows the possible consequences for physics yet, despite a few vague speculations.

And then, it asks: “And might the answer unlock the secret of ‘dark energy’?” I see no reason to expect it will.

I could write a much better list myself, especially with the help of other mathematicians — starting with the readers of this blog. But, I won’t unless DARPA pays me to.

Re: Darpa’s challenges for the future

I dislike posting anonymously, but I’m doing so now because I like flame wars even less; and that’s what this discussion seems perilously close to. I’d urge anyone interested to step back and look at this more calmly.

One of Darpa’s charges is to survey matters of scientific and technological interest, with an eye toward gaps and blind spots in the conventional wisdom. It has recently funded research into the geometric Langlands program, which (among other things) turns out to have nontrivial applications in physics, and I believe that program was regarded as highly successful. I would read Mann’s challenges as an invitation for mathematicians, physicists, computer scientists and related workers to look for important mathematical questions which deserve wider attention (and more funding).

Possible connections between the 4D smooth Poincaré conjecture and cosmology seems to me a quite reasonable example of this sort of question.: it’s arguable that the standard Big Bang model, successful as it is, politely disguises the fact that we really don’t understand the boundary conditions in Einstein’s model for gravitation. The Cauchy problem for GR has an enormous literature, but there isn’t much concern there for the uncountably many smooth structures on Euclidean four-space. Smooth h-cobordisms of three-manifolds are not well-understood either. D-brane models for dark energy given serious attention in physics seem to be extremely speculative; this whole area may be an opportunity for mathematicians, physicists, and computer sciences (interested eg in spin foam models) to benefit from talking to each other.

There is limited funding for mathematical research, and I would read these challenges as a serious attempt to widen those possibilities. It might be most constructive if interested researchers who perceive the questions as off the mark (or not!) would respond to them with considered proposals as to how these questions (or, perhaps, better ones) could be approached. I don’t see these proposals as anything more than an attempt to open a door.

I agree with Anonymous #1 in thinking that this list of 23 supposed “challenges” is an “insult and embarrassment to the mathematical community”, and deserves tremendous negativity.

But isn’t DARPA the Research and Development organisation for the Department of Defense? I wouldn’t want their money anyway.

Hmmm… I often wondered what happened to the Hominidissiumus-project . Its founder Ew. Ger. Seeliger performed the first two experiments, but died too early to complete the third.

The first of them, performed 1922 in Bavaria, started with the “Handbook of Humbug”, a book designed to be confiscated. The following three years long trial caused nationwide excitement when Seeliger ordered “shitpeople” for witnesses like the Bavarian president of state, the catholic cardinal-archbishop and “the writer” A. Hitler. An interrogation of them should have proved the correctness of Seeliger’s naming of them. The court declared Seeliger as crazy and ordered him to be send into a mental hospital – where the physician begged him to leave because he was “too crazy for my institution”. According to Seeliger, the first Hominidissimus-Experiment was a complete success.

In the following years he traveled a lot, founded in New York the “Messias Foundation”, worked on a movie-adaption of “Peter Voss – the Thief of Millions” (one of the most successful films then in Germany) and observed sharply the rise of the Nazis (“Tratschional-Kotzialisten”). 1932 the 2nd Hominidissimus-Experiment was made. Seeliger distributed a parody (“The Howling for the Heil”) of a nazi-song in 10.000 copies in Berlin. After their seisure of power, Seeliger sheltered persecuted people and helped them to emigrate, e.g. the Feuchtwanger family. When he, at mayday 1933, put instead of the prescribed flag a flag-parody made of brownish paper and satirical comments (e.g. “every madness needs a flag”) onto his house, he himself became persecuted and fled into Switzerland. His friends in the resistance persuaded him to go back to Hamburg, where the Nazis were too frightened by his popularity to do him worse things than a publication-interdictum.

Until 1953, Seeliger lived very reclusive, turned towards past centuries – esp. the renaissance – and wrote never published novels about Erasmus v. Rotterdam. But the Hominidissimus-Project was not forgotten:

1953 - instead of 1957 as announced 1931 in New York – the third Hominidissimus-Experiment began with the publication of the baroque picaresque novels “Muchbeloved Falsette” and “Junker Schlörk”. Local district attorneys jumped forcefully into the trap and had to regret it. 1957 the Hominidissimus-Project should have led into a worldwide humorous scandal, set to trigger a “humorous world-revolution”. This step was never taken, Seeliger died 1959. Most of his manuscripts went lost, among them his autobiography and seven “Metacartoonfilms”. His movie mentioned above has been remade three or four times; the rest is forgotten.

Recent rumors suggest that his foundation in NYC still exists and pursuits the project in secrecy. May they have infiltrated governmental organisations?

Sigh. I suppose might as well identify myself as yesterday’s unindexed anonymous poster.

I’m afraid this whole thread reminds me a little too much of the following story, from Tristes Tropiques, by Claude Lévi-Strauss : Part VIII, the Tupi-Kawahib

A widower had an only son, who was already almost grown up. One day he sent for him and told him that it was high time he got married. “What must I do to get married?” the boy asked. “It’s very simple,” his father said. “All you have to do is go and see our neighbors and try to get their daughter to like you.” “But I don’t know how to make a girl like me!” “Well then – play the guitar, and laugh, and sing her a song or two!” The son did as he was told. But as he arrived just as the girl’s father was dying, his behavior was thought to be most unsuitable and they drove him away and threw stones after him. He went home and complained to his father, who told him how he ought to behave in such cases. The boy went off to his neighbors again and arrived just as they were killing a pig. Remembering the latest of his lessons he burst into tears: “How sad! How good he was! How we loved him! We shall never find a better.” Once again the neighbors drove him away in exasperation. He described all this to his father, and once again was told exactly how to behave in such circumstances. When he paid his third visit, his neighbors were busy clearing the caterpillars from their garden. Always one lesson behind, he burst out with, “What an abundance of good things! May you have more and more such animals on your property! May they never be lacking!” And he was chased away again.

After this third rebuff the father told his son to build himself a hut. He went into the forest and cut down the necessary trees. A werewolf passed by in the night, thought the site a good one for himself to settle in, and went to work. The next day the boy came back to the clearing and found the work well advanced. “God is giving me a hand!” he thought to himself delightedly. And so they worked in double shifts, the boy by day and the werewolf by night. Before long the house was ready.

By way of house-warming the boy decided to feast off a roebuck. The werewolf preferred a human body. The one brought the buck by day, the other a corpse during the night. And when the boy’s father came along to join in the feast he saw a dead man on the table as pièce de résistance and said to his son, “Ah, my boy, I’m afraid you’ll never be up to anything much …”

One might have hoped that the philosophy of science could have offered some suggestions as to the most rational way to distribute funds. But I can’t say philosophy’s own funding practices are too clever.

I can see that there’s a danger in wording proposals in a way that encourages what John calls ‘blather’ that you’ll tend to pick ‘low risk’ candidates, i.e., the well-established.

On the brighter side, perhaps you’re doing better than you did in the early nineteenth century when support for Abel and Galois was lacking (although the latter was always going to be an awkward case). Are there any cases of what we would now see as neglected mathematical geniuses of the twentieth century?

Allyn Jackson discusses the Darpa program (and mentions us) in April’s Notices.