The intrinsic curvature of a space is a subtle concept; it took a lot of hard work to formulate, clarify and crystallize its most prominent measure, the Riemann curvature tensor. Now that that work is done, of course, you can learn about it even in some undergraduate courses. Perhaps you start with a rigorous definition, then see a geometric interpretation, and compute the curvature tensor for a few example spaces. You may get a little mystified in the beginning, but as you come across it in various contexts, read about alternative definitions and interpretations, and see theorems where it plays a fundamental role, the mystery slowly disappears—or appears to. Perhaps at some point, you start to think it doesn’t seem that scary anymore.

Here are two quotes from famous geometers—two people who probably have as intimate an understanding of curvature as anybody—to help you keep your guard up.

The curvature tensor of a Riemannian manifold is a little monster of (multi)linear algebra whose full geometric meaning remains obscure.

–Mikhail Gromov,

Sign and Geometric Meaning of Curvature

Do not despair if the curvature tensor does not appeal to you. It is frightening for everybody.

–Marcel Berger,

A Panoramic View of Riemannian Geometry

I came across the first quote recently, but had seen the second one a while ago, while browsing Berger’s delightful book (which is great for leisure reading, if you are into the subject—lots of informal and intuitive comments, puzzles, history, all written in a fun and refreshing style).

“Milnor’s octahedron”

Berger also has a similar (and intriguing) comment about the Levi-Civita connection in his book. I had a fun email exchange with him about that comment once, but let me leave that to another time.

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(See also: 1, 2)

The video above shows Myrmegraph, a screen-based artwork by Scott Snibbe. The picture, by Victor Juhasz, is from *The Mind’s I*, a book by D. Hofstadter and D. Dennett.

From Snibbe’s page:

As you move the mouse over Myrmegraph’s screen, you release a stream of simulated ants and simulated pheromone: the chemicals ants use to communicate. These ants obey a simple set of rules to follow the pheromone gradients stored invisibly in the image. From moment to moment, ants can change their heading to better pursue the trail of pheromone; and, like real ants, they sometimes lose their way and wander off.

Among the four combinations of the form *<building blocks>* → *<emergent structure>*,

(1) nonsentient → nonsentient

(2) nonsentient → sentient

(3) sentient → nonsentient

(4) sentient → sentient,

the possibilities (1) and (3) do not cause much cognitive dissonance, and to some degree, one learns to live with (2), however uncomfortably. The possibility (4), though, is rather unfamiliar, and confusing/fascinating in its own unique way.

]]>Important though the general concepts and propositions may be with which the modern industrious passion for axiomatizing and generalizing has presented us, . . . I am convinced that the special problems in all their complexity constitute the stock and core of mathematics; and to master their difficulties requires on the whole the harder labor. . . . The general theories are shown here as springing forth from special problems[.]

–Hermann Weyl, The Classical Groups

Oh, no, no, no. I think fiction, and biography and history, are

theforms. I think one can say much more about general abstract ideas in terms of concrete characters and situations, whether fictional or real, than one can in abstract terms. . . . And I must say I think that probably all philosophy ought to be written in this form; it would be much more profound and much more edifying. It’s awfully easy to write abstractly, without attaching much meaning to the big words. But the moment you have to express ideas in the light of a particular context, in a particular set of circumstances, although it’s a limitation in some ways, it’s also an invitation to go much further and much deeper.–Aldoux Huxley, interviewed by Raymond Fraser and George Wickes, The Paris Review

I have collected a few other quotes along these lines over the years on my pedagogy page, including one by Michael Atiyah, who expresses great admiration for Weyl and his book. Here is another, beautiful bit from the latter’s preface:

The stringent precision attainable for mathematical thought has led many authors to a mode of writing which must give the reader an impression of being shut up in a brightly illuminated cell where every detail sticks out with the same dazzling clarity, but without relief. I prefer the open landscape under a clear sky with its depth of perspective, where the wealth of sharply defined nearby details gradually fades away towards the horizon.

In the same preface, we also find a complaint:

]]>The gods have imposed upon my writing the yoke of a foreign tongue that was not sung at my cradle.

- Take an elastic ball and suck some air out of it. What happens to its surface?Experiments by P. Reis and colleagues show that it can do fun things like this:

and that it can assume other strange shapes that resemble wrinkled fingers or grapes:

The question then is, what aspects of this physical system determine the qualitative properties of the surface? (Or, in the language of a theoretical physicist, how could one obtain the phase diagram for the relevant symmetry-breaking transition?) Sarah Lewin and Jennifer Chu have nice popular-level pieces in the Quanta magazine and MIT News, respectively, that talk about the relevant experiments and a collaboration with mathematicians that resulted in a general answer to this question, where *curvature* plays a key role. (I am a fan of questions and insights at the intersection of geometry and mechanics, and have done a bit of work around the area with my friend A. Yavari, but more on that in another post, hopefully.)

- Discussions on the future of academic publishing in the age of the internet have been going on for a long time, but progress has been slow. The famous algebraic geometer David Mumford has a recent blog post where he shares his views and experience, and he doesn’t mince words. To quote a couple of striking paragraphs:

I want to make a plea to my colleagues to spend more time considering how we should shape this aspect of our profession and then being open to radical changes: you have nothing to lose but the chains that are binding you to capitalist exploitation and you can gain a freer, simpler world to work in.

If you think a large part of our professional life is not mortgaged to capitalists, perhaps you have spent too much time thinking only about theorems. Private equity buys a firm for one and only one reason: they believe they can squeeze more profits out of their operations, i.e. out of us mathematicians (and our societies and libraries).

See also this Facebook discussion where my friend and mentor J. Bellissard makes some comments (that confused me a bit), and if you haven’t already, take a look at this powerful speech by Ursula Le Guin:

- Physics does a good job of describing the behavior of individual, ideal/ized building blocks like point particles and solid bodies, or large, uniform-ish collections of them like fluids, gasses, crystals, and planetary rings. But when things get more complicated, things get complicated.

As dreamy-eyed undergraduates some 20 years ago, some of us were intrigued by the vaguely-defined field of complex systems, and fantasized about studying at the famous Santa Fe Institute, trying to understand not just the important but “dead” systems of traditional physics, but also more “alive” ones like sandpiles, self-organizing chemical reactions, and even life itself. Most of us later chose more traditional, well-established paths, but still can’t help keeping an eye towards the work being done around this intriguing area.

All this being a preamble for a link to a nice popular-level piece by Jordan Ellenberg on one of the simplest complex, self-organized systems, namely, an idealized version of a sandpile. This system, called the abelian sandpile model, displays certain patterns of complex behavior familiar from ordinary sandpiles, and also other, strange and beautiful structures such as the one above.

- Alexey Chervonenkis was a co-founder (eşbaşkan) of Vapnik-Chervonenkis theory, which is one of the fundamental pieces of modern statistical learning theory, i.e., the mathematical theory of learning from data. (If you have heard about Occam’s razor and wondered whether one could formulate a precise, quantitative version of this idea that would be amenable to rigorous proof, VC-theory is a fascinating attempt in this direction.)

A few years ago, I had the great opportunity to meet and have lunch with V. Vapnik and A. Chervonenkis, and talk to them about their views on the field of machine learning. Together with my collaborator A. Gray, we prodded them with questions and heard some surprising commentary from V. Vapnik on Occam’s razor, but unfortunately couldn’t get A. Chervonenkis to talk a lot. He was the more reserved of the two, and mostly listened to the conversation with polite attention, with the occasional friendly, mischievous grin. He showed some interest in our work on density estimation on manifolds, which Vapnik saw as a step in the wrong direction, trying to solve a harder problem than necessary.

I recently came across this obituary of A. Chervonenkis by Bernhard Schölkopf, and only then realized that he had passed away last year. This saddened me quite a bit despite not having spent a lot of time with him, and I realized his polite, friendly, but reserved demeanor had left quite a strong impression. My memory of him from that day is almost exactly as in the picture above, taken from here.

B. Schölkopf describes A.C. as an avid hiker, mentions a close call where they ended up swimming against strong currents together in some dangerous waters, and expresses the great surprise he felt seeing that the old man, despite having a non-standard and presumably very inefficient swimming style, was not afraid to take on the adventure. In outdoor sports, elite athletes get most of the glory, but there is a lot to be said about the amateur adventurer who has a good understanding of his or her limits, and goes on to explore and have private adventures no less exciting, rewarding, and inspiring than those that find their ways into newspapers.

That’s it for now. Hoping to collect posts like this under the tag WoP—Wonders of Procrastination. Hat tip to Peter Woit for Mumford’s post.

]]>Two examples from Parchman Farm, “the oldest prison and the only maximum security prison for men in the state of Mississippi”, according to WP:

The first one is by Henry (Jimpson) Wallace, and the second one is by Dan Barnes and “unidentified prisoners”, according to the Cultural Equity page, where you can find other recordings. See the source of the picture above for a quote about Parchman Farm.

A happier example from Tanzania:

Frank Gunderson, who made this recording, asks, “How do people from around the world draw the line between work and play?”, and quotes one of the workers:

]]>Without the music you just get tired, it is as if you are being tortured.

- Henry Segerman has various math-related creations, one of which is this fascinating visualization of the stereographic projection with a 3D-printed object.

This is a candidate in NSF’s visualization challenge; you can see some other candidates competing in the same section here. Segerman’s website has a lot of other interesting things. - Wikipedia says that the term
*stereographic projection*was coined by François d’Aguilon. A search for his*Six Books of Optics*gives an archive.org page where you can flip through the book, and find the following illustration, which is another visualization of the stereographic projection.

Here is a direct link to the page. This was apparently drawn for the book by the famous painter Rubens. How cool is that? - Another famous projection of the sphere is onto a cylinder, instead of the plane, as in the image below.

Archimedes realized that a region on the sphere and its projected version on the cylinder have the same area, which allows one to calculate the area of the sphere using the area of the cylinder (which is easier to figure out). He liked his discovery so much that he requested to have a picture of a sphere inscribed in a cylinder on his tomb. (Amusingly, in his book on PDEs, V.I. Arnold calls this projection*the Archimedes symplectomorphism*.)The Roman philosopher/politician Cicero writes about finding Archimedes’ tomb in Sicily, 137 years after Archimedes’ death. Search for “obscure mathematician” (ha!) here to read a translation. Here is a painting by Pierre-Henri de Valenciennes, depicting Cicero’s (claimed) discovery:

- Finally, another one of Archimedes’ findings, demonstrated beautifully in an animation by Zachary Abel, who summarizes it as “hemisphere + cone = cylinder”.

Taken from Abel’s blog.

More on Archimedes some other time, hopefully.

]]>The illustration above is from *The Kentucky Derby Is Decadent and Depraved*, the article that is said to have marked the beginning of gonzo journalism. Hunter S. Thompson, the author of the piece, tells how it came to be:

THOMPSON: I’d gone down to Louisville on assignment for Warren Hinkle’s [magazine]

Scanlan’s. A freak from England named Ralph Steadman was there—first time I met him—doing drawings for my story. The lead story. Most depressing days of my life. I’d lie in my tub at the Royalton. I thought I had failed completely as a journalist. I thought it was probably the end of my career. Steadman’s drawings were in place. All I could think of was the white space where my text was supposed to be. Finally, in desperation and embarrassment, I began to rip the pages out of my notebook and give them to a copyboy to take to a fax machine down the street. When I left I was a broken man, failed totally, and convinced I’d be exposed when the stuff came out. It was just a question of when the hammer would fall. I’d had my big chance and I had blown it.INTERVIEWER: How did

Scanlan’sutilize the notebook pages?THOMPSON: Well, the article starts out with an organized lead about the arrival at the airport and meeting a guy I told about the Black Panthers coming in; and then it runs amok, disintegrates into flash cuts, a lot of dots.

INTERVIEWER: And the reaction?

THOMPSON: This wave of praise. This is wonderful . . . pure gonzo. I heard from friends—Tom Wolfe, Bill Kennedy.

INTERVIEWER: So what, in fact, was learned from that experience?

THOMPSON: I realized I was on to something

Food for thought.

The only book by Thompson I had attempted to read was *Hell’s Angels*, which I didn’t finish. A PDF version of *The Kentucky Derby* is here, if you are interested (found here).

On a plane trip a couple of months ago, I watched a documentary about Ralph Steadman, the cartoonist Thompson mentions above. The two are/were apparently very different characters—Thompson being the more spontaneous, crazy one—but had a fruitful partnership that spanned years. A remark from the film had stuck in my mind, looking for a transcript, I found this:

The basis of Ralph and Hunter’s friendship was that they saw kindred spirits in each other. [I think] Hunter realized that Ralph was crazier than him. Ralph was willing to go to extremes that Hunter was not willing to, and you’d think Hunter would be the one who was the, you know, more outrageous and reckless and the one who would go out on a limb on something. But Ralph was the one who’d actually go there. I’m not talking about physical safety. But I’m talking about sort of, you know, mental, moral, philosophical

The limbs Steadman was willing to go out on mentioned in the film included the Vietnam War and Watergate.

I got the image at the top from Steadman’s website. He is still active, apparently, and recently contributed to *Ghosts of Gone Birds*, a project “dedicated to breathing life back into the birds we have lost – so we don’t lose anymore”.

The trailer of the documentary is here.

The emotional impact of music is so incommensurate with what people can say about it, and that seems to be very illustrative of something fundamental—that very powerful emotional effects often can’t be articulated. You know something’s happened to you but you don’t know what it is. You’ll find yourself going back to certain poems again and again. After all, they are only words on a page, but you go back because something that really matters to you is evoked in you by the words. And if somebody said to you, Well, what is it? or What do your favorite poems mean?, you may well be able to answer it, if you’ve been educated in a certain way, but I think you’ll feel the gap between what you are able to say and why you go on reading.

]]>A few years ago I came across a press release about a TV program by Glenn Gould, with the provocative title *How Mozart Became a Bad Composer*. I attempted to get a copy of it from the Glenn Gould Foundation, but was told that it was being prepared as part of a DVD set (which was supposed to be released last year). After exchanging a few emails, I ended up ordering an issue of the Glenn Gould magazine where the full transcript of the program was published. The following paragraph from that transcript comes to my mind every now and then, and I decided to post it here and maybe return to it sometime. If nothing else, it is pretty good food for thought on creative work.

Perhaps it all comes down to this: within every creative person there’s an inventor at odds with a museum curator. And most of the extraordinary and moving things that happen in art are the result of a momentary gain by one at the expense of the other. In Mozart’s case, the inventor was endowed with the most precocious gifts, and the curator, who manufactured all those sequences and arpeggiated links and passages of scale padding, zealously carried out

hisduties as well. But what I object to is that Mozart tries to cover up the conflict between them. Time and again, the curator wins out over the inventor, as he has every right to do. But I’d like to find some evidence of protest, some frantic, disruptive, unsyntactical attempt on the part of the inventor to get free of the curator’s control. Or, in the absence of that desperation move, I’d like Mozart to feel guilty, and because of that guilt to sacrifice something of the charm and courtesy which mask the humanity of his work.

What did Gould’s Mozart playing sound like? Here is one example.

Wikipedia says,

]]>The A minor sonata is the first of only two Mozart piano sonatas to have been composed in a minor key . . . It was written in one of the most tragic times of his life: his mother had just died, and his father blamed him for his wife’s death. Mozart was devastated, and poured his constant torment into his sonata, one of the darkest. The last movement in particular has an obsessive, haunted quality about it, heightened near the end by the interruption of the relentless drive to the conclusion by repeated and chilling quiet falling passages.

**1. L1 distance between probability density functions**

If and are two random variables with densities and , respectively, the density of the scaled variable is

and similarly for , where . Using this transformation rule, we see that the distance between and satisfies

Setting , we get

This latter result also holds for -dimensional vector random variables (for which the transformation rule is ).

In their book on the approach to nonparametric density estimation, Luc Devroye and László Györfi prove a much more general version of this result, which includes as a special case the fact that distance between PDFs is invariant under continuous, strictly monotonic transformations of the coordinate axes. In the one-dimensional case, this means that one can have a visual idea about the distance between two PDFs by bringing the real line into a finite interval like by a monotonic transformation.

The figure below shows the transformed version of the PDFs above, via the transformations , . The area of the gray region inside the interval below is the same as the area of the gray region in the third plot above (which is spread over the whole real line).

You can read the proof of the general version of the theorem in the introductory chapter of the Devroye-Györfi book, which you can download here. Luc Devroye has many other freely accessible resources on his website, and he offers the following quote as an aid in understanding him.

**2. L1 penalty and sparse regression: A mechanical analogy**

Suppose we would like to find a linear fit of the form to a data set , where , , and . Ordinary least-squares approach to this problem consists of seeking an -dimensional vector that minimizes

One sometimes considers regularized versions of this approach by including a “penalty” term for , and minimizing the alternative objective function

where is a tuning parameter, and is a function that measures the “size” of the vector . When

,

we have what is called *ridge regression*, and when

,

we have the so-called *LASSO*. Clearly, in both cases, increasing the scale of the penalty term results in solutions with “smaller” , according to the appropriate notion of size.

Perhaps a bit surprisingly, for the case of penalty (LASSO), one often gets solutions where some are not just small, but exactly zero. I recently came across an intuitive explanation of this fact based on a mechanical analogy, on a blog devoted to compressed sensing and related topics. The following three slides reproduced from a presentation by Julien Mairal perhaps do not exactly constitute a “proof without words“, but are really helpful nevertheless. The terms in the figures represent the and versions of the penalty function as “spring” and “gravitational” energies, respectively. Increasing the spring constant makes smaller, but not zero. Increasing the gravity (or the mass), on the other hand, eventually makes zero.

In case you haven’t seen it before, here is a visual from the original LASSO paper by Tibshirani, comparing and penalties in a two-dimensional problem ( denotes the solution of the ordinary least squares problem, without any constraint or penalty):

This provides another intuitive explanation of the sparsity of solutions obtained from LASSO. In order to make sense of this figure, you should keep in mind that the regularized problems above are equivalent to the constrained problem

where is analogous to the tuning parameter , and stands for or penalty, as above. (If this was a bit too cryptic, you can take look at the original LASSO paper, which, according to Google Scholar, has been cited more than 9000 times.)

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