Wrinkles, sandpiles, capitalists, and A. Chervonenkis

Quick links to a few things I’ve read recently.

  • 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.

Abelian sandpile

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.

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Musical labor

Parchman Penitentiary, Parchman, MS 1997

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.

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Just a few geometry-related things I like.

    • 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.
      Stereographic_ProjectionHere 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.

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