Especially before the fall of the USSR, the best Russian mathematicians would often meet and discuss their work at seminars.
Gelfand's seminar in Moscow was especially famous, since he would stop speakers any time they said something unclear. In fact, sometimes he'd appoint an audience member to play the role of arbiter: if this guy in the audience doesn't understand it, the speaker has to explain it better!
As a result, the seminar would often go on until late at night, even after the building was locked up. But everyone learned a lot of math.
With such exhaustive seminars, publishing proofs sometimes became a mere afterthought. You'll often see short papers from this era making important claims with just a tiny sketch of an argument to back them up.
That annoyed Western mathematicians. And I've bumped into a few mysteries that I'm having trouble with, thanks to these short Russian papers without clear proofs. Here is one.
This image by Greg Egan shows the set of points \((a,b,c)\) for which the quintic $$ x^5 + ax^4 + bx^2 + c $$
has repeated roots... with the plane \(c = 0\) removed. You'll notice this surface crosses over itself in a cool way, creating lines of sharp cusps.
Vladimir Arnol'd, who ran one of these famous seminars, says that one O. V. Lyashko studied this surface in 1982 with the help of a computer - a very primitive computer by our standards, I'm sure. And he says Lyashko proved this surface looks the same as another surface defined using the icosahedron.
Arnol'd doesn't mention removing the plane \(c = 0\), so his claim is technically wrong. But if you remove that plane, it looks right! So I'd like to see a proof that these surfaces are the same after a smooth change of coordinates. The icosahedron and the quintic equation are connected in many ways, so there should be a nice explanation. But I don't know it!
For more details on this surface, see this blog post:
The icosahedron is connected to some of the most fascinating symmetrical structures in the mathematical universe, such as \(E_8\) and the Golay code. I'm trying to get to the bottom of this, so every clue helps.
Here is a longer description of Gelfand's seminar, as told by Simon Gindikin:
The Gelfand seminar was always an important event in the very vivid mathematical life in Moscow, and, doubtless, one of its leading centers. A considerable number of the best Moscow mathematicians participated in it at one time or another. Mathematicians from other cities used all possible pretexts to visit it. I recall how a group of Leningrad students agreed to take turns to come to Moscow on Mondays (the day of the seminar, to which other events were linked), and then would retell their friends what they had heard there. There were several excellent and very popular seminars in Moscow, but nevertheless the Gelfand seminar was always an event.
I would like to point out that, on the other hand, the seminar was very important in Gelfand's own personal mathematical life. Many of us witnessed how strongly his activities were focused on the seminar. When, in the early fifties, at the peak of antisemitism, Gelfand was chased out of Moscow University, he applied all his efforts to seminar. The absence of Gelfand at the seminar, even because of illness, was always something out of the ordinary.
One cannot avoid mentioning that the general attitude to the seminar was far from unanimous. Criticism mainly concerned its style, which was rather unusual for a scientific seminar. It was a kind of a theater with a unique stage director playing the leading role in the performance and organizing the supporting cast, most of whom had the highest qualifications. I use this metaphor with the utmost seriousness, without any intention to mean that the seminar was some sort of a spectacle. Gelfand had chosen the hardest and most dangerous genre: to demonstrate in public how he understood mathematics. It was an open lesson in the grasping of mathematics by one of the most amazing mathematicians of our time. This role could be only be played under the most favorable conditions: the genre dictates the rules of the game, which are not always very convenient for the listeners. This means, for example, that the leader follows only his own intuition in the final choice of the topics of the talks, interrupts them with comments and questions (a privilege not granted to other participants) [....] All this is done with extraordinary generosity, a true passion for mathematics.
Let me recall some of the stage director's strategems. An important feature were improvisations of various kinds. The course of the seminar could change dramatically at any moment. Another important mise en scene involved the "trial listener" game, in which one of the participants (this could be a student as well as a professor) was instructed to keep informing the seminar of his understanding of the talk, and whenever that information was negative, that part of the report would be repeated. A well-qualified trial listener could usually feel when the head of the seminar wanted an occasion for such a repetition. Also, Gelfand himself had the faculty of being "unable to understand" in situations when everyone around was sure that everything is clear. What extraordinary vistas were opened to the listeners, and sometimes even to the mathematician giving the talk, by this ability not to understand. Gelfand liked that old story of the professor complaining about his students: "Fantastically stupid students - five times I repeat proof, already I understand it myself, and still they don't get it."
It has remained beyond my understanding how Gelfand could manage all that physically for so many hours. Formally the seminar was supposed to begin at 6 pm, but usually started with an hour's delays. I am convinced that the free conversations before the actual beginning of the seminar were part of the scenario. The seminar would continue without any break until 10 or 10:30 (I have heard that before my time it was even later). The end of the seminar was in constant conflict with the rules and regulations of Moscow State University. Usually at 10 pm the cleaning woman would make her appearance, wishing to close the proceedings to do her job. After the seminar, people wishing to talk to Gelfand would hang around. The elevator would be turned off, and one would have to find the right staircase, so as not to find oneself stuck in front of a locked door, which meant walking back up to the 14th (where else but in Russia is the locking of doors so popular!). The next riddle was to find the only open exit from the building. Then the last problem (of different levels of difficulty for different participants) - how to get home on public transportation, at that time in the process of closing up. Seeing Gelfand home, the last mathematical conversations would conclude the seminar's ritual. Moscow at night was still safe and life seemed so unbelievably beautiful!
One of the world's largest insects lives in Australia. It looks like a stick and it's called Ctenomorpha gargantua. It's very hard to find, because it lives in the highest parts of the rainforests in Queensland, and it's only active at night!
In 2014 one fell down. Scientists found it hanging on a bush. They took
it to the Museum Victoria, in Melbourne. They named it 'Lady Gaga-ntuan'.
And now it has a daughter that's 0.56 meters long — that is, 22.2
June 7, 2016
By now there's a lot of evidence that dark matter exists, but not so much about what it is. The most popular theories say it's some kind of particles that don't interact much with ordinary matter, except through gravity. These particles would need to be fairly massive — as elementary particles go — so that despite having been hot and energetic shortly after the Big Bang, they'd move slow enough to bunch up thanks to gravity. Indeed, the bunching up of dark matter seems necessary to explain the formation of the visible galaxies!
Searches for dark matter particles have not found much. The DAMA experiment, a kilometer underground in Italy, seemed to detect them. Even better, it saw more of them in the summer, when the Earth is moving faster relative to the Milky Way, than in the winter. That's just what you'd expect! But other similar experiments haven't seen anything. So most physicists doubt the DAMA results.
Maybe dark matter is not made of massive weakly interacting particles. Maybe it's a superfluid made of light but strongly interacting particles. Maybe there are lot more 25-solar-mass black holes than most people think! There are lots of theories, and I don't have time to talk about them all.
I just want to tell you about a cool idea which assumes that dark matter is made of massive weakly interacting particles. It's still the most popular theory, so we should take it seriously and ask: if they exist, what would these particles do?
In the early Universe they'd attract each other by gravity. They'd bunch up, helping seed the formation of galaxies. But after stars and planets formed, they'd pull at the dark matter, making it thicker in some places, thinner in others.
And this is something we can simulate using computers! After all, the relevant physics is well-understood: just Newton's law of gravity, applied to stars, planets and zillions of tiny dark matter particles.
Gary Prezeau of NASA's Jet Propulsion Laboratory did these simulations and discovered something amazing.
When dark matter flows past the Earth, it gets deflected and focused by the Earth's gravity. Like light passing through a lens, it gets intensely concentrated at certain locations!
This creates long thin 'hairs' where the density of dark matter is enhanced by a factor of 10 million. Each hair is densest at its 'root'. At the root, the density of dark matter is about a billion times greater than average!
The hairs in this picture are not to scale: the Earth is drawn too big. The roots of the hairs would be about a million kilometers from Earth, while the Earth's radius is only 6,400 kilometers.
Of course we don't know dark matter particles exist. What's cool is that if they exist, it forms such beautiful structures! And if we could do a dark matter search in space, near one of these possible roots, we might have a better chance of finding something.
Let me paraphrase Prezeau, because the real beauty is in the details. From his abstract:
It is shown that compact bodies form strands of concentrated dark matter filaments henceforth simply called 'hairs'. These hairs are a consequence of the fine-grained stream structure of dark matter halos surrounding galaxies, and as such they constitute a new physical prediction of the standard model of cosmology. Using both an analytical model of planetary density and numerical simulations (a fast way of computing geodesics) with realistic planetary density inputs, dark matter streams moving through a compact body are shown to produce hugely magnified dark matter densities along the stream velocity axis going through the center of the body. Typical hair density enhancements are 107 for Earth and 108 for Jupiter. The largest enhancements occur for particles streaming through the core of the body that mostly focus at a single point called the root of the hair. For the Earth, the root is located at about 106 kilometers from the planetary center with a density enhancement of around 109 while for a gas giant like Jupiter, the root is located at around 105 kilometers with a enhancement of around 1011. Beyond the root, the hair density precisely reflects the density layers of the body providing a direct probe of planetary interiors.
The mathematicians and physicists among you may enjoy even more detail. Again, I'll paraphrase:
According to the standard model of cosmology, the velocity dispersion of cold dark matter (CDM) is expected to be greatly suppressed as the universe expands and the CDM collisionless gas cools. In particular, for a weakly interacting mass particle with mass of 100 GeV that decoupled from normal matter when the Universe cooled to an energy of 10 MeV per particle, the velocity dispersion is only about 0.0003 meters per second.
As the Universe cools and the nonlinear effects of gravity become more prominent and galactic halos grow, the dispersion of velocities will increase somewhat, but 10 kilometers per second is an upper limit on the velocity dispersion of the resulting dark matter streams.
Dark matter starts out having a very low spread in velocities, but its location can be anywhere. So, it forms a 3-dimensional sheet in the 6-dimensional space of position-velocity pairs, called 'phase space'.
As time passes this sheets gets bent, but it can never be broken. When this sheet gets folded enough, we get a 'caustic where lots of different dark matter particles have almost the same position, though different velocities. You can see a caustic by shining light into a reflective coffee cup, or shining light through a magnifying glass. The same math applies here:
A phase-space perspective sheds additional light on the processes affecting the CDM under the influence of gravity. When the CDM decouples from normal matter, the CDM occupies a 3-dimensional sheet in the 6-dimensional phase space since it has a tiny velocity dispersions. The process of galactic halo formation cannot tear this hypersurface, thanks to generalization of Liouville's theorem. Under the influence of gravity, a particular phase space volume of the hypersurface is stretched and folded with each orbit of the CDM creating layers of fine-grained dark matter streams, each with a vanishingly small velocity dispersion. These stretches and folds also produce caustics: regions with very high CDM densities that are inversely proportional to the square root of the velocity dispersion.Here are some more pictures:
In May he tweeted:
Good news is that my campaign has perhaps more cash than any campaign in the history of politics.But this was a lie. By the end of May his campaign had less than $1.3 million. At least, that's what he reported to the Federal Election Commission.
That may sound like a lot if you don't know US politics. But Clinton, by comparison, had $42 million. Even Ben Carson - remember that guy, the nutty candidate who claimed the pyramids were built for storing grain? - had $1.7 million when he quit back in March.
So, by US standards, Trump's campaign is broke.
And he keeps putting campaign money back into his own pocket!
Throughout his campaign, up to the end of May, he has given $6.2 million of campaign funds to companies he owns. That's roughly 10% of his campaign spending so far. And in May this rose to almost 20%: he spent $6.7 million on his campaign, but over $1 million of that went to his own companies.
According to the Huffington Post:
The most striking expenditure in the new filings was $423,372, paid by the Trump campaign for rentals and catering at Trump.s 126-room Palm Beach, Florida, mansion, Mar-A-Lago, which Trump operates as a private club.So, Trump has given a whole new meaning to the term 'self-funding'. In 2000, he said:
Other Trump-owned recipients of campaign funds include Trump Restaurants, which raked in $125,080 in rent and utilities; Trump Tower Commercial, which charged $72,800 in rent and utilities in the building that houses Trump.s campaign headquarters; the Trump National Golf Club, in Jupiter, Florida, which collected $35,845 for facilities rental and catering; and the Trump International Golf Club in West Palm Beach, Florida, which billed the campaign for $29,715, for facilities rentals and catering.
It's very possible that I could be the first presidential candidate to run and make money on it.It seems that Trump plans to let the Republican National Committee pay for most of his campaign. They've got some money: they started June with $20 million in cash. But four years ago at this time, they had more than $60 million. Their big donors are shying away from Trump.
I would love to get money out of US politics. I hadn't expected Trump to take the lead.
Here is his May report to the Federal Election Commission:
This is the small stellated dodecahedron. It's like a star made of stars. It has 12 pentagrams, 5-pointed stars, as faces. These stars cross over each other. Five meet at each sharp corner.
But here's the really cool part: you should think of each pentagram as a pentagon that's been mapped into space in a very distorted way, with a 'branch point of order 2' at its center.
What does that mean?
Stand at the center of a pentagon! Measure the angle you see between two corners that are connected by an edge. You'll get \(2\pi/5\). But now stand at the center of a pentagram. Measure the angle you see between two corners that are connected by an edge. You get \(4\pi/5\). Twice as big!
So, to map a pentagon into space in a way that makes it look like a pentagram, you need to wrap it twice around its central point. That's what a branch point of order 2 is all about.
That's the cool way to think of this shape you see spinning before you. It's a surface made of 12 pentagons, each wrapped twice around its center, with 5 meeting at each sharp corner.
There's another way to think about this surface! Any equation of this sort $$ z^5 + pz + q = 0 $$ has 5 solutions, or 'roots'. To make this true we need to bend the rules a bit. First, we let the solutions be complex numbers... so let \(p\) and \(q\) be complex too. Second, we must allow for the possibility of 'repeated roots': when you factor \(z^4 + pz + q\), the same root may show up twice.
Now here's the cool part: the small stellated dodecahedron is the set of all lists of 5 numbers that are roots of some equation of this form: $$ z^5 + pz + q = 0 $$
So it's not just a pretty star-shaped thing. It's a serious mathematical entity! It's actually a Riemann surface, the most symmetrical Riemann surface with 4 holes! You can build it starting from a tiling of the hyperbolic plane by pentagons. In this tiling 5 pentagons meet at each corner — just like 4 squares meet at each corner in a square tiling of the ordinary plane.
It's all about the number 5, which has a lot of star power. To understand more, read my blog article:
Most of this was discovered by Felix Klein in 1877. He discovered lots of cool facts like this. It's almost annoying. I keep learning cool things about Riemann surfaces and the hyperbolic plane... and it keeps turning out they were discovered by Klein. He found more than his fair share.
By the way, this post and many others are now part of my "geometry" collection on G+. If you want to binge on beauty, try these collections:
But beware: next morning you may wake up in a gutter with a headache, seeing stars.
This image was created by someone named 'Cyp' and placed on
June 24, 2016
A lynx kitten bounds forward, confident and focused.
I need this picture today, to cheer myself up. I don't like the Brexit. The very best possible interpretation I can put on it is that it's ordinary folks poking a stick in the eye of the elite, demanding more local control of government, more democracy. Maybe the elite will wake up, stop trying to hog all the wealth, and realize that in the long run it pays to help the downtrodden.
Maybe London will become less dominated by corrupt financiers. Maybe Scotland will become independent and join the EU.
I can imagine a wave of decentralization and localization being a good thing.... if it's balanced by the right larger-scale structures, allowing plenty of free trade, free movement of people, and so on. But I don't get any sense that the Brexiters have a constructive vision for the future.
Back to the theme of youth:
The young are generally bolder, less careful, less fearful. It's got pros and cons.
75% of British people between ages 18 and 24 said they voted for Britain to stay in the EU. For people 25-49 it was 56%. For people 50-64 it was 44%. For people above 65, just 39%.
So this is an interesting case. Perhaps the old are more fearful — of refugees, of Polish plumbers, of EU bureaucrats — but in this case they were more eager to do something rash. It's quite amazing how little is known about what will happen next! About all we be sure about is that it will create a big mess.
Good luck, Britain! Good luck, EU!
June 27, 2016
The fuzz on bees helps them collect pollen. But it may also help them detect electric fields!
The surprising part — to me — is that flowers have electric fields. And different kinds of flowers have noticeably different fields.
Gregory Sutton, a biomechanical engineer who is studying this, says that flower petals tend to accumulate electric charge. So, they produce an electric field just like when you rub a balloon on a woolly sweater — but smaller:
"It's a very small electrical field, which is why we're quite astounded that bees can actually detect it," Sutton says. "And there is different charge distribution at different locations on the petals of different species of flowers. So two flowers of the same species will have a similar electric field, whereas two flowers of a different species will have different electric fields."
Together with biophysics researcher Erica Morley and some other scientists, Sutton did experiments to test the theory that bees use electric fields to help find food.
They built 10 flowers with the same shape, size and smell. They put sugar water on some of the flowers and then added small static electric fields to those flowers. On the rest of the flowers, they put bitter water and no electric field. They let the bees loose among the flowers and kept moving the flowers around so the bees couldn.t learn the location of the sugar water.
"As they forage, they start to go to the flowers with the sugar water 80 percent of the time," Sutton says. "So you know they've figured out the difference between the two sets of flowers. The last step is you just turn off the voltage and then check to see if they can continue telling the difference. And when we turned off the voltage, they were unable to tell the difference. And that's how we knew it was the voltage itself that they were using to tell the difference between the flowers."
It's good that they did this last step, because otherwise I'd be unconvinced. They also studied the mechanism that bees use to detect electric fields. Basically, bee hairs get pulled by an electric field, and the bee can feel it:
"What we found in bees is that they're using a mechanic receptor," Morley says. "It's not a direct coupling of this electrical signal to the sensory system. They're using mechanical movement of hair in a very non-conductive medium. Air doesn't conduct electricity very well — it's very resistive. So these hairs have moved in response to the field, which then causes the nerve impulses from the cells at the bottom of the hair."I love results like this, which show the world is bigger and more interesting than I thought. But I'm a bit suspicious too, so I hope more scientists try to replicate these experiments or poke holes in them.
The paper is open-access, so if you have questions you can read it yourself!
I got my quotes from here:
Metaculus is a website where you can ask about future events and predict their probabilities. The "wisdom of crowds" says that this is a pretty reasonable way to divine the future. But some people are better predictors than others, and this skill can be learned.
Metaculus was set up by two professors at U.C. Santa Cruz. Anthony Aguirre, a physicist, is a co-founder of the Foundational Questions Institute, which tries to catalyze breakthrough research in fundamental physics, and the Future of Life Institute, which studies disruptive technologies like AI. Greg Laughlin, an astrophysicist, is an expert at predictions from the millisecond predictions relevant to high-frequency trading to the ultra-long-term stability of the solar system.
I've asked and answered a few questions there. It's fun, and it will get more fun as more people take it seriously! Here's some stuff from their latest report:
Dear Metaculus Users,I recently predicted — not on Metaculus — that Hillary Clinton has a 99% chance of getting the Democratic nomination. Maybe I should have said 98%. But I definitely should put my prediction on Metaculus! This could develop into a useful resource.
We recently logged our 10,000th prediction. Not quite Big Data (which will take lots more growth), but we're making progress! With this milestone passed, it seems like a good time to share an overview of our results.
First, the big picture. This can be summarized with a single histogram that shows the distribution of the first 10,042 predictions on our first 146 questions. Unambiguously, the three most popular predictions are 1%, 50% and 99%, with spikes of varying strength at each multiple of 5%. There.s a definite overall skew toward lower percentages. This phenomenon stems in part from the fact that the subset of provocative low-probability questions is most naturally worded in a way that the default outcome is negative, e.g., Question: Will we confirm evidence for megastructures orbiting the star KIC 8462852? (Answer: No.) The histogram also makes the point that while 99% confidence — the equivalent of complete confidence — is very common, it's very rare that anyone is ever 98% sure about anything. One takeaway from the pileup at 1% and 99% is that we could use more possible values there, so we plan to introduce an expanded range, from 0.1% to 99.9% soon — but as cautioned below, be careful in using it. Excluding the 1% and 99% spikes and smoothing a bit, the prediction distribution turns out to be a pretty nice gaussian, illustrating the ubiquitous effect of the law of large numbers.
The wheels of Metaculus are grinding slowly, but they grind very fine. Almost 80% of the questions that have been posed on site are still either active (open), or closed (pending resolution) We are starting, however, to get meaningful statistics on questions that have resolved to date — a collection that spans a wide range of topics (from Alpha Go to LIGO and from VIX to SpaceX). We've been looking at different metrics to evaluate collective predictive success. A simple approach is to chart the fraction of outcomes that actually occurred, after aggregating over all of the predictions in each percentage bin. In the limit of a very large number of optimally calibrated predictions on a very large number of questions, the result would be the straight line shown in gold on Figure 2 below. It's clear that the optimal result compares quite well to the aggregation produced by the Metaculus user base. Error bars are 25% and 75% confidence intervals, based on bootstrap resampling of the questions. The only marginally significant departure from the optimal result comes at the low end: as a whole, the user base has been slightly biased toward pessimism, assigning a modest overabundance of low probabilities to events that actually wound up happening. In particular, the big spike in the 1% bin in Figure 1 isn't fully warranted. (This is also somewhat true at 99%: these predictions have come true 90% of the time.) Take-away: if you're inclined to pull the slider all the way to the left or even right, give it a second thought...
It has been demonstrated that the art of successful prediction is a skill that can be learned. Predictors get better over time, and so it's interesting to look at the performance of the top predictors on Metaculus, as defined by users with a current score greater than 500. The histogram of predictions for the subset of top users shows some subtle differences with the histogram of all the predictions. The top predictors tend to be more equivocal. The 50% bin is still highly prominent, whereas the popularity of 1% votes is quite strongly diminished.
If you want to become a "super-forecaster", you need to learn about the Good Judgment Project. Start here:
For the past three years, Rich and 3,000 other average people have been quietly making probability estimates about everything from Venezuelan gas subsidies to North Korean politics as part of the Good Judgment Project, an experiment put together by three well-known psychologists and some people inside the intelligence community.Then read Philip Tetlock's books Expert Political Judgment and Superforecasting: The Art and Science of Prediction. I haven't! But I would like to become a super-forecaster.
According to one report, the predictions made by the Good Judgment Project are often better even than intelligence analysts with access to classified information, and many of the people involved in the project have been astonished by its success at making accurate predictions.
For a nice discussion of Metaculus and related issues, check out the comments on my G+ post.