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Shaun Martin
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Maryam Mirzakhani won the Fields medal yesterday.

As a child in Tehran, she didn't intend to become a mathematician - she just wanted to read every book she could find!  She also watched television biographies of famous women like Marie Curie and Helen Keller.  She started wanting to do something great... maybe become a writer.

She finished elementary school while the Iran-Iraq war was ending, and took a test that got her into a special middle school for girls.  She did poorly in math her first year, and it undermined her confidence.  “I lost my interest in math," she said.

But the next year she had a better teacher, and she fell in love with the subject.  She and a friend became the first women on Iranian math Olympiad team.  She won a gold medal the first year, and got a perfect score the next year.

After getting finishing her undergraduate work at Sharif University in Tehran in 1999, she went on to grad school at Harvard.  There she met Curtis McMullen, a Fields medalist who works on hyperbolic geometry and related topics.

Hyperbolic geometry is about curved surfaces where the angles of a triangle add up to less than 180 degrees, like the surface of a saddle.  It's more interesting than Euclidean geometry, or the geometry of a sphere.  One reason is that if you have a doughnut-shaped thing with 2 or more holes, there are many ways to give it a hyperbolic geometry where its curvature is the same at each point.  These shapes stand at the meeting-point of many roads in math.  They are simple enough that we can understand them in amazing detail - yet complicated enough to provoke endless study.

Maryam Mirzakhani took a course from McMullen and started asking him lots of questions.  “She had a sort of daring imagination,” he later said.  “She would formulate in her mind an imaginary picture of what must be going on, then come to my office and describe it. At the end, she would turn to me and say, ‘Is it right?’ I was always very flattered that she thought I would know.”

Here's a question nobody knew the answer to.  If an ant walks on a flat Euclidean plane never turning right or left, it'll move along a straight line and never get back where it started.  If it does this on a sphere, it will get back where it started: it will go around a circle.  If it does this on a hyperbolic surface, it may or may not get back where it started.  If it gets back to where it started, facing the same direction, the curve it moves along is called a closed geodesic.  

The ant can go around a closed geodesic over and over.  But say we let it go around just once: then we call its path a simple closed geodesic.    We can measure the length of this curve.  And we can ask: how many simple closed geodesics are there with length less than some number L?

There are always only finitely many - unlike on the sphere, where the ant can march off in any direction and get back where it started after a certain distance.  But how many?

In her Ph.D. thesis, Mirzakhani figured out a formula for how many.  It's not an exact formula, just an 'asymptotic' one, an approximation that becomes good when L becomes large.  She showed the number of simple closed geodesics of length less than L is asymptotic to some number times L to the power 6g-6, where g is the number of holes in your doughnut. 

She boiled her proof down to a 29-page argument, which was published in one of the most prestigious math journals:

• Maryam Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces, Annals of Mathematics 168 (2008), 97–125, http://annals.math.princeton.edu/wp-content/uploads/annals-v168-n1-p03.pdf.

This is a classic piece of math: simple yet deep.  The statement is simple, but the proof uses many branches of math that meet at this crossroads. 

What matters is not just knowing that the statement is true: it's the new view of reality you gain by understanding why it's true.   I don't understand why this particular result is true, but I know that's how it works.  For example, her ideas also gave here a new proof of a conjecture by the physicist Edward Witten, which came up in his work on string theory!  

This is just one of the first things Mirzakhani did.  She's now a professor at Stanford.

"I don't have any particular recipe," she said.  "It is the reason why doing research is challenging as well as attractive. It is like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck you might find a way out."

She has a lot left to think about.  There are problems she has been thinking about for more than a decade. "And still there’s not much I can do about them," she said.

"I can see that without being excited mathematics can look pointless and cold. The beauty of mathematics only shows itself to more patient followers."

I got some of my quotes from here:

http://www.simonsfoundation.org/quanta/20140812-a-tenacious-explorer-of-abstract-surfaces/

and some from here:

http://www.theguardian.com/science/2014/aug/13/interview-maryam-mirzakhani-fields-medal-winner-mathematician

They're both fun to read.

#spnetwork doi:10.4007/annals.2008.168.97 #geometry #mustread
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The quantum geometry of water

You may have heard that water has a pH of 7.  This means that for every 10,000,000 water molecules, one splits into a negatively charged OH⁻ and a positively charged H⁺.  There are 7 zeros in that number - so, pH 7.

An H⁺ is a hydrogen atom missing its electron: a proton, all by itself! 

What happens when you've got a lone proton in water?   It quickly attracts two water molecules, as shown in the middle of this picture.  Then you've got H₅O₂⁺.  Then each of these water molecules attracts two more, giving H₁₃O₆⁺.  That's what you see inside the dashed line!

The game doesn't stop there: more water molecules get attracted to this thing.  But the dashed line roughly marks the edge of the region where the positive charge can roam.

What?  Didn't I say the lone proton was right in the middle?  Isn't that what the picture shows - the H in the middle? 

Well, the picture is a bit misleading!   First, everything is wiggling around a lot, very quickly.  And second, quantum mechanics says we don't know the position of that proton precisely!  Instead, it's a 'probability cloud' smeared over a large region, ending roughly at the dashed line.  You can't say precisely where a cloud ends....

At least this is what these scientists claimed in 2010, based on infrared spectroscopy:

• Evgenii S. Stoyanov, Irina V. Stoyanova, Christopher A. Reed, The unique nature of H⁺ in water, Chemical Science 2 (2011), 462-472, http://pubs.rsc.org/en/content/articlelanding/2011/sc/c0sc00415d#!divAbstract.

(Unfortunately the paper is not free.) 

The structure of water is complicated, and there have been many arguments about it.  But we know the basic laws that govern it, and I think we're finally getting enough computational power to simulate it and settle some of these arguments.

But it's not easy.  +Kieron Taylor did his PhD work on simulating water, and he wrote:

It's a most vexatious substance to simulate in useful time scales. Including the proton exchange or even flexible multipoles requires immense computation.

It would be very interesting if the computational complexity of water were higher, in some precise sense, than many other liquids.  It's a weird liquid in other ways.  It takes a lot of energy to heat water, it expands when it freezes, and water molecules have a large 'dipole moment' - meaning the electric charge is distributed in a very lopsided way, thanks to the 'Mickey Mouse' way the two H's are attached to the O.

Happy Thanksgiving!  I'm thankful for the weird properties of water, which make life on this planet possible.

#chemistry  
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A brief tribute to a great and influential scientist. I'll explain later. [Update: Okay, let me explain. That is a DNA sequence. Every set of three letters codes for an amino acid, and each amino a...
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Sous-vide pork A/B test:

On the left: Iberico - slightly more gamey, awesome texture,

On the right: Twee Bak - slightly more meaty, gelatinous, good bite.

Which one wins at the end of the day? Iberico, but only by the slimmest of margins :-)
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Third wave coffee reaches my neighbourhood.
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Room temperature semiconductors will enable all kinds of awesome technology. Can't wait!
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Thanks for that reference! Yes, it does appear that their material is unstable. My reading is that they're inferring superconductivity in a "small volume fraction" of the material they constructed by measuring specific heat, and that their measurements needed to be performed "immediately after annealing" (because the material is strongly hygroscopic.) What do you think of this interpretation?
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An inspirational quote needs no explanation...
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"... Giving up can be a natural — indeed, a rational — response to a time frame that wasn’t properly framed to begin with, according to a series of new studies conducted by Mr. Kable’s decision neuroscience lab at the University of Pennsylvania and published in Cognition and Psychological Review.

“There are lots of situations, probably the majority of situations, in the real world,” Mr. Kable told me, “where waiting longer is actually a valid cue that the reward is getting further and further away.”

Mr. Kable and Mr. McGuire tested this logic on a group of shoppers in a mall in New Jersey. As people went about their usual routine, some of them were asked to take part in a 10-minute study during which they could make between $5 and $10. Study participants would see a yellow light appear on a computer screen and could choose to do one of two things: Keep their mouse cursor over a box marked “wait for 15 cents” or move the cursor to a second box marked “take one cent.” What they didn’t know was how long they would have to wait if they opted for the promise of more money. In some cases, the larger rewards were given at relatively regular intervals. In others, however, the timing was more uncertain: the longer you waited, the larger the chance you’d have to keep right on waiting.

The researchers found that while the shoppers seeing the regular intervals looked like the very model of persistence and self-control, those seeing the erratic intervals grew increasingly less persistent over time — even if they had initially been quite patient. The uncertainty of the reward timing was itself enough to push them toward behavior that looked increasingly impulsive."
Is our sense of time, not our lack of willpower, the real issue?
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Quantum corrections to Newton's law of gravity

Newton said the force of gravity drops off as the inverse square of the distance.  But Einstein's theory of gravity says that's not quite true.  And quantum mechanics gives further corrections.  That's why this formula has three terms in the brackets!  And there are more, which we can't calculate yet...

We don't fully understand how quantum mechanics and gravity fit together.   But with a lot of hard work, people have been able to figure out the first quantum correction to Newton's inverse square force law.   It's a hard calculation - you can see that from the weird number 41/10π.   People got it wrong a few times before getting it right.

The formula here is not for the force of gravity.  It's the formula for the potential energy due to gravity when you have two masses, m1 and m2, at rest a distance r from each other.  G is Newton's constant describing the strength of gravity.

Newton's theory says the potential energy is:

V = - G m1 m2 / r

This is the first term in the formula.  This gives Newton's inverse square force law.

The second term involves the speed of light, c.   This comes from Einstein's theory - general relativity.  This term has an extra r in the denominator, so it gives an inverse cube correction to Newton's force law.

The third term also involves Planck's constant, ħ.  This comes from quantum mechanics!   When we take quantum mechanics into account, the force of gravity is carried by particles called gravitons.   One mass can send a single graviton to another; this gives Newton's law.  But fancier things can happen, too!  These give a quantum correction to Newton's law - the third term here.

These fancier things are hard to explain, but let me try.  A mass can send two gravitons to another, one after another, and this gives part of the third term, with the number -47/3π in front.  A mass can also send off two gravitons at the same time!  If they reach the other mass at different times we get another part of the third term, with 28/π in front.  If they reach the other mass at the same time we get -22/π.  And a graviton can also split into two gravitons en route.  This gives another part of the answer, namely 7/π.  And finally, if a graviton splits in two and the two gravitons rejoin to form a single graviton we get yet another part of the third term, with the number -43/30π in front.

Adding these up we get

-47/3π + 28/π - 22/π + 7/π - 43/30π = 41/10π

You can see pictures of the different graviton processes here:

• N. E. J. Bjerrum-Bohr, John F. Donoghue and Barry R. Holstein, Quantum gravitational corrections to the nonrelativistic scattering potential of two masses, http://arxiv.org/abs/hep-th/0211072.

If you look at earlier papers you'll see different answers.  People made lots of mistakes.  But in 2002 two separate teams redid the calculations and got the same answer!  The other team is:

• I. B. Khriplovich and G. G. Kirilin, Quantum power correction to the Newton law, http://arxiv.org/abs/gr-qc/0207118

I believe the higher quantum corrections to Newton's law of gravity cannot be computed without choosing a theory of quantum gravity.  These will be proportional to higher powers of Planck's constant, ħ. It's a minor miracle that the correction proportional to ħ can be computed using what we know today.

#spnetwork #recommend arXiv:hep-th/0211072
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Alfred Russell Wallace died 100 years ago today. Wallace discovered the principle of evolution by natural selection, independently of, and at the same time as Charles Darwin.

From the ages of 31 to 39, Wallace explored what is today Singapore, Malaysia and Indonesia. He returned to England in 1862 with more than 126,000 specimens, including the gliding tree frog Rhacophorus nigropalmatus, known as Wallace's flying frog.

In his 1869 book "The Malay Archipelago" Wallace writes:

"The island of Singapore consists of a multitude of small hills, three or four hundred feet high, the summits of many of which are still covered with virgin forest. The mission-house at Bukit-tima was surrounded by several of these wood-topped hills, which were much frequented by woodcutters and sawyers, and offered me an excellent collecting ground for insects. Here and there, too, were tiger pits, carefully covered over with sticks and leaves, and so well concealed, that in several cases I had a narrow escape from falling into them. They are shaped like an iron furnace, wider at the bottom than the top, and are perhaps fifteen or twenty feet deep so that it would be almost impossible for a person unassisted to get out of one. Formerly a sharp stake was stuck erect in the bottom; but after an unfortunate traveller had been killed by falling on one, its use was forbidden. There are always a few tigers roaming about Singapore, and they kill on an average a Chinaman every day, principally those who work in the gambir plantations, which are always made in newly-cleared jungle. We heard a tiger roar once or twice in the evening, and it was rather nervous work hunting for insects among the fallen trunks and old sawpits when one of these savage animals might be lurking close by, awaiting an opportunity to spring upon us."
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