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John Baez
Works at Centre for Quantum Technologies
Attended Massachusetts Institute of Technology
Lives in Riverside, California
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John Baez

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I don't like posting about politics.  I prefer math.  Math is the beautiful game of truth.  Politics is the ugly game of lies. 

But I woke up in the middle of the night and realized that if I didn't say the obvious, I might regret it:

We've got to fight Trump with everything we've got.

The US is dangerously close to electing a buffoon and would-be dictator: our very own Berlusconi, our very own Putin.  Elizabeth Warren explains it clearly, so please reshare this video:

Unfortunately, if you’ve been watching the presidential race, you know that we need to stand up now more than ever. Just yesterday, it came out that Donald Trump had said back in 2007 that he was “excited” for the real estate market to crash because, quote, “I’ve always made more money in bad markets than in good markets.” That’s right. The rest of us were horrified by the 2008 financial crisis, by what happened to the millions of families like Mr. Estrada’s that were forced out of their homes. But Donald Trump was drooling over the idea of a housing meltdown — because it meant he could buy up a bunch more property on the cheap.

What kind of a man does that? Root for people to get thrown out on the street? Root for people to lose their jobs? Root for people to lose their pensions? Root for two little girls in Clark County, Nevada, to end up living in a van? What kind of a man does that? I’ll tell you exactly what kind — a man who cares about no one but himself. A small, insecure moneygrubber who doesn’t care who gets hurt, so long as he makes some money off it. What kind of man does that? A man who will NEVER be President of the United States.

Sometimes Trump claims he is tough on Wall Street – tough on the guys who cheated people like Mr. Estrada. I’m sure you’ve heard him say that. But now he’s singing a very different song. Last week, he said that the new Dodd-Frank financial regulations have, and I’m quoting here, “made it impossible for bankers to function” and he will put out a new plan soon that “will be close to dismantling Dodd-Frank.” Donald Trump is worried about helping poor little Wall Street? Let me find the world’s smallest violin to play a sad, sad song.

Can Donald Trump even name three things that Dodd-Frank does? Seriously, someone ask him. But this much he should know: If he’s so tough on Wall Street, he should be cheering on Dodd-Frank’s capital and leverage requirements that have made big banks less likely to fail. If he’s so tough on Wall Street, he should be cheering on Dodd-Frank’s living wills process, which is helping push big banks to become safer. If he’s so tough on Wall Street, he should be cheering on the Consumer Financial Protection Bureau, which has already returned over $11 billion to families who were cheated.

He SHOULD be, but he’s not. Now that he’s sewn up the Republican nomination, Donald Trump is dropping the pretense. Now he’s kissing the fannies of poor, misunderstood Wall Street bankers. But the American people are a whole lot smarter than Donald Trump thinks they are. The American people are NOT looking for a bait and switch. They are NOT looking for a man so desperate for power he will say and do anything to get elected. Take the hint, Donald: the time for letting big banks call all the shots in Washington is coming to an end.

And I want to make just one last point about Donald Trump that won’t fit into a Twitter war. One last point that sums up what Donald Trump is all about – his taxes.

We don’t know what Trump pays in taxes because he is the first Presidential nominee in 40 years to refuse to disclose his tax returns. Maybe he’s just a lousy businessman who doesn’t want you to find out that he’s worth a lot less money than he claims.

But we know one thing: the last time his taxes were made public, Donald Trump paid nothing — zero. Zero taxes before, and for all we know he’s paying zero taxes today. And he’s proud of it. Two weeks ago he said he’s more than happy to dodge taxes because he doesn’t want to throw his money “down the drain.”

Trump likes being a billionaire, and doesn’t think the rules that apply to everyone else should apply to him. But let’s be clear: Donald Trump didn’t get rich on his own. His businesses rely on the roads and bridges the rest of us paid for. His businesses rely on workers the rest of us paid to educate and on police-forces and fire fighters who protect all of us and the rest of us pay to support. Donald Trump and his businesses are protected by a world-class military that defends us abroad and keeps us safe at home and that the rest of us pay to support. When anyone builds something terrific, they should get to keep a big hunk of it. But they should also pay a fair share forward so the next kid and the next kid and the next kid who come along gets their chance to build something too. That’s how we build a future that works for everyone.

And that goes double for Donald Trump, because he didn’t even get rich by building something terrific. He inherited a fortune from his father, and kept it going by scamming people, declaring bankruptcy, and skipping out on what he owed.

Nurses, teachers, and dockworkers pay their fair share for all the services that keep Trump’s businesses going. Programmers and engineers and small business owners pay their fair share to support our military who show courage and sacrifice every single day. Donald Trump thinks supporting them is throwing money “down the drain.”

I say we just throw Donald Trump down the drain.

Let’s face it: Donald Trump cares about exactly one thing — Donald Trump. It’s time for some accountability because these statements disqualify Donald Trump from ever becoming President. The free ride is over.

I am going to disable comments because I don't see a need for discussion.  +Elizabeth Hahn gets the last word.
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I think you just made me realize why I've always loved math so much. :) but I agree, this is very important. 

John Baez

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Logic hacking

In mathematics, unlike ordinary life, the boundary between the knowable and the unknowable is a precisely defined thing.   But finding it isn't easy.  Its exact location could itself  be unknowable.  But we don't even know that! 

This month, a bunch of 'logic hackers' have stepped up to the plate and made a lot of progress.  They've sharpened our estimate of where this boundary lies.  How?   By writing shorter and shorter computer programs for which it's unknowable whether these programs run forever, or stop.

A Turing machine is a simple kind of computer whose inner workings have N different states, for some number N = 1,2,3,...

The Busy Beaver Game is to look for the Turing machine with N states that runs as long as possible before stopping.  Machines that never stop are not allowed in this game. 

We know the winner of the Busy Beaver Game for N = 1,2,3 and 4.  Already for N = 5, the winner is unknown.  The best known contestant is a machine that runs for 47,176,870 steps before stopping.  There are 43 machines that might or might not stop - we don't know. 

When N is large enough, the winner of the Busy Beaver Game is unknowable

More precisely, if you use the ordinary axioms of mathematics, it's impossible to prove that any particular machine with N states is the winner of the Busy Beaver Game... as long as those axioms are consistent.

How big must N be, before we hit this wall?

We don't know. 

But earlier this month, Adam Yedidia and Scott Aaronson showed that it's 7910 or less. 

And by now, thanks to a group of logic hackers like Stefan O’Rear, we know it's 1919 or less. 

So, the unknowable kicks in - the winner of the Busy Beaver Game for N-state Turing machines becomes unknowable using ordinary math - somewhere between N = 5 and N = 1919. 

The story of how we got here is is fascinating, and you can read about it on my blog post:

https://johncarlosbaez.wordpress.com/2016/05/21/the-busy-beaver-game/

Anything that I didn't make clear here, should be explained there.  If it ain't clear there, ask me!

#bigness  
This month, a bunch of ‘logic hackers’ have been seeking to determine the precise boundary between the knowable and the unknowable. The challenge has been around for a long time. But on…
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+John Baez Side note: Shouldn't MathOverflow have a more appropriate name, like DivideByZero?
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John Baez

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Gently down the stream - life is but a dream

A sea otter and her sleeping pup float downstream. 

For the whole adorable video see:

https://www.youtube.com/watch?v=de6uTMEiZf0
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Here's a new news item that might melt your heart. A local otter pup has been re-united with its family after it fell into a canal [ http://www.straitstimes.com/singapore/environment/six-week-old-otter-pup-rescued-reunited-with-family-after-it-almost-drowned ].
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John Baez

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Global warming pause?

Umm, not really.  You can see how the Earth is heating up rapidly.   This great image was made by Ed Hawkins, a climate scientist at the University of Reading in the United Kingdom. 

He points out some features:

1877-78: strong El Nino event warms global temperatures

1880s-1910: small cooling, partially due to volcanic eruptions

1910-1940s: warming, partially due to recovery from volcanic eruptions, small increase in solar ouput and natural variability

1950s-1970s: fairly flat temperatures as cooling sulphate aerosols mask the greenhouse gas warming

1980-now: strong warming, with temperatures pushed higher in 1998 and 2016 due to strong El Nino events

He used temperature data from January 1850 – March 2016.  The numbers give the temperature above the average of 1850-1900.    The temperatures are from a British data set called HadCRUT4.4.  You can get that data here:

http://www.metoffice.gov.uk/hadobs/hadcrut4/

For more details, read this article on The Guardian:

http://www.theguardian.com/environment/2016/may/10/see-earths-temperature-spiral-toward-2c-rise-graphic

and check out Hawkin's website and blog:

http://www.climate-lab-book.ac.uk/2016/spiralling-global-temperatures/

Thanks to +rasha kamel for pointing this out.

#climate  
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+Johnathan Chung  One caveat. This is a big El Nino year. All expectations will be broken, but part of it is due to the ocean temperature fluctuation.

The good news is, the "Hiatusers" will have to wait for a really strong La Nina before they can do their hijinx again. Which is then officially the only good thing about the massive El Nino surge we're getting.
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The Earth is flat - and accelerating upwards

It's fun to read the frequently asked questions on the Flat Earth Society wiki.  First question:

Is this site a joke?

Answer: no, we're just diagonally parked in a parallel universe.  

Sorry - that's my answer, not theirs.  This site is not a joke.  But it's sure funny. 

How do you explain day/night cycles and seasons?

Day and night cycles are easily explained on a flat earth. The sun moves in circles around the North Pole. When it is over your head, it's day. When it's not, it's night. The sun acts like a spotlight and shines downward as it moves. The picture below illustrates how the sun moves and also how seasons work on a flat earth. The apparent effect of the sun rising and setting is usually explained as a perspective effect.

And all that stuff about the Moon's phases, and lunar and solar eclipses, was apparently set up just to fool us into thinking the Earth, Moon and Sun are round objects, with the Earth able to come between the Sun and Moon, and the Moon able to come between the Earth and Sun.

But what I really like is the explanation of gravity.  Wouldn't gravity pull the Earth into a round ball?   No:

The earth is constantly accelerating up at a rate of 32 feet per second squared (or 9.8 meters per second squared). This constant acceleration causes what you think of as gravity. Imagine sitting in a car that never stops speeding up. You will be forever pushed into your seat.

That's brilliant!   But wait a minute...

Objects cannot exceed the speed of light. Doesn't this mean that the Earth can't accelerate forever?

They've got an answer to that too:

Due to special relativity, this is not the case. At this point, many readers will question the validity of any answer which uses advanced, intimidating-sounding physics terms to explain a position. However, it is true. The velocity can never reach the speed of light, regardless of how long one accelerates for and the rate of the acceleration.

Fantastic!

What I like about this is that people can understand special relativity, yet not believe the Earth is round.  I had never encountered that combination.  I know more people who go the other way.

Of course there's the problem of what's powering this eternal acceleration.  But they have an answer to that too: it's the Universal Accelerator, also known as dark energy or the "aetheric wind".

Here's the site:

https://wiki.tfes.org/Frequently_Asked_Questions

Do not be angry.  Enjoy.
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Plato's student Aristotle knew perfectly well that the Earth was a sphere.  In his book On the Heavens he wrote:

The evidence of the senses further corroborates this. How else would eclipses of the moon show segments shaped as we see them? As it is, the shapes which the moon itself each month shows are of every kind - straight, gibbous, and concave - but in eclipses the outline is always curved: and, since it is the interposition of the earth that makes the eclipse, the form of this line will be caused by the form of the earth's surface, which is therefore spherical.

Again, our observations of the stars make it evident, not only that the earth is circular, but that it is a circle of no great size. For quite a small change of position to south or north causes a manifest alteration of the horizon. There is much change, I mean, in the stars which are overhead, and the stars seen are different, as one moves northward or southward. Indeed there are some stars seen in Egypt and in the neighborhood of Cyprus which are not seen in the northerly regions; and stars, which in the north are never beyond range of observation, in those regions rise and set.

All of which goes to show not only that the earth is circular in shape, but also that it is a sphere of no great size: for otherwise the effect of so slight a change of place would not be so quickly apparent. Hence one should not be too sure of the incredibility of the view of those who conceive that there is continuity between the parts about the pillars of Hercules and the parts about India, and that in this way the ocean is one.

That is, the Atlantic Ocean could be connected to the Indian Ocean - a possibility later explored by Columbus.

It's worth reading this:

https://en.wikipedia.org/wiki/Myth_of_the_flat_Earth

The myth of the flat Earth is the modern misconception that the prevailing cosmological view during the Middle Ages in Europe saw the Earth as flat, instead of spherical.  During the early Middle Ages, virtually all scholars maintained the spherical viewpoint first expressed by the Ancient Greeks.  From at least the 14th century, belief in a flat Earth among the educated was almost nonexistent...
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The longest G+ post I'll ever write

Sometimes math is just downright weird.  Here's an example.

A shelf is a set with a binary operation ▷ that distributes over itself:

a ▷ (b ▷ c) = (a ▷ b) ▷ (a ▷ c)

They're important in knot theory, as you can see from the picture.  But there's a strange connection between shelves, extremely large infinities, and extremely large finite numbers!

It goes like this.  Let's write 2^n for 2 to the nth power.  For each n we can make the numbers {1,2, ...,2^n} into a shelf by defining

a ▷ 1 = a + 1  mod  2^n

So, the elements of our shelf are

1
1 ▷ 1 = 2
2 ▷ 1 = 3

and so on, until we get to

2^n ▷ 1 = 1

However, we can now calculate

1 ▷ 1
1 ▷ 2
1 ▷ 3

and so on.  You should try it yourself for some simple examples!  You'll need to use the self-distributive law.  It's quite an experience.

You'll get a list of 2^n numbers, but this list will not contain all the numbers {1, 2, ... 2^n}  Instead, it will repeat with some period P(n).

And here is where things get weird.  The numbers P(n) form this sequence:

1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, ...

It may not look like it, but a logician named Richard Laver proved the numbers in this sequence approach infinity!

... if we assume an extra axiom, which goes beyond the usual axioms of set theory, but so far seems consistent! 

This axiom asserts the existence of an absurdly large infinity, called an I3 rank-into-rank cardinal..  If you read my blog post on this stuff, I'll explain what that is.  So, this is an example of how an axiom about large infinite numbers can have implications for down-to-earth math.

On the other hand, a mathematician named Randall Dougherty has proved a lower bound on how far you have to go out in this sequence to reach the number 32.

And, it's an incomprehensibly large number!  If you read my blog post, I'll explain that too:

https://johncarlosbaez.wordpress.com/2016/05/06/shelves-and-the-infinite/

So, what we've got here is a very slowly growing sequence... which is easy to define but grows so slowly that - so far, at least -mathematicians need new axioms of set theory to settle the most basic questions about it!

Like alien spores floating down from the sky, large infinite numbers can come down and contaminate the study of down-to-earth questions about ordinary finite numbers!
A shelf is a set with a binary operation $latex \rhd$ that distributes over itself: $latex a \rhd (b \rhd c) = (a \rhd b) \rhd (a \rhd c)$ There are lots of examples, the simplest being any group, …
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Linas Vepstas's profile photoHervé Perdry's profile photoJohn Baez's profile photofabian lizcano's profile photo
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+Laura Caro Sáenz Here is a great example of a weid thing with the topics you love.
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A windy day

This is a tornado that occurred in south-central Oklahoma on May 9th.   It's impressive what moist air can do when it's not in thermal equilibrium.

At first I couldn't remember where I found this video.  But Chris Greene found it:

https://www.youtube.com/watch?v=K1R_N_pysRs

Watch the whole thing!  At the end you'll see a car driving dangerously close to an oncoming tornado.  I wonder how that worked out.
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Michael Verona's profile photoJohn Baez's profile photoAllen Knutson's profile photoKram Einsnulldreizwei's profile photo
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fair enough :)
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Mountain biking is fun, but this is ridiculous!

This is Danny MacAskill on the Inaccessible Pinnacle on the Isle of Skye.

He is a great mountain biker,  but he had to carry the bike up the last part of this scary peak.

The Isle of Skye is an island off the west coast of Scotland.   It's the largest of the Inner Hebrides, and the most northerly of the large islands in this group.    In the center of this island is a mountain range called the Cuillin, and the Inaccessible Pinnacle sits among these.

Skye has been occupied since Mesolithic times, and it appears in Norse poetry, for example in this romantic line:

"The hunger battle-birds were filled in Skye with blood of foemen killed."

Almost a third of the inhabitants still speak Gaelic, and apart from a few bigger towns, the population lives in crofting townships scattered around the coastline.  "Crofting"?  Yeah, a croft is a small farm with a wall around it.  

The only distillery on the Isle of Skye is the Talisker Distillery, which makes a rather famous single malt Scotch whisky.  It's in a village on the south shore.

I've always been fascinated by the Inner Hebrides and the even more exotic-sounding Outer Hebrides.  I'm annoyed at how all my visits to the British Isles have only taken me to the lofty centers of academe, not places like this.  I don't know much about them, but anything remote appeals to me: inaccessible pinnacles, inaccessible cardinals, the Taklamakan desert, the underground oceans of Europa....

Danny Macaskill is actually from the Isle of Skye!  You can see his whole journey along the Cuillin Ridgeline here:

https://www.youtube.com/watch?v=xQ_IQS3VKjA

Pretty impressive!  Beautiful scenery, too!
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Tom Lowe's profile photoJohn Baez's profile photoViv Kendon's profile photoRoss Duncan's profile photo
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+John Baez That is shame. Do get in touch if you're in Scotland. (btw next QPL will probably be in North America....)
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A mathematical mystery

Russian mathematicians have discovered a mysterious connection between waves and a shape called the icosahedron.   You can see it in this image by Marshall Hampton.   It shows a wave going around an obstacle. 

The obstacle, in red, has an inflection point in the middle.  In other words, it curves down  on one side of this point, but curves up  on the other side. 

The blue curve below the red curve is the front of a wave.  As this wave front moves past the inflection point, something interesting happens.  It crosses over itself, and two sharp points form!

Together with some friends, the famous Russian mathematician Vladimir Arnol'd discovered that these two sharp points are different.  One is more pointy than the other!   The top one is a cusp of order 3/2, meaning it's described by an equation like

y³ = x²

The bottom one is a cusp of order 5/2, meaning it's described by an equation like

y⁵ = x²

More precisely, you can get these equations after you do a change of coordinates. 

Arnol'd and friends also discovered something else.  The blue curve, as it moves along, traces out a surface that you can also get starting from an icosahedron!   As Arnol'd wrote:

Thus the propagation of the waves is controlled by an icosahedron hidden at the inflection point of the boundary. This icosahedron is hidden, and it is difficult to find it even if its existence is known.

Here's a hint of how it works.  An icosahedron has 5 triangles meeting at each corner.  The "5" here gives the cusp of order 5/2, while the "3" hiding in the word "triangle" gives the cusp of order 3/2!

You can read a much better explanation here:

http://blogs.ams.org/visualinsight/2016/05/01/involutes-of-a-cubical-parabola/

and here:

http://blogs.ams.org/visualinsight/2016/05/15/discriminant-of-the-icosahedral-group/

But it's still mysterious to me! 

First of all, it seems the proof was never written up.  Secondly, there's the question of "what does it all really mean?"  I don't think anyone knows. 

So I plan to get to the bottom of this....

#geometry  
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Abdelaziz Nait Merzouk's profile photoJohn Baez's profile photo
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+Abdelaziz Nait Merzouk - yes, that's where we figured out a lot of the stuff I wrote about on Visual Insight.  There's a third Visual Insight article coming on this set of ideas: Arnold also claims there's a connection to a certain class of quintics.
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Fecht Yeah

Kevin Kelly has claimed that "tools never die" - that any tool ever made is still being made somewhere.   There are interesting arguments about this online.  You can find videos on how to make stone hand axes.  You can find instructions on how make a calcium oxide light - the old-fashioned "limelight" used in theaters until it was replaced by electric arc lamp in the 1890s.  

And you can certainly buy a longsword.  That's a sword with a long double-edged blade and a cross-shaped handle, as shown here.  They reached their height of popularity from 1350 to 1550.  But people still fight with them - mainly for fun.

In fact, this weekend on Staten Island there's a course for women who want to fight with longswords!   And there's a tournament, too!  It's called Fecht Yeah, and it's probably not too late to register.  Bring your weapon.

It's part of the Historical European Martial Arts movement, or HEMA.  Here's the ad:

A weekend of training, learning, and collaboration for women who study HEMA and other sword arts.

This is an event for women of all skill levels with varied interests to come together and develop their skills. Workshops for beginners will be available. Free from tournament pressure and the constraints of classes, we have the ability to workshop teaching methods, rulesets, and learning strategies with other dedicated practitioners.

We will have laurel tournaments in longsword, sword and buckler, rapier, and saber. Prizes will be modest. Attend to learn, not win.

I'm an absurdly nonviolent guy, who will pick up a spider and take it outside rather than squash it.   But I admire skills like sword-fighting, and I'm glad people are keeping those skills alive.   Why?  I'm not completely sure.  I could theorize about it, but never mind.

Check out this video of German longsword fighting:

https://www.youtube.com/watch?v=5zueF4Mu2uM

Register here:

http://www.hema.events/aboutfy/

As you might expect, female swordfighters get flack from some male ones.  There's a nice article about Fecht Yeah here, and it get into that a bit:

http://www.villagevoice.com/news/fecht-club-new-yorks-women-warriors-kick-ass-8601021

Tiby Kantorowitz, one of the women running Fecht Yeah, treats swordfighting as a spiritual exercise:

"It's the flip side to yoga.   It's easy to Zen out with twinkly music, incense, and soft light. But can I maintain the same equanimity when there's some six-foot guy" — she's four-ten — "with a sword who's trying to brain me?"

The woman in this picture is Laura McBride, and she was photographed by Brad Trent.

For Kevin Kelly's claim, try:

http://www.npr.org/templates/transcript/transcript.php?storyId=133188723

KRULWICH: And then he made this ridiculous bet. He said: I bet you can't find any tool, any machine - go back to any century you like - that still isn't being made and made new today. So all I have to do is find a single tool that's not being made anymore, and I win.

(Soundbite of laughter)

KELLY: Yes, that's right.

KRULWICH: You're so going to lose this.

And then the show explores this....
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Matthew Rapaport's profile photoJohn Baez's profile photoAndreas Geisler's profile photoPrime Lemur's profile photo
63 comments
 
I think its curious that we retain such knowledge ... but it is perhaps our very curiosity that lends itself to the preservation of such artisanal techniques.

Look at flint-knapping. The art has been re-invigorated by archaeologists keen to understand the requirements ... the material, the techniques, the costs and benefits of knapping ... in order to place it into the context of early tool manufacturing.

At different times, the technologies of tool making have been fiercely guarded by the practitioners ... in The Making of the English Working Class, historian EP Thompson discusses how the cutlery makers of Sheffield protected their trade, and instead of cutlery making spreading to the growing hubs of mechanisation, steel making and processing were brought to Sheffield, changing the city and industry from a cottage economy to a national, then global powerhouse.
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Weapons of math destruction

This Thursday evening, a 40-year-old man with dark, curly hair, olive skin and an exotic foreign accent boarded a plane in the United States.  It was a regional jet making a short hop from Philadelphia to nearby Syracuse.

The curly-haired man tried to keep to himself, intently scribbling mysterious symbols on a notepad he’d brought aboard. His seatmate, a blonde, 30-something woman sporting flip-flops and a red tote bag, looked him over. He was wearing navy Diesel jeans and a red Lacoste sweater – but something about him didn’t seem right.

She decided to try out some small talk.

"Is Syracuse home?"

"No," he replied curtly.

He similarly deflected further questions. He appeared laser-focused — perhaps too laser-focused — on the task at hand, those strange scribblings.

She became suspicious.   Maybe it was code, or something in in Arabic — maybe the details of a plot to destroy the dozens of innocent lives aboard their flight!    Just to be safe, she decided, it was her duty to alert the authorities.

And so she did.  

And so the suspicious-looking man was taken off the plane, and interrogated. 

[I've paraphrased part of the article.  To hear what happened, read the rest.   Thanks to +Jenny Meyer for pointing this out.   My title is stolen from Cathy O'Neil's book, which you can obtain here:

http://www.amazon.com/Weapons-Math-Destruction-Increases-Inequality/dp/0553418815

It's about the abuses of math in the financial system.  There are economists who should be in big trouble for what they do with math.]
Move over, Clock Boy. Another swarthy-looking nerd is alarming the authorities.
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John Baez's profile photoMichael Tetelman's profile photoBenjamin Hardisty's profile photoRongmin Lu's profile photo
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+Benjamin Hardisty​​​ Sounds more like liability aversion and a class action waiting to happen.
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Canadian oil-mining town experiences global warming

The city of Fort McMurray, Canada, mainly exists thanks to petroleum mining.   And thanks to humans burning carbon, this year is the hottest on record, especially in the far north.  The climate around Fort McMurray is borderline Arctic... but on Tuesday the temperature soared over 32 Celsius (91 Fahrenheit).   Thanks to the heat and wind, a huge fire engulfed the city.   A resident said:

"It was the most terrifying feeling looking straight ahead at a wall of flames 10 times higher than us.   I was in a complete state of shock and fear.   The streets were in a panic, people were abandoning their vehicles and hitchhiking."

Most news stories aren't pointing out the connection here, or the sad irony.

http://www.cnn.com/2016/05/04/world/fort-mcmurray-fire-canada/
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I'm a mathematical physicist.
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  • Centre for Quantum Technologies
    Visiting Researcher, 2011 - present
  • U.C. Riverside
    Professor, 1989 - present
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I teach at U. C. Riverside and work on mathematical physics — which I interpret broadly as ‘math that could be of interest in physics, and physics that could be of interest in math’. I’ve spent a lot of time on quantum gravity and n-categories, but now I want to work on more practical things, too.

Why? I keep realizing more and more that our little planet is in deep trouble! The deep secrets of math and physics are endlessly engrossing — but they can wait, and other things can’t.

So, I’ve cooked up a plan to get scientists and engineers interested in saving the planet: it's called the Azimuth Project.  It includes a wiki, a blog, and a discussion forum.  I also have an Azimuth page here on Google+, where you can keep track of news related to energy, the environment and sustainability.

Check them out, and join the team!  Or drop me a line here.
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  • Massachusetts Institute of Technology
    Mathematics, 1982 - 1986
  • Princeton University
    Mathematics, 1979 - 1982
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