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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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Canadian oil-mining town experiences global warming

The city of Fort McMurray, Canada, mainly exists thanks to petroleum mining.   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 (92 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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Bad physics

You may have heard of the "EmDrive", a gadget that supposedly provides thrust by bouncing microwaves around in a metal can.  It's sort of like trying to power a spaceship by having the crew play ping-pong

Now there's a new "theoretical explanation" of this quite possibly nonexistent effect.  It appeared on the arXiv in an unpublished paper by someone named Michael E. McCulloch.  It's completely flaky, and normally I'd ignore it, but for some reason the normally respectable mag Technology Review decided to mention it.  So people are starting to talk about it, not realizing how goofy it actually is!

McCulloch talks a lot about the Unruh effect, so you should learn a bit about that.   It's never been detected, but most physicists believe in it, because it's a consequence of special relativity and quantum mechanics.   When you put these theories together, they predict that an accelerating observer will see a faint glow of thermal radiation.

Why hasn't it been detected?   Because it's predicted to be very, very  weak.   Absurdly weak!

For example, suppose you accelerate at a trillion gee - a trillion times more than a falling object on Earth.  Then the theory predicts you'll see thermal radiation at a temperature of 40 billionths of a degree Celsius above absolute zero.   That's so faint nobody knows how to detect it!

What if you sit there watching someone else accelerate past you?  What will you see then? 

There are arguments about this, but whatever happens, it'll be too small to detect under most circumstances.  Chen and Tajima have proposed an experiment to accelerate a single electron at 10 septillion gee  (that is, 10^25 gee).  That might be enough for something interesting to happen.  However, the EmDrive gadget is nowhere near as intense. The version NASA built is weaker than a typical microwave oven.
 
This has not stopped McCulloch from claiming that the Unruh effect "explains" the EmDrive! 

He also claims it explains the rotations of galaxies, eliminating the need for dark matter.  He also claims that it explains the accelerating expansion of the Universe, eliminating the need for dark energy.  He also claims that it explains the Pioneer anomaly - a small mysterious acceleration that some spacecraft have encountered as they go far out into the Solar System. 

None of this makes any sense.  In fact, I can barely believe I'm even talking about it!  But it fooled the folks at Technology Review, so let me quote a bit of McCulloch's paper, and comment on it:

McCulloch (2007) has proposed a new model for inertia (MiHsC) that assumes that the inertia of an object is due to the Unruh radiation it sees when it accelerates [...]

So the inertial mass of an object is caused  by the Unruh radiation?   Okay... yup, that's certainly new.   Let me just say there's no evidence for this.

[...] radiation which is also subject to a Hubble-scale Casimir effect.

Oh, good, the Casimir effect!  As if things weren't confused enough already.  The Casimir effect is a very real thing: a force between very nearby metal plates, caused by the fact that the electric field can't easily penetrate a conductor.  It's a reasonably large force when the plates are a few nanometers apart, but it rapidly becomes weaker as you move them farther apart.   So now imagine they're as far apart as most distant galaxies we can see....

In this model only Unruh wavelengths that fit exactly into twice the Hubble diameter are allowed, so that a greater proportion of the waves are disallowed for low accelerations (which see longer Unruh waves) leading to a gradual new loss of inertia as accelerations become tiny.

The Hubble diameter is very roughly the size of the observable Universe.  Now he's saying that at rather small accelerations the Unruh effect is so tiny that the thermal radiation has wavelengths even larger than the size of the observable Universe.  That's true.  And that of course means that this effect is even more absurdly weak than in the example I gave. 

But he's also saying that something like the Casimir effect takes place, where the size of Universe plays the role of distance between the metal plates in the usual Casimir effect.   In other words, when an object accelerates fast enough that the Unruh radiation it sees fits inside the Universe, the Unruh effect "kicks in" and gives the object a kick, or makes its mass get bigger, or something.

Again, two things stand out: 1) it doesn't work like this, and 2) even if it did, the effect would be so tiny that... why are we even talking about it?  Even the pathetically weak thrusts the EmDrive supposedly creates - less than 100 micronewtons in the latest experiments - are like a thundering herd of giant elephants compared to what we're talking about here. 

The difficulty of demonstrating MiHsC on Earth is the huge size of [the Universe] in Eq. 1 which makes the effect very small unless the acceleration is tiny, as in deep space. One way to make the effect more obvious is to reduce the distance to the horizon and this is what the emdrive may be doing since the radiation within it is accelerating so fast that the Unruh waves it sees will be short enough to be limited by the cavity walls in a MiHsC-like manner.

So now it's the radiation inside the can that's "accelerating so fast" that it sees Unruh radiation... which is limited in wavelength by the size of the can... which somehow makes the whole can get a push when the Unruh radiation fits into the can.

In short, we've got a Rube Goldberg machine where all the parts involve brand new theories of physics with nothing backing them up, and all the actual effects cited are absurdly tiny.

But that's not all!   One amusing thing is that while the Unruh effect involves quantum mechanics, Planck's constant - the number that shows up in every calculation in quantum mechanics - never shows up in this paper.  So McCulloch is not actually doing anything with the Unruh effect!  Instead, he's making up brand new stuff, like this:

Normally, of course, photons are not supposed to have inertial mass in this way, but here this is assumed.

So his photons have mass - and on top of that, the mass changes with time: see his Equation 4!

Verdict: this paper is a stew of nonsense served with a hefty helping of warmed-over baloney.   And yet we see in the Daily Mail:

Have scientists cracked the secret of NASA’s 'impossible' fuel-free thruster? New theory could explain the EmDrive that may one day take man to Mars in 10 weeks
The same theory that explains the puzzling fly-by anomalies could also explain how the controversial EmDrive produces thrust.
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+Alex S. - yes, I have now.  It claims all the important calculations are done in another paper.
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Points at infinity

Math tells us three of the saddest love stories:

1) of parallel lines, who will never meet.
2) of tangent lines, who were together once, then parted forever.
3) and of asymptotes, who come closer and closer, but can never truly be together.

But mathematicians invented projective geometry to provide a happy ending to the first story.   In this kind of geometry, parallel lines do meet - not in ordinary space, but at new points, called "points at infinity". 

The Barth sextic is an amazing surface with 65 points that look like the place where two cones meet - the most possible for a surface described using polynomials of degree 6.  But in the usual picture of this surface, which emphasizes its symmetry, 15 of these points lie at infinity.  

In this picture by +Abdelaziz Nait Merzouk, the Barth sextic has been rotated to bring some of these points into view!  It's also been sliced so you can see inside.

You can see more of his images here:

https://plus.google.com/114982179961753756261/posts/B6zWUjNTaVr

and learn more about the Barth sextic here:

http://blogs.ams.org/visualinsight/2016/04/15/barth-sextic/
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Looks like some serious polyamory going on here!
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Barth sextic

Some mathematical objects look almost scary, like alien artifacts.  The Barth sextic, drawn here by +Craig Kaplan, is one.

In school you learned to solve quadratic equations.  Then come cubics, then quartics, then quintics.  Then come sextics, which are more sexy, and then come septics, which are downright stinky.

A sextic surface is a surface defined by a polynomial equation of degree 6. The Barth sextic is the one with the biggest possible number of ordinary double points, meaning points where it looks like a cone.  It has 65 of them! 

Even better, it has the symmetries of a dodecahedron!  20 of the double points lie at the vertices of a regular dodecahedron, and 30 lie at the midpoints of the edges of another regular dodecahedron.

Puzzle: where are the rest?  I honestly don't know.

For more pictures of this beautiful beast, including some rotating views, visit my blog Visual Insight:

http://blogs.ams.org/visualinsight/2016/04/15/barth-sextic/

The proof that the Barth sextic has the maximum possible number of ordinary double point uses the theory of codes!
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+Abdelaziz Nait Merzouk Awesome, thanks! So much harder for me to visualize than the symmetric version, but if I just keep on looking at it and trying to comprehend it...
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The crystal that nature forgot: the triamond

Carbon can form diamonds, and the geometry of the diamond crystal is stunningly beautiful.  But there's another crystal, called the triamond, that is just as beautiful.  It was discovered by mathematicians, but it doesn't seem to exist in nature.

In a triamond, each carbon atom would be bonded to three others at 120° angles, with one double bond and two single bonds. Its bonds lie in a plane, so we get a plane for each atom
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But here’s the tricky part: for any two neighboring atoms, these planes are different.  And if we draw these bond planes for all the atoms in the triamond, they come in four kinds, parallel to the faces of a regular tetrahedron!

The triamond is extremely symmetrical.  But it comes in left- and right-handed forms, unlike a diamond.

In a diamond, the smallest rings of carbon atoms have 6 atoms.  A rather surprising thing about the triamond is that the smallest rings have 10 atoms!   Each atom lies in 15 of these 10-sided rings.

When I heard about the triamond, I had to figure out how it works.  So I wrote this:

https://johncarlosbaez.wordpress.com/2016/04/11/diamonds-and-triamonds/

The thing that got me excited in the first place was a description of the 'triamond graph' - the graph with carbon atoms as vertices and bonds as edges.  It's a covering space of the complete graph with 4 vertices.  It's not the universal cover, but it's the 'universal abelian cover'. 

I guess you need to know a fair amount of math to find that exciting.  But fear not - I lead up to this slowly: it's just a terse way to say a lot of fun stuff. 

And while the triamond isn't found in nature (yet), the mathematical pattern of the triamond may be.
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really trivial
  :-)
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The right to bear arms

As you know, a lot of conservatives in the US support the right to bear arms.  It's in the Bill of Rights, after all:

"A well regulated militia being necessary to the security of a free state, the right of the people to keep and bear arms shall not be infringed."

The idea is basically that if enough of us good guys are armed, criminals and the government won't dare mess with us.

In this they are in complete agreement with the Black Panthers, a revolutionary black separatist organization founded in the 1960s by Huey P. Newton.   Later it became less active, but in 1989 the New Black Panther Party was formed in South Dallas, a predominantly black part of Dallas, Texas.  They helped set up the Huey P. Long Gun Club, "uniting five local black and brown paramilitary organizations under a single banner." 

Here you see some of their members marching in a perfectly legal manner down the streets of South Dallas.  They started doing this after the killing of Michael Brown by a policeman in Ferguson. 

From last year:

On a warm fall day in South Dallas, ten revolutionaries dressed in kaffiyehs and ski masks jog the perimeter of Dr. Martin Luther King Jr. Park bellowing "No more pigs in our community!" Military discipline is in full effect as the joggers respond to two former Army Rangers in desert-camo brimmed hats with cries of "Sir, yes, sir!" The Huey P. Newton Gun Club is holding its regular Saturday fitness-training and self-defense class. Men in Che fatigues run with weight bags and roll around on the grass, knife-fighting one another with dull machetes." I used to salute the fucking flag!" the cadets chant. "Now I use it for a rag!"

You'd think that white conservatives would applaud this "well-regulated militia", since they too are suspicious of the powers of the government.   Unfortunately they have some differences of opinion. 

For one thing, there's that white versus black business, and the right-wing versus left-wing business.  To add to the friction, the Black Panthers are connected to the Nation of Islam, a black Muslim group, while the white conservatives tend to be Christian.

It was thus not completely surprising when a gun-toting right-wing group decided to visit a Nation of Islam mosque in South Dallas.  This group has an amusingly bureaucratic name: The Bureau of American Islamic Relations.  They said:

“We cannot stand by while all these different Anti American, Arab radical Islamists team up with Nation of Islam/Black Panthers and White anti American Anarchist groups, joining together in the goal of destroying our Country and killing innocent people to gain Dominance through fear!”

So, yesterday, the so-called Bureau showed up at the Nation of Islam mosque in South Dallas.   They were openly carrying guns.

But the Huey P. Newton Gun Club expected this.  So they showed up in larger numbers, carrying more guns. 

Things became tense.  People stood around holding guns, holding signs, yelling at each other,  exercising all their constitutional freedoms like good Americans: the right of free speech, the right of assembly, the right to bear arms.

In the end, no shots were fired.  The outgunned Bureau went home. 

One of the co-founders of the Huey P. Newton Gun Club was interviewed while this was going on.  He said:

Those banditos are out of their minds if they think they're going to come to South Dallas like this.

See?  This is how the 2nd Amendment works.   For more:

https://www.rawstory.com/2016/04/armed-hate-group-backs-out-of-texas-mosque-protest-when-faced-with-gun-toting-worshipers/

http://www.vice.com/read/huey-does-dallas-0000552-v22n1
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The Tagish Lake meteorite

On January 18, 2000, at 8:43 in the morning, a meteor hit the Earth's atmosphere over Canada and exploded with the energy of a 1.7 kiloton bomb.  Luckily this happened over a sparsely populated part of British Columbia. 

It was over 50 tons in mass when it hit the air, but 97% of it vaporized.  Just about a ton reached the Earth.  It landed on Tagish Lake, which was frozen at the time.  Local inhabitants said the air smelled like sulfur.

Only about 10 kilograms was found and collected.   Except for a gray crust, the pieces look like charcoal briquettes. 

And here is where things get interesting.

Analysis of the Tagish Lake fragments show they're very primitive.   They contain dust granules that may be from the original cloud of material that created our Solar System and Sun!  They also have a lot of of organic chemicals, including amino acids.

It seems this rock was formed about 4.55 billion years ago.

Scientists tried to figure out where it came from.  They reconstructed its direction of motion and compared its properties with the spectra of various asteroids.  In the end, they guessed that it most likely came from 773 Irmintraud.

773 Irmintraud is a dark, reddish asteroid from the outer region of the asteroid belt.  It's about 92 kilometers in diameter.   It's just 0.034 AU away from a chaotic zone associated with one of the gaps in the asteroid belt created by a resonance with Jupiter.  So, if a chunk got knocked off, it could wind up moving chaotically and make it to Earth!

And here's what really intrigues me.  773 Irmintraud is a D-type asteroid - a very dark and rather rare sort.  One model of Solar System formation says these asteroids got dragged in from very far out in the Solar System - the Kuiper Belt, out beyond Pluto.   (Some scientists think Mars' moon Phobos is also a D-type asteroid.) 

So, this chunk of rock here may have been made out in the Kuiper Belt, over 4.5 billion years ago!

For more, see:

https://en.wikipedia.org/wiki/Tagish_Lake_(meteorite)
https://en.wikipedia.org/wiki/773_Irmintraud
https://en.wikipedia.org/wiki/D-type_asteroid

#astronomy  
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shift at ~22,369 mph; difficult to imagine, vital for space travel.
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Flying over Antarctica

There are lots of flights that go near the North Pole.  When you fly from California to Europe, for example, that's an efficient route!  Are there flights that go near the South Pole?   If not, why not?

A friend of mine asked this question, and I promised I'd try to get an answer.  When she flew from Argentina to New Zealand she took a very long route.  Why, she wondered, don't airplanes take a southerly route?  Is the weather too bad? 

My guess is that maybe there's not enough demand to fly from South America to New Zealand for there to be direct flights.  Or from South America to South Africa, or Madagascar. 

But I haven't even checked!  Maybe there are such flights!

Does anyone here know about this? 

(Yes, I could look it up on Google.  I thought a conversation would be more fun.  If you want to look it up, go ahead.)
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+Carsten Führmann - it still works fine here in the US.
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"And then we wept."

The chatter of gossip distracts us from the really big story: the Anthropocene, the new geological era we are bringing about.   Pay attention for a minute.  Most of the Great Barrier Reef, the world's largest coral reef system, now looks like a ghostly graveyard.  

Most corals are colonies of tiny genetically identical animals called polyps.   Over centuries, their skeletons build up reefs, which are havens for many kinds of sea life.  Some polyps catch their own food using stingers.  But most get their food by symbiosis!  They cooperate with algae called zooxanthellae.  These algae get energy from the sun's light.   They actually live inside the polyps, and provide them with food.  Most of the color of a coral reef comes from these zooxanthellae.

When a polyp is stressed, the zooxanthellae living inside it may decide to leave.  This can happen when the sea water gets too hot.  Without its zooxanthellae, the polyp is transparent and the coral's white skeleton is revealed - as you see here.  We say the coral is bleached.

After they bleach, the polyps begin to starve.  If conditions return to normal fast enough, the zooxanthellae may come back.   If they don't, the coral will die.

The Great Barrier Reef, off the northeast coast of Australia, contains over 2,900 reefs and 900 islands.  It's huge: 2,300 kilometers long, with an area of about 340,000 square kilometers.  It can be seen from outer space!

With global warming, this reef has been starting to bleach.  Parts of it bleached in 1998 and again in 2002.  But this year, with a big El Niño pushing world temperatures to new record highs, is the worst.

Scientists have being flying over the Great Barrier Reef to study the damage, and divers have looked at some of the reefs in detail.  Of the 522 reefs surveyed in the northern section, over 80% are severely bleached and less than 1% are not bleached at all.    Of 226 reefs surveyed in the central section, 33% are severely bleached and 10% are not bleached.  Of 163 reefs in the southern section, 1% are severely bleached and 25% are not bleached. 

The top expert on coral reefs in Australia, Terry Hughes, wrote:

“I showed the results of aerial surveys of bleaching on the Great Barrier Reef to my students.  And then we wept.”

Some of the bleached reefs may recover.  But as oceans continue to warm, the prospects look bleak.  The last big El Niño was in 1998.  With a lot of hard followup work, scientists showed that in the end, 16% of the world’s corals died in that event. 

This year is quite a bit hotter.

So, global warming is not a problem for the future: it's a problem now.   It's not good enough to cut carbon emissions eventually.   We've got to get serious now.  

I need to recommit myself to this.  For example, I need to stop flying around to conferences.  I've cut back, but I need to do much better.  Future generations, living in the damaged world we're creating, will not have much sympathy for our excuses.
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+Pancho Eliott - that's pretty interesting!  Letting whales flourish probably isn't enough to stop global warming, but it sounds like it will help ocean ecosystems.
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The inaccessible infinite

In math there are infinite numbers called cardinals, which say how big sets are.  Some are small.  Some are big.  Some are infinite.  Some are so infinitely big that they're inaccessible - very roughly, you can't reach them using operations you can define in terms of smaller cardinals. 

An inaccessible cardinal is so big that if it exists, we can't prove that using the standard axioms of set theory! 

The reason why is pretty interesting.  Assume there's an inaccessible cardinal K.  If we restrict attention to sets that we can build up using fewer than K operations, we get a whole lot of sets.   Indeed, we get a set of sets that does not contain every set, but which is big enough that it's "just as good" for all practical purposes.

We call such a set a Grothendieck universe.   It's not the universe - we reserve that name for the collection of all sets, which is too big to be a set.  But all the usual axioms of set theory apply if we restrict attention to sets in a Grothendieck universe.  

In fact, if an inaccessible cardinal exists, we can use the resulting Grothendieck universe to prove that the usual axioms of set theory are consistent!   The reason is that the Grothendieck universe gives a "model" of the axioms - it obeys the axioms, so the axioms must be consistent.

However, Gödel's first incompleteness theorem says we can't use the axioms of set theory to prove themselves consistent.... unless they're inconsistent, in which case all bets are off.

The upshot is that we probably can't use the usual axioms of set theory to prove that it's consistent to assume there's an inaccessible cardinal.  If we could, set theory would be inconsistent!

Nonetheless, bold set theorists are fascinated by inaccessible cardinals, and even much bigger cardinals.  For starters, they love the infinite and its mysteries.   But also, if we assume these huge infinities exist, we can prove things about arithmetic that we can't prove using the standard axioms of set theory!

I gave a very rough definition of inaccessible cardinals.  It's not hard to be precise.  A cardinal X is inaccessible if you can't write it as a sum of fewer than X cardinals that are all smaller X, and if any cardinal Y is smaller than X, 2 to the Yth power is also smaller than X. 

Well, not quite.   According to this definition, 0 would be inaccessible - and so would the very smallest infinity.   Neither of these can be gotten "from below".  But we don't count these two cardinals as inaccessible.

https://ncatlab.org/nlab/show/inaccessible+cardinal

#bigness
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Oh really +John Baez​? Fascinating! I wish you success, and lots of fun working with tech folks!
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Computing the uncomputable

Last month the logician +Joel David Hamkins proved a surprising result: you can compute uncomputable functions!  

Of course there's a catch, but it's still interesting.

Alan Turing showed that a simple kind of computer, now called a Turing machine, can calculate a lot of functions.  In fact we believe Turing machines can calculate anything you can calculate with any fancier sort of computer.  So we say a function is computable if you can calculate it with some Turing machine.

Some functions are computable, others aren't.  That's a fundamental fact.

But there's a loophole.

We think we know what the natural numbers are:

0, 1, 2, 3, ...

and how to add and multiply them.  We know a bunch of axioms that describe this sort of arithmetic: the Peano axioms.  But these axioms don't completely capture our intuitions!  There are facts about natural numbers that most mathematicians would agree are true, but can't be proved from the Peano axioms.

Besides the natural numbers you think you know - but do you really? - there are lots of other models of arithmetic.  They all obey the Peano axioms, but they're different.  Whenever there's a question you can't settle using the Peano axioms, it's true in some model of arithmetic and false in some other model.

There's no way to decide which model of arithmetic is the right one - the so-called "standard" natural numbers.   

Hamkins showed there's a Turing machine that does something amazing.  It can compute any function from the natural numbers to the natural numbers, depending on which model of arithmetic we use. 

In particular, it can compute the uncomputable... but only in some weird "alternative universe" where the natural numbers aren't what we think they are. 

These other universes have "nonstandard" natural numbers that are bigger than the ones you understand.   A Turing machine can compute an uncomputable function... but it takes a nonstandard number of steps to do so.

So: computing the computable takes a "standard" number of steps.   Computing the uncomputable takes a little longer.

This is not a practical result.  But it shows how strange simple things like logic and the natural numbers really are.

For a better explanation, read my blog post:

https://johncarlosbaez.wordpress.com/2016/04/02/computing-the-uncomputable/

And for the actual proof, go on from there to the blog article by +Joel David Hamkins.
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Lovely how the posted jpg headline resonates with the topic.   A little longer (than immediately) allows achieving the impossible, provided some adequate non-standard "liitle".
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A new polyhedron

The rectified truncated icosahedron is a surprising new polyhedron discovered by +Craig Kaplan.  It has 60 equilateral triangles, 12 regular pentagons and 20 regular hexagons as faces.

It came as a shock because it's a brand-new Johnson solid - a convex polyhedron whose faces are all regular polygons. 

Johnson solids are named after Norman Johnson, who in 1966 published a list of 92 such solids. He conjectured that this list was complete, but did not prove it.

In 1969, Victor Zalgaller proved that Johnson’s list was complete, using the fact that there are only 92 elements in the periodic table. 

It thus came as a huge shock to the mathematical community when Craig Kaplan, a computer scientist at the University of Waterloo, discovered an additional Johnson solid!

At the time, he was compiling a collection of ‘near misses’: polyhedra that come very close to being Johnson solids.  In an interview with the New York Times, he said:

When I found this one, I was impressed at how close it came to being a Johnson solid. But then I did some calculations, and I was utterly flabbergasted to discover that the faces are exactly regular! I don’t know how people overlooked it.

It turned out there was a subtle error in Zalgaller’s lengthy proof.

Or maybe not: for details see

http://blogs.ams.org/visualinsight/2016/04/01/rectified_truncated_icosahedron/
The rectified truncated icosahedron is a surprising new polyhedron discovered by Craig S. Kaplan. Two triangles, a regular pentagon and a regular hexagon meet at each vertex, and it has a total o…
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+John Baez Sure, no problem.
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I'm a mathematical physicist.
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    Visiting Researcher, 2011 - present
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    Professor, 1989 - present
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Introduction
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.
Education
  • Massachusetts Institute of Technology
    Mathematics, 1982 - 1986
  • Princeton University
    Mathematics, 1979 - 1982
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Male