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Number Theory
Primed for Success, Smithsonian
Special Issue, Fall 2007, 74-75.
Terence Tao has made one of the most
difficult of all transitions ¾
he is the rare prodigy who lives up to his
early promise. At age 32, he has already won a MacArthur "genius grant"
and the most prestigious prize in math, the Fields Medal, and has made
major advances in several branches of mathematics. And he's still just
beginning.
'Cranky' Proof Reveals Hidden
Regularities, Science, 1 April 2005, 36-37.
The fallout from Andrew Wiles' 1994
proof of Fermat's Last Theorem will probably last for decades. A group
of three mathematicians uses Wiles-like methods, together with an
elementary "binning" procedure, to uncover some neat patterns in a
classical function of number theory, the partition function.
Hardy's Prime Problem Solved, New
Scientist, 8 May 2004, 13.
A prime progression is a
sequence of prime numbers that are all equally spaced -- such as 5, 17, 29,
41, 53. Mathematicians have have never found one with more than 22 terms.
Now, two young number theorists have proved that prime progressions of
any length exist.
Prime Proof Helps
Mathematicians Mind the Gaps,
Science, 4 April 2003, 32.
Dan Goldston of the U.S. and
Cem Yildirim of Turkey prove that prime numbers are like weeds -- they occur
in clumps. Unfortunately, since this article was published a gap has
appeared in their proof, so its current status is unclear. Come back to this
page for updates. [July 2005 update: Goldston's work was finally
vindicated a month or two ago, with an even simpler proof.]
Geometry/Topology
A Singular Career,
Swarthmore Alumni Bulletin, March 2007
Bob MacPherson's
career as a mathematician has been "singular" in three
different ways. He invented intersection homology; he
helped rescue Russian mathematics in the economic crisis
after the fall of the Soviet regime; and he and Mark
Goresky (the Rogers and Hammerstein of mathematics) have
become one of the most successful gay couples in
mathematics. Too bad a lot of gays aren't aware of their
accomplishments.
Mapping the 248-Fold Way,
Science, 23 March 2007.
Mathematicians, with a
huge help from computers, succeed in computing a
representation table for E8, the most
mysterious and exotic of all Lie (pronounced "lee")
groups. This may be relevant to physical theories of the
universe ¾ we probably won't
know for a long time. For mathematicians interested in
different kinds of symmetry, it is something like the
scaling of Mt. Everest or the completion of the Human
Genome Project.
The Poincaré Conjecture
— PROVED, Science, 22 December 2006.
Science's
editors picked Grigori Perelman's proof of the Poincaré Conjecture as
the Breakthrough of the Year for 2006. I wrote the cover article about
it. This gave me a chance to survey and put into context the events of
the preceding four years. You'd think that mathematicians would be
overjoyed at seeing one of their greatest unsolved problems conquered
(and they are), but it has been a bittersweet harvest because of the
personal rivalries that have been stirred up.
Perelman Declines
Math's Top Prize; Three Others Honored in Madrid, Science, 25 August
2006.
Grigori
Perelman created a huge stir when he declined the Fields Medal (the
mathematical equivalent of a Nobel Prize). The reclusive Russian
mathematician was the first person in history to do so. However, three
other young mathematicians accepted their well-deserved medals from
Spain's King Juan Carlos: Andrei Okounkov of Princeton University,
Terence Tao of UCLA, and Wendelin Werner of Université de Paris-Sud.
Taming the Hyperbolic Jungle by Pruning
its Unruly Edges,
Science, 24 December 2004.
Are you a fan of fractals? Find out
what the first fractals were invented for, back in the 1800s. Then learn
how topologists are now using them to pin down the geometry of
hyperbolic (negatively curved) 3-dimensional universes. Or if that is a
little too mind-boggling, just look at the amazing pictures.
Mathematics World Abuzz Over Possible
Poincaré Proof, Science, 18 April 2003, 417.
A mathematician goes into
relative seclusion for seven years and emerges with a claim that he can
prove one of the most famous problems in mathematics. Sound familiar? It
happened 10 years ago with Fermat's Last Theorem, and now it has happened
again with the Poincaré Conjecture. But mathematicians, burned often in the
past, are being verrry verrry cautious about this claim.
A Fine Mess,
New
Scientist, 25 May 2002, 32-35.
Packing oranges
together is pretty simple in ordinary Euclidean space, but in hyperbolic
space it gets seriously weird. Charles Radin of the University of Texas
figures out how weird.
Applied Mathematics
All Stirred Up, New Scientist, 1
December 2007, 54-57.
Turbulently flowing fluids appear to
be complex and unpredictable. But they aren't, if you can detect the "Lagrangian
coherent structures" concealed in them, which direct where fluid
particles go. Physicists can now do this in near-real time. Applications
include pollution control, detection of clear-air turbulence by LIDAR,
and detection of abnormal blood flow that leads to the formation of
athereosclerotic plaques.
Mathematicians Confront
Climate Change, SIAM News, June 2007, 1.
A symposium on climate change at the
Mathematical Sciences Research Institute, in Berkeley, CA, asks what
contributions mathematicians can make towards understanding the process
of climate change and finding solutions to the problem of global
warming.
Tilt!,
Discover, July 2005, 36-37.
A mathematician thinks he knows why
certain buildings topple over during earthquakes and others don't. The
answer lies partly in an ancient book of Archimedes, and partly in
modern catastrophe theory.
Topologists Take Scalpel to Brain Scans,
SIAM News, September 2004, 1-11.
Computer-rendered 3-dimensional MRI
images are a valuable tool to brain surgeons. Unfortunately, the scans
often contain glitches that mess up the spherical topology of the image,
and wreak havoc with automated brain-mapping programs. A group of
mathematicians has now found and proved a simple algorithm for cleaning
the images up.
Ensemble Kalman Filters Bring Weather Models Up to Date,
SIAM NEWS, October 2003.
Meteorology isn't rocket
science, is it? You bet it is! This article explains how Kalman filters, a
"data assimilation" method used to keep rockets on course, can help predict
the weather. In 1999, the winter storm Lothar wreaked havoc on the gardens
of Versailles and killed more than 100 people in Europe. It caught the
British and French weather services totally off guard. But the possibility of such a severe storm could have been anticipated with
ensemble Kalman filters.
Tailor-Made Vision Descends to
the Eye of the Beholder, Science, 14 March 2003, 1654-5.
Adaptive optics, designed to
help telescopes see more sharply through atmospheric distortions, are now
being used to correct subtle aberrations in the eye. One application you may
have heard of -- or will soon -- is "wavefront LASIK." Adaptive optics can
also be used to look into the eye and study eye disease at a cellular
level.
The Science of Surprise,
Discover, February 2002, 58-63.
Complexity theory
enables biologists to model the flocking of birds, companies to control
tightly interwoven supply chains, ... and insurers to deal with the
ramifications of an unforeseen catastrophe like September 11. This was one
of my most difficult articles to write, as the events of September 11
dramatically changed what it was going to be about.
Wavelets: Seeing the Forest -- and the Trees,
Beyond Discovery, National Academy of Sciences.
Wavelets are a
beautiful example of mathematics with significance far beyond its original
context. They are used to process a signal into components that are compact
both in time and in frequency. The original applications were in audio, but
the really exciting ones are in video, where you can build up an image at
different scales of resolution. If you've seen any recent digitally animated
movie (e.g., A Bug's Life), you've seen what wavelets can do.
Mathematical Biology
Ramping Up To
Multiscale, Biomedical Computation Review, Spring 2006.
Computer
scientists and biologists have made huge strides in designing realistic
models of the human body, from individual cells to whole organs. The
most difficult thing to do, though, is to integrate models that work at
different scales. For example, how does the flow of ions through individual cells
precipitate an arrhythmia of the whole heart? This article describes
some approaches that work.
No Age Limits: Can Mathematical Models of Fish Shed Light on Human Aging?
SIAM News, April 2005, 4-6.
The rockfish is an amazing fish: Some
species live up to 200 years, but others live for only a dozen years or
so. Mathematical models suggest that the variety of life spans evolved
because they enabled the species to avoid competing with one another.
Not too much here about humans, actually, in spite of the title.
Making Sense of Stents,
SIAM News, May
2004, 1-3.
In less than 20 years, stents
(which prop open an occluded artery) have gone from an experimental
procedure to a routine treatment for cardiovascular disease. But about a
quarter of stented arteries eventually clog up again. Mathematicians are
developing models that explain why and what we can do about it.
Making Sense of a Heart Gone Wild,
Science, 6 February 2004.
You've seen automatic
defibrillators in medical dramas on TV. You know that they bring people back
to life who would otherwise die in minutes. But did you realize that no one
understands why they work? Mathematical models have identified two possible
explanations. The models may also help engineers design pain-free
implantable defibrillators for people who are susceptible to arrhythmias.
Cryptology
Communication Speed Nears Terminal
Velocity, New Scientist, 9 July 2005.
All communications, whether you're
downloading files over the Internet or radioing instructions to a
satellite, have to obey a fundamental speed limit known as the Shannon
limit. Try to go faster and your data will be garbled. Engineers have
now developed not one but two codes that allow communication speeds
within an eyelash of the Shannon limit.
The Code War,
Beyond Discovery, National
Academy of Sciences
With the arrival of the
Internet and electronic commerce, codes are no longer the exclusive province
of spies and generals: They are part of our everyday life. Today's
hardest-to-crack codes are based on the mathematics of number theory.
Tomorrow's may exploit the quantum structure of the universe.
Combinatorics
Graph Theory Uncovers
the Roots of Perfection, Science, 5 July 2002, 38.
Some graphs (or
networks) are easier to "color" than others. Mathematicians now have a
simple test for perfect graphs, a large class of graphs that are optimal for
coloring problems. Claude Berge, who originally proposed this test and was
one of the big names in graph theory, heard about the proof of his
conjecture on his deathbed. (He died just before this article came out.)
Statistics
Vital Statistics, New Scientist, 26
June 2004, 36-41.
The "hard sciences" of physics
and astronomy seem like the last place where you would have to deal with the
ambiguities of statistics. But physicists and astronomers are gradually (and
sometimes reluctantly) learning from the "soft sciences" how to deal with
the ambiguities in their data. Three case studies: the Higgs particle, dark
matter, and neutrino oscillations.
Computer Science
What in the Name of Euclid is Going On
Here? Science, 4 March 2005, 1402-3.
Computerized proof verifiers have been
mostly scorned or ignored by mathematicians, but they are starting to
get to the point where they can check the proofs of serious theorems.
They may be just in time, because more and more papers in recent years
have grown to gargantuan sizes (several hundred pages), making it
impossible for anyone to be completely sure they are right.
Games and Recreations
The Cycling Speed Freaks of Battle
Mountain, New Scientist, 4 December 2004.
Once a year, the fastest human-powered
vehicles in the world come to a flat stretch of highway outside Battle
Mountain, Nevada, and their riders strive to become the world's fastest
human. Forget Lance Armstrong--these bikes go almost twice as fast as he
does.
Non-Trivial Pursuits, SIAM News,
December 2004, 1-4.
To be a successful mathematician, do
you have to give up all your other interests? Not necessarily! Read
about five people people who have balanced math with other hobbies:
swing dancing, soaring, photography, musical composition, scrapbooking,
and taking care of zoo animals. (This may be the first time you will
ever see a mathematician and a giraffe in the same photo.)
The Mathematics of ... Shuffling: The Stanford Flip,
Discover, October 2002, 22-23.
Persi Diaconis, a
former magician and blackjack card counter, defects to the other side and
helps the casinos figure out why their shuffling machines aren't producing
random shuffles.
Top Young Problem
Solvers Vie for Quiet Glory, Science, 27 July 2001, 596-599.
The International
Mathematics Olympiad came to the U.S. in 2001. The Chinese stole the show
but the Americans did pretty well, especially Reid Barton, who became the
first person to win four gold medals in four years.
Exploring Origami,
Exploratorium, Summer 1999, 20-24.
This one is a
little bit older than the others, but it's worth checking out because the
Exploratorium did such a fantastic job of Web-ifying my article. See Jeremy
Shafer fold a paper crane while the paper is on fire! See his "flasher" hat
that can instantly expand to two feet wide or collapse so that it can fit
into your pocket! Also see a gallery of other fascinating and unusual
origami creations, with some helpful explanatory text by yours truly.
Mathematics in Arts and the
Media
Beautiful Mind's Math Guru Makes Truth = Beauty,
Science, 1 February 2002, 789-791.
A profile of Dave
Bayer, who was responsible for making sure the math in the Oscar-winning
movie would pass mathematicians' sniff test. The link points to a slightly
condensed version of the article that was reprinted, with permission, on the
Swarthmore College website. (Bayer and I both studied at Swarthmore and took
one class together.)
Economics
Boomtown, New Scientist, 26 May 2007, 48-51.
Ever since Frank Lloyd Wright, people have compared
cities to living organisms. Turns out they're not like
living creatures in one important way: as cities get
bigger, their pace of life speeds up. This has troubling
implications for sustainable growth, because it seems to
imply mathematically that cities must go through boom
and bust cycles.
Whose Gift is it
Anyway? New Scientist, 24-31 December 2005.
For the holiday
issue, I took a look at gift cards, a hugely popular marketing tool in
America that was just beginning to establish a foothold in England. The
trouble with gift cards is the fine print: cards that expire after a
certain time, or that have hidden fees. Some US states have banned these
practices, but in the UK it was still very much "Recipient beware."
Soft Cash,
New
Scientist, 23 July 2005.
Frequent-flier
miles. Loyalty cards. Subway cards that can be used to buy coffee, too.
Cell phones that can pay for concert tickets. The line between cash and
other forms of payment is fast becoming blurred. Some economists have
predicted the "death of cash" for years. It probably won't happen, but
it does seem that many parallel "currencies" are emerging that will
exist alongside the previously monolithic dollar (or pound sterling).
Voting Theory and Practice
Plurality Rules,
Swarthmore College Bulletin,
September 2004, 17-19.
Learn about instant runoff
voting, a way of choosing a winner in multi-candidate elections that
gives minority groups a better chance to make their voices heard,
according to Swarthmore alumna Cynthia Terrell. Also in the same issue,
read my article
Voting Power, which explains how to find the
"right" number of at-large seats for a city council or any
representative body, according to Swarthmore alumnus and
mathematician/lawyer Paul Edelman.
Hacking the Ballot Box, Discover, May
2004, 22-23.
When you cast your ballot, will
your vote really go to the candidate you intended it to? If you're voting on
an electronic voting machine with no paper trail, you're taking your
chances.
May the Best Man Lose,
Discover, November 2000, 84-91.
My semi-prophetic
piece about why the most popular candidate doesn't necessarily win an
election. It has nothing to do with hanging chads, but with the vagaries of
plurality-based elections. Find out why John McCain might have won in 2000
if we had a different election system!
Making Sense Out of Consensus,
SIAM News, October 2000.
Another article
about the mathematics of elections. This one focuses especially on the work
of Don Saari (University of California, Irvine) and assumes a little bit
more mathematical background.
Particle Physics
Free Fall, New Scientist, 10 February
2007.
A team of physicists at Stanford
prepares a new experiment to measure the force of gravity on an atomic
scale. Think of it as a cross between quantum physics and Galileo's
experiment of dropping cannonballs off the Tower of Pisa. The goal:
confirm some never-before-tested predictions of Einstein's theory of
general relativity, or else discover new physics.
When God Plays
Dice, New Scientist, 4 November 2006.
Twistors, a
mathematical gadget invented by Roger Penrose in the 1960s, are now
dramatically simplifying the computation of processes that go on in
particle colliders. Twistor theory, which was the best candidate for a
"theory of everything" before physicists dumped it in favor of string
theory, may turn out to be the right way to look at spacetime after all.
Super Computer vs.
Particle Accelerator: The Great Physics Heavyweights, New Scientist,
13 August 2005.
Once you get
past the overblown prize-fight metaphor (sorry, my editors made me do
it), this is an article about lattice QCD (quantum chromodynamics),
which is a way to make the equations of particle physics work in a
universe divided into pixels. This is perfect, of course, for computers.
Physicists have now managed to use lattice QCD to predict the results of
certain experiments to within 1 or 2 percent, before the
experimenters actually announced their results.
Spinning Into Posterity,
SIAM News, March 2003.
This year marks the 75th
anniversary of one of the most fundamental equations in physics: Dirac's
equation for the electron, which explained the phenomenon of electron spin
and predicted the existence of antimatter.
Solar System
The Gritty Problem of Moon Dust,
New
Scientist, 28 May 2005.
When the Apollo astronauts
walked on the Moon, they could not have ventured out for a fourth Moon
walk because their space suits were so fouled up by Moon dust. If we
ever want to go back for longer stays, scientists will have to figure
out how to control the stuff. That means they need some fake Moon
dust to experiment with, because there isn't enough real Moon dust to go
around. But how do you make fake Moon dust? Easier said than done!
Here Comes the Sun, Discover, May
2004, 62-69.
Our seemingly mild-mannered Sun
can kick up a fuss sometimes, as it did last fall when a spectacular solar
storm released two of the most powerful X-ray flares on record. The effects
on Earth were minor -- this time -- but out in space there was a long roster
of damaged or dead satellites. This article, along with Discover's
usual gorgeous graphics, explains how solar researchers are working to
understand and predict the Sun's behavior.
The Loneliest Lab, New Scientist, 31
January 2004, 32-33.
In January, George W. Bush
proposed a multi-year plan to send astronauts to the Moon and Mars. Mars
gets most of the publicity, but there is lots of good science that could be
done at a Moon base: exploring the South Pole-Aitken basin, looking for ice
deposits at the poles, and building a telescope under perpetually dark
skies.
Astronomy
Anselm's Question,
Swarthmore College Bulletin,
June 2003, 16-21.
A profile of Sandra Faber, a
multi-talented astronomer who found the first evidence that the universe is
not expanding uniformly, and helped identify the largest known structure in
the universe, the so-called "Great Attractor."
Geology
Journey to the Center of the Earth, SIAM News,
December 2007, 1.
A
geologist and an applied mathematician collaborate on a
new way of creating images of the core-mantle boundary,
and find an unexpectedly complex pattern of reflecting
layers. They might be signals of a phase transition that
was previously only theoretical, and if so they will
help to pin down how rapidly heat flows from Earth's
core.
What Killed the
Dinosaurs? Ask, October 2005.
My first foray
into writing for kids, this article describes the widely accepted theory
that a meteorite impact caused the catastrophic climate change that
killed the dinosaurs. I also mention an alternative theory, that
volcanic mega-eruptions did the dirty work of climate change, and the
meteorite impact was only the straw that broke the dinosaur's back.
Seismic Shift,
Discover, September 2001, 60-67.
Seismologist Terry
Wallace was the first person to tell what really happened in the Kursk
submarine disaster -- even though he was thousands of miles away.
Archaeology
Ahead of Its Time?,
Smithsonian, January
2005, 26-28.
New Philadelphia, a town on the
Illinois prairie, prospered from 1836 to about 1870, when the railroad
passed it by. Frank McWorter, the first black man ever to found a town
legally in America, used the proceeds from land sales to buy his family
out of slavery. Archaeologists are now excavating New Philadelphia's
remains, and hope to learn how blacks and whites lived together in one
of America's most racially mixed pre-Civil War communities.
Memoir
The
Tenure Chase Papers, in Starting Our Careers (Amer.
Math. Society, 1999), pp. 79-100. Also available
right here!
In an
category by itself is my memoir on how I was denied tenure at Kenyon
College. It's full of unexpected twists and turns and good lessons for young
professors on the tenure track. The website
www.phds.org called it
"required reading for all academics and would-be
academics." I have gotten far more reader feedback from
this article than any other
article that I have ever written.
Chess
The
Hook and Ladder Trick, Chess Life, July 2007, pp.
44-45.
A
little-known middlegame trap catches amateurs and
grandmasters alike. This was one of my earliest and most
popular lectures for
ChessLecture.
The "Hook and Ladder Trick" can go by other names (it's
basically a deflection sacrifice), but it's a
distinctive enough pattern that I felt it deserved its
own name.
Sac
Your Queen on Move Six! (A New Anti-Computer Variation),
Chess Life, March 2007, pp. 30-33.
I
spent two years working out an amazingly early sacrifice
of the queen (normally the most valuable piece) in the
Sicilian Defense. Eventually I got to the point where I
could beat the strongest computer chess program, set to
maximum strength, with this variation. And
then I got my first chance to
play it against a human...
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