Math Matters, December 2006

Unfortunately, that is the only kind of math that I have given any thought to lately.

But here are some things to think about over break, (while I am finishing a referee job)

Geometric Topology: there is a way to form a wild knot as an intersection of nested solid tori, and I think that I can prove that if two knots from this class are equivalent, then the defining sequences must also be equivalent up to some sort of cofinality condition. If I can do this, then perhaps I can come up with some invariants that distinguish very ill behaved will knots (that is, knots that are so wild that they fail to pierce a disk anywhere)

College level mathematics: suppose one has the differential equation y”+y = f(t) and f(t) is both bounded and periodic of period 2pi as well as piecewise smooth. Then any solution to this differential equation is bounded on [0, infinity) if and only if the k = 1 term of the Fourier series expansion vanishes. Sufficiency is easy to show; necessity is where the work lies.
Of course, this is just a mild extension of the concept of resonance; for example the solution to y” + y = sin(t), y(0)=0, y’(0)= -1/2 is y(t) = -(1/2)tcos(t), which has unbounded oscillations as t goes to infinity, even though the driving function is bounded. For those who like to see examples, I suggest surfing to the interactive differential equations website. Click enter, and then go to the menu; there is tons of stuff there. You can use sliders to vary parameters and you can watch the phase plane as the systems evolve with time.

Ok, now I’ve said it. Now what am I going to do about it?

Topic: math teaching

Math and People Skills

It appears that the students consider me to be a bit more approachable this year. Hence, they ask more questions and therefore I can see what confuses them.

Of course, some of that startles me.

I am teaching differential equations and a current topic is linear second order differential equations.

For those who have had some calculus, a differential equation is an equation involving a function and its derivatives (possibly some of higher order).

Example: y' + y = 0 is an example of a differential equation; this reads: a function plus its derivative is always zero. A general solution is y = a*exp(-t) because
y' + y = -a*exp(-t) + a*exp(-t) = 0.

Now if we had instead: y' + y = cos(t) we would obtain:

y = a*exp(-t) + (1/2)(cos(t) + sin(t)) as a solution. The solution with the arbitrary constant is called the homogenous solution, and the one without is (the trig part, in this case) is called the particular solution.

There are many techniques for solving these; you know that you are done by the various existance and uniqueness theorems (e. g. differential equations meeting certain criteria have a unique solution, so if you get a solution by "any means necessary" (e. g. good guessing) then you have all of the solutions.)

So, we need to teach "good guessing" which is sometimes known as "the method of undetermined coefficients".

The basic principle is that

if yp(t) = exp(a*t) then all derivatives are of (something)*exp(a*t)
if yp(t) = (b*cos(r*t) + d* sin(r*t)) then all derivatives are sums of sin(r*t) and cos(r*t)
if yp(t) = polynomial, then all derivatives are polynomials.

That is, certain types of functions formed a closed class with respect to taking derivatives.

To see this, consider a function that does not fall into this category: y(t) = sec(t)
taking derivatives yields tan(t)sec(t), then sec^3(t) + tan^2(t)sec(t), etc., which are not of the form (some constant) * sec(t)

So the good guessing technique doesn't work with sec(t).

This is something I understood from day one as a student.

Most of my students (or at least many) don't have a natural understanding of this.

I never knew that.

Math Fun From An Old Friend

One of my old graduate student buddies is a professor at Pittsburg State University in Kansas.
He calls himself The Okie in Exile.

His reasearch area is geometric topology; he has published in the area of non-compact three manifolds. He is one of the world's experts in Whitehead manifolds.

He has put out a couple of Youtube intervals in which he teaches math topics via folksy, funny stories.

The first of these deals with the Chinese Remainder Theorem in a clever way; the second one talks about Egyptian fractions. Enjoy!

Moral: When you write an exam, check your problems.

Mathematics: Differential Equations

I almost had some egg on my face. In differential equations class, we had just covered the basic existance and uniqueness theorems for linear first order differential equations. That is, if one is given y' = f(t, y) where f is continuous at (t0, y0) and continuous on some open rectangle containing (to, yo) then the differential equation y'=f(t,y), y(t0) = y0 has at least one solution. Furthermore, if the partial derivative of f(t,y) with respect to y is similarly continuous at (t0, y0), then the solution is unique.

Example: y' = t*exp(y), y(t0) = y0 has a unique solution for any initial condition,
y' = t*y^(2/3) has a soluton of all initial conditions, but fails to have a unique solution for any initial condition y(t0) = 0, as the partial derivative with respect to y is t*(2/3)*y^(-1/3) which fails to be continuous when y = 0.

So, on the test I gave y' = y^(2/3)*ln(t)*tan(y) and asked if there was a guarantee for a unique solution at y(2) = 0.

Answer: not that obvious! taking the partial derivative with respect to y we get:

ln(t) (y^(2/3) * sec^2(y) + (2/3)^y(-1/3)*tan(y)). Pluggin in t = 2 causes no difficulty, but what about y = 0? The first term gives 0 (no problems) but what about the second?

Actually, if you go through the proof (of the uniqueness theorem), it is the fact that we have some boundedness condition for the partial with respect to y (or a Lipshitz condition, at least).

Note what happens if we apply L'Hopital's rule to the second term:
lim as y goes to zero of (2/3)*(tan(y)/y^(1/3)) = (2/3)(sec^2(y)/ (1/3)y^(-2/3)) = (2/3)(3)sec^2(y)y^(2/3) = 2*1*0 = 0 !!!

In other words, technically speaking, the partical derivative may not have been continuous, but the singularity is removable and hence the uniqueness condition holds.

I didn't anticipate this at all and was lucky that I caught this prior to passing out the exams. Moral: test your problems ahead of time.

New Calculators: Brain still required

I teach lots of calculus as I am a small-time college mathematics professor.

By "small time" I mean that my research never put me in contention for an endowed professorship anywhere!

One of the biggest changes we've had to calculus teaching over the past 15 years or so is the introduction of calculators that do symbolic calculus; for exampe, if you wanted to find:
integral(e^2*x dx) you could just enter it and obtain (1/2)e^(2x). You'd have to add the "c", of course.

So, after a recent class, one student brought me the following:

integral((e^x )*sin(2*x) dx) and obtained:
(1/pi^2 + 8100)*e^x*(-90*pi*cos(2*x) + 8100*sin(2*x)) + C

What did the student do wrong?

Daily Kos and Mathematics?

What does the Daily Kos have to do with mathematics? Well, we have a "Science Friday" and this past Friday we had a gem of an article:

http://www.dailykos.com/story/2006/8/18/10571/4335

The Poincaré Conjecture

by angrytoyrobot [Subscribe]

Fri Aug 18, 2006 at 07:57:01 AM PDT

It's Science Friday!
We are here to blow up the brains of any Republicans who may be sneaking, yes SSSNEAKING Precious, around this board.

It was mentioned earlier, in another post (on cats) that there had been no mention of the Poincaré Conjecture Proved--This Time for Real

In mathematics, a 3-manifold is a 3-dimensional manifold. A manifold is an abstract mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be more complicated. In discussing manifolds, the idea of dimension is important. For example, lines are one-dimensional, and planes two-dimensional.

• angrytoyrobot's diary :: ::
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In a one-dimensional manifold (or one-manifold), every point has a neighborhood that looks like a segment of a line. Examples of one-manifolds include a line, a circle, and a pair of circles. In a two-manifold, every point has a neighborhood that looks like a disk. Examples include a plane, the surface of a sphere, and the surface of a torus.

Manifolds are important objects in mathematics and physics because they allow more complicated structures to be expressed and understood in terms of the relatively well-understood properties of simpler spaces.

Additional structures are often defined on manifolds. Examples of manifolds with additional structure include differentiable manifolds on which one can do calculus, Riemannian manifolds on which distances and angles can be defined, symplectic manifolds which serve as the phase space in classical mechanics, and the four-dimensional pseudo-Riemannian manifolds which model space-time in general relativity.

http://mathworld.wolfram.com/...

Russian mathematician Dr. Grigori (Grisha) Perelman of the Steklov Institute of Mathematics (part of the Russian Academy of Sciences in St. Petersburg) gave a series of public lectures at the Massachusetts Institute of Technology last week. These lectures, entitled "Ricci Flow and Geometrization of Three-Manifolds," were presented as part of the Simons Lecture Series at the MIT Department of Mathematics on April 7, 9, and 11. The lectures constituted Perelman's first public discussion of the important mathematical results contained in two preprints, one published in November of last year and the other only last month.

Perelman, who is a well-respected differential geometer, is regarded in the mathematical community as an expert on Ricci flows, which are a technical mathematical construct related to the curvatures of smooth surfaces. Perelman's results are clothed in the parlance of a professional mathematician, in this case using the mathematical dialect of abstract differential geometry. In an unusally explicit statement, Perleman (2003) actually begins his second preprint with the note, "This is a technical paper, which is the continuation of [Perelman 2002]." As a consequence, Perelman's results are not easily accessible to laypeople. The fact that Perelman's preprints are intended only for professional mathematicians is also underscored by the complete absence of a single reference to Poincaré in either paper and by the presence of only a single reference to Thurston's conjecture.

Stripped of their technical detail, Perelman's results appear to prove a very deep theorem in mathematics known as Thurston's geometrization conjecture.

LINK

Thurston's conjecture has to do with geometric structures on mathematical objects known as manifolds, and is an extension of the famous Poincaré conjecture. Since Poincaré's conjecture is a special case of Thurston's conjecture, a proof of the latter immediately establishes the former.

Thurston's conjecture proposed a complete characterization of geometric structures on three-dimensional manifolds.

Before stating Thurston's geometrization conjecture in detail, some background information is useful. Three-dimensional manifolds possess what is known as a standard two-level decomposition. First, there is the connected sum decomposition, which says that every compact three-manifold is the connected sum of a unique collection of prime three-manifolds.

The second decomposition is the Jaco-Shalen-Johannson torus decomposition, which states that irreducible orientable compact 3-manifolds have a canonical (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the 3-manifold removed by the tori is either "atoroidal" or "Seifert-fibered."

Thurston's conjecture is that, after you split a three-manifold into its connected sum and the Jaco-Shalen-Johannson torus decomposition, the remaining components each admit exactly one of the following geometries:

1. Euclidean geometry,

2. Hyperbolic geometry,

3. Spherical geometry,

4. The geometry of S^2xR,

5. The geometry of H^2xR,

6. The geometry of the universal cover SL_2R^~ of the Lie group SL_2R,

7. Nil geometry, or

8. Sol geometry.

Here, S^2 is the 2-sphere (in a topologist's sense) and H^2 is the hyperbolic plane. If Thurston's conjecture is true, the truth of the Poincaré conjecture immediately follows. Thurston shared the 1982 Fields Medal for work done in proving that the conjecture held in a subset of these cases.

Six of these geometries are now well understood, and there has been a great deal of progress with hyperbolic geometry (the geometry of constant negative scalar curvature). However, the geometry of constant positive curvature is still poorly understood, and in this geometry, the Thurston elliptization conjecture extends the Poincaré conjecture (Milnor).

Results due to Perelman (2002, 2003) appear to establish the geometrization conjecture, and thus also the Poincaré conjecture. Unlike a number of previous manuscripts attempting to prove the Poincaré conjecture, mathematicians familiar with Perelman's work describe it as well thought-out and expect that it will be difficult to locate any mistakes (Robinson 2003).

In the form originally proposed by Henri Poincaré in 1904 (Poincaré 1953, pp. 486 and 498), Poincaré's conjecture stated that every closed simply connected three-manifold is homeomorphic to the three-sphere. Here, the three-sphere (in a topologist's sense) is simply a generalization of the familiar two-dimensional sphere (i.e., the sphere embedded in usual three-dimensional space and having a two-dimensional surface) to one dimension higher. More colloquially, Poincaré conjectured that the three-sphere is the only possible type of bounded three-dimensional space that contains no holes. This conjecture was subsequently generalized to the conjecture that every compact n-manifold is homotopy-equivalent to the n-sphere if and only if it is homeomorphic to the n-sphere. The generalized statement is now known as the Poincaré conjecture, and it reduces to the original conjecture for n = 3.

The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical (and was known even to 19th century mathematicians), n = 3 has remained open up until now, n = 4 was proved by Freedman in 1982 (for which he was awarded the 1986 Fields Medal), n = 5 was proved by Zeeman in 1961, n = 6 was demonstrated by Stallings in 1962, and n >= 7 was established by Smale in 1961 (although Smale subsequently extended his proof to include all n >= 5).

Renewed interest in the Poincaré conjecture was kindled among the general public when the Clay Mathematics Institute included the conjecture on its list of million-dollar-prize problems. According to the rules of the Clay Institute, any purported proof must survive two years of academic scrutiny before the prize can be collected. A recent example of a proof that did not survive even this long was a five-page paper presented by M. J. Dunwoody in April 2002 (MathWorld news story, April 18, 2002), which was quickly found to be fundamentally flawed.

Almost exactly a year later, Perelman's results appear to be much more robust. While it will be months before mathematicians can digest and verify the details of the proof, mathematicians familiar with Perelman's work describe it as well thought out and expect that it will prove difficult to locate any significant mistakes.
References

Clay Mathematics Institute. "The Poincaré Conjecture."
http://www.claymath.org/...

Johnson, G. "A Mathematician's World of Doughnuts and Spheres." The New York Times, April 20, 2003, p. 5.

Perelman, G. "Ricci Flow and Geometrization of Three-Manifolds." Massachusetts Institute of Technology Department of Mathematics Simons Lecture Series.
http://www-math.mit.edu/...

Perelman, G. "The Entropy Formula for the Ricci Flow and Its Geometric Application." November 11, 2002.
http://www.arxiv.org/...

Perelman, G. "Ricci Flow with Surgery on Three-Manifolds." March 10, 2003.
http://www.arxiv.org/...

Poincaré, H. Oeuvres de Henri Poincaré, tome VI. Paris: Gauthier-Villars, pp. 486 and 498, 1953.

Robinson, S. "Russian Reports He Has Solved a Celebrated Math Problem." The New York Times, April 15, 2003, p. D3.

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Go to the link to see the rest of the article, where the author applies this to politics and to see the comments.

Class Size: whose perspective?

I was reading the most recent College Mathematics Journal and found an article written by a former professor of mine, Allen Schwenk.

Yes, I had this guy for third semester calculus, freshman year. And yeah, I made a "C"; "weak" academic performance he said.

Why? Well, at that time, I simply wasn't intellectually mature enough to learn from the way I was being taught; I'd have loved him for my upper division courses.

Lesson: don't judge a professor by the opinion of a freshman! Other students did quite well in his classes.

Eventually, he moved from Annapolis to Western Michigan University.

Anyway, back to his article (and I always enjoy his articles):

When one shops for universities, one often encounters the statistic "average class size"; here is an example.

So, what does this mean?

Let's look at a hypothetical example: we have a small school of 500 students. Each student takes 5 classes: math, language, history, political science, and science.

Math is taught in 25 sections of 20 students each (500)
Language is taught in 50 sections of 10 students each (500)
History is taught in 20 sections of 25 students each (500)
Political Science is taught in 5 sections of 100 students each (500)
Science is taught in 2 sections of 250 students each (500)

So the school would say: we have 25 + 50 + 20 + 5 + 2 = 102 sections
And the class enrollment is: 500 * 5 = 2500
So the "average class size" is 2500/102 = 24.5 students per class.

And, this is what the faculty (one per section) would experience.

But what is it like for the student? In this simple example, the student would see:
5 classes, of size 20, 10, 25, 100, and 250, or each student "sees" 405/5 = 81 students per class.
If this seems strange, think of it this way: those 2 sections of 250 students each are experienced by 500 students, but only by 2 faculty members. Hence this receives "heavier weight" when one does the calculation from the student's perspective.

In the article, Dr. Schwenk takes this beyond the hypothetical example, and shows that the student will always see a class size at least as large, or larger than the faculty does.

He gives a couple of proofs; if you wish to try this yourself (and this is an elementary, but tricky problem; it requires some cleverness to set up) and you want a hint, think: "Cauchy Schwartz" inequality, or think of the proof from statistics that says the sum of the squares is always at least as great, or greater than the square of the sums.

Two cool articles

I commend Mark Johnson for alerting me to two excellent articles that contain stuff about mathematics.

The first one deals with my research area: topology.

The second one is an interview with Field's Medalist Edward Witten on String Theory.

Topology: the Poincare Conjecture and Pearlman's proof

http://www.nytimes.com/2006/08/15/science/15math.html?_r=1&pagewanted=print&oref=slogin

August 15, 2006

Elusive Proof, Elusive Prover: A New Mathematical Mystery

By DENNIS OVERBYE

Grisha Perelman, where are you?

Three years ago, a Russian mathematician by the name of Grigory Perelman, a k a Grisha, in St. Petersburg, announced that he had solved a famous and intractable mathematical problem, known as the Poincaré conjecture, about the nature of space.

After posting a few short papers on the Internet and making a whirlwind lecture tour of the United States, Dr. Perelman disappeared back into the Russian woods in the spring of 2003, leaving the world’s mathematicians to pick up the pieces and decide if he was right.

Now they say they have finished his work, and the evidence is circulating among scholars in the form of three book-length papers with about 1,000 pages of dense mathematics and prose between them.

As a result there is a growing feeling, a cautious optimism that they have finally achieved a landmark not just of mathematics, but of human thought.

“It’s really a great moment in mathematics,” said Bruce Kleiner of Yale, who has spent the last three years helping to explicate Dr. Perelman’s work. “It could have happened 100 years from now, or never.”

In a speech at a conference in Beijing this summer, Shing-Tung Yau of Harvard said the understanding of three-dimensional space brought about by Poincaré’s conjecture could be one of the major pillars of math in the 21st century.

Quoting Poincaré himself, Dr.Yau said, “Thought is only a flash in the middle of a long night, but the flash that means everything.”

But at the moment of his putative triumph, Dr. Perelman is nowhere in sight. He is an odds-on favorite to win a Fields Medal, math’s version of the Nobel Prize, when the International Mathematics Union convenes in Madrid next Tuesday. But there is no indication whether he will show up.

Also left hanging, for now, is $1 million offered by the Clay Mathematics Institute in Cambridge, Mass., for the first published proof of the conjecture, one of seven outstanding questions for which they offered a ransom back at the beginning of the millennium.

“It’s very unusual in math that somebody announces a result this big and leaves it hanging,” said John Morgan of Columbia, one of the scholars who has also been filling in the details of Dr. Perelman’s work.

Mathematicians have been waiting for this result for more than 100 years, ever since the French polymath Henri Poincaré posed the problem in 1904. And they acknowledge that it may be another 100 years before its full implications for math and physics are understood. For now, they say, it is just beautiful, like art or a challenging new opera.

Dr. Morgan said the excitement came not from the final proof of the conjecture, which everybody felt was true, but the method, “finding deep connections between what were unrelated fields of mathematics.”

William Thurston of Cornell, the author of a deeper conjecture that includes Poincaré’s and that is now apparently proved, said, “Math is really about the human mind, about how people can think effectively, and why curiosity is quite a good guide,” explaining that curiosity is tied in some way with intuition.

“You don’t see what you’re seeing until you see it,” Dr. Thurston said, “but when you do see it, it lets you see many other things.”

Depending on who is talking, Poincaré’s conjecture can sound either daunting or deceptively simple. It asserts that if any loop in a certain kind of three-dimensional space can be shrunk to a point without ripping or tearing either the loop or the space, the space is equivalent to a sphere.

The conjecture is fundamental to topology, the branch of math that deals with shapes, sometimes described as geometry without the details. To a topologist, a sphere, a cigar and a rabbit’s head are all the same because they can be deformed into one another. Likewise, a coffee mug and a doughnut are also the same because each has one hole, but they are not equivalent to a sphere.

In effect, what Poincaré suggested was that anything without holes has to be a sphere. The one qualification was that this “anything” had to be what mathematicians call compact, or closed, meaning that it has a finite extent: no matter how far you strike out in one direction or another, you can get only so far away before you start coming back, the way you can never get more than 12,500 miles from home on the Earth.

In the case of two dimensions, like the surface of a sphere or a doughnut, it is easy to see what Poincaré was talking about: imagine a rubber band stretched around an apple or a doughnut; on the apple, the rubber band can be shrunk without limit, but on the doughnut it is stopped by the hole.

With three dimensions, it is harder to discern the overall shape of something; we cannot see where the holes might be. “We can’t draw pictures of 3-D spaces,” Dr. Morgan said, explaining that when we envision the surface of a sphere or an apple, we are really seeing a two-dimensional object embedded in three dimensions. Indeed, astronomers are still arguing about the overall shape of the universe, wondering if its topology resembles a sphere, a bagel or something even more complicated.

Poincaré’s conjecture was subsequently generalized to any number of dimensions, but in fact the three-dimensional version has turned out to be the most difficult of all cases to prove. In 1960 Stephen Smale, now at the Toyota Technological Institute at Chicago, proved that it is true in five or more dimensions and was awarded a Fields Medal. In 1983, Michael Freedman, now at Microsoft, proved that it is true in four dimensions and also won a Fields.

“You get a Fields Medal for just getting close to this conjecture,” Dr. Morgan said.

In the late 1970’s, Dr. Thurston extended Poincaré’s conjecture, showing that it was only a special case of a more powerful and general conjecture about three-dimensional geometry, namely that any space can be decomposed into a few basic shapes.

Mathematicians had known since the time of Georg Friedrich Bernhard Riemann, in the 19th century, that in two dimensions there are only three possible shapes: flat like a sheet of paper, closed like a sphere, or curved uniformly in two opposite directions like a saddle or the flare of a trumpet. Dr. Thurston suggested that eight different shapes could be used to make up any three-dimensional space.

“Thurston’s conjecture almost leads to a list,” Dr. Morgan said. “If it is true,” he added, “Poincaré’s conjecture falls out immediately.” Dr. Thurston won a Fields in 1986.

Topologists have developed an elaborate set of tools to study and dissect shapes, including imaginary cutting and pasting, which they refer to as “surgery,” but they were not getting anywhere for a long time.

In the early 1980’s Richard Hamilton of Columbia suggested a new technique, called the Ricci flow, borrowed from the kind of mathematics that underlies Einstein’s general theory of relativity and string theory, to investigate the shapes of spaces.

Dr. Hamilton’s technique makes use of the fact that for any kind of geometric space there is a formula called the metric, which determines the distance between any pair of nearby points. Applied mathematically to this metric, the Ricci flow acts like heat, flowing through the space in question, smoothing and straightening all its bumps and curves to reveal its essential shape, the way a hair dryer shrink-wraps plastic.

Dr. Hamilton succeeded in showing that certain generally round objects, like a head, would evolve into spheres under this process, but the fates of more complicated objects were problematic. As the Ricci flow progressed, kinks and neck pinches, places of infinite density known as singularities, could appear, pinch off and even shrink away. Topologists could cut them away, but there was no guarantee that new ones would not keep popping up forever.

“All sorts of things can potentially happen in the Ricci flow,” said Robert Greene, a mathematician at the University of California, Los Angeles. Nobody knew what to do with these things, so the result was a logjam.

It was Dr. Perelman who broke the logjam. He was able to show that the singularities were all friendly. They turned into spheres or tubes. Moreover, they did it in a finite time once the Ricci flow started. That meant topologists could, in their fashion, cut them off, and allow the Ricci process to continue to its end, revealing the topologically spherical essence of the space in question, and thus proving the conjectures of both Poincaré and Thurston.

Dr. Perelman’s first paper, promising “a sketch of an eclectic proof,” came as a bolt from the blue when it was posted on the Internet in November 2002. “Nobody knew he was working on the Poincaré conjecture,” said Michael T. Anderson of the State University of New York in Stony Brook.

Dr. Perelman had already established himself as a master of differential geometry, the study of curves and surfaces, which is essential to, among other things, relativity and string theory Born in St. Petersburg in 1966, he distinguished himself as a high school student by winning a gold medal with a perfect score in the International Mathematical Olympiad in 1982. After getting a Ph.D. from St. Petersburg State, he joined the Steklov Institute of Mathematics at St. Petersburg.

In a series of postdoctoral fellowships in the United States in the early 1990’s, Dr. Perelman impressed his colleagues as “a kind of unworldly person,” in the words of Dr. Greene of U.C.L.A. — friendly, but shy and not interested in material wealth.

“He looked like Rasputin, with long hair and fingernails,” Dr. Greene said.

Asked about Dr. Perelman’s pleasures, Dr. Anderson said that he talked a lot about hiking in the woods near St. Petersburg looking for mushrooms.

Dr. Perelman returned to those woods, and the Steklov Institute, in 1995, spurning offers from Stanford and Princeton, among others. In 1996 he added to his legend by turning down a prize for young mathematicians from the European Mathematics Society.

Until his papers on Poincaré started appearing, some friends thought Dr. Perelman had left mathematics. Although they were so technical and abbreviated that few mathematicians could read them, they quickly attracted interest among experts. In the spring of 2003, Dr. Perelman came back to the United States to give a series of lectures at Stony Brook and the Massachusetts Institute of Technology, and also spoke at Columbia, New York University and Princeton.

But once he was back in St. Petersburg, he did not respond to further invitations. The e-mail gradually ceased.

“He came once, he explained things, and that was it,” Dr. Anderson said. “Anything else was superfluous.”

Recently, Dr. Perelman is said to have resigned from Steklov. E-mail messages addressed to him and to the Steklov Institute went unanswered.

In his absence, others have taken the lead in trying to verify and disseminate his work. Dr. Kleiner of Yale and John Lott of the University of Michigan have assembled a monograph annotating and explicating Dr. Perelman’s proof of the two conjectures..

Dr. Morgan of Columbia and Gang Tian of Princeton have followed Dr. Perelman’s prescription to produce a more detailed 473-page step-by-step proof only of Poincaré’s Conjecture. “Perelman did all the work,” Dr. Morgan said. “This is just explaining it.”

Both works were supported by the Clay institute, which has posted them on its Web site, claymath.org. Meanwhile, Huai-Dong Cao of Lehigh University and Xi-Ping Zhu of Zhongshan University in Guangzhou, China, have published their own 318-page proof of both conjectures in The Asian Journal of Mathematics (www.ims.cuhk.edu.hk/).

Although these works were all hammered out in the midst of discussion and argument by experts, in workshops and lectures, they are about to receive even stricter scrutiny and perhaps crossfire. “Caution is appropriate,” said Dr. Kleiner, because the Poincaré conjecture is not just famous, but important.

James Carlson, president of the Clay Institute, said the appearance of these papers had started the clock ticking on a two-year waiting period mandated by the rules of the Clay Millennium Prize. After two years, he said, a committee will be appointed to recommend a winner or winners if it decides the proof has stood the test of time.

“There is nothing in the rules to prevent Perelman from receiving all or part of the prize,” Dr. Carlson said, saying that Dr. Perelman and Dr. Hamilton had obviously made the main contributions to the proof.

In a lecture at M.I.T. in 2003, Dr. Perelman described himself “in a way” as Dr. Hamilton’s disciple, although they had never worked together. Dr. Hamilton, who got his Ph.D. from Princeton in 1966, is too old to win the Fields medal, which is given only up to the age of 40, but he is slated to give the major address about the Poincaré conjecture in Madrid next week. He did not respond to requests for an interview.

Allowing that Dr. Perelman, should he win the Clay Prize, might refuse the honor, Dr. Carlson said the institute could decide instead to use award money to support Russian mathematicians, the Steklov Institute or even the Math Olympiad.

Dr. Anderson said that to some extent the new round of papers already represented a kind of peer review of Dr. Perelman’s work. “All these together make the case pretty clear,” he said. “The community accepts the validity of his work. It’s commendable that the community has gotten together.”

PBS inverview with Edward Witten on String Theory

http://www.pbs.org/wgbh/nova/elegant/view-witten.html

Viewpoints on String Theory Edward Witten		The Elegant Universe homepage
Many physicists consider Ed Witten to be Einstein's true successor. A mathematical physicist at the Institute for Advanced Study in Princeton, New Jersey, Witten has been awarded everything from a MacArthur "genius grant" to the Fields Medal, the highest honor in the world of mathematics. His contributions to string theory have been myriad, including the time in 1995 when he gave the then somewhat moribund field a much-needed boost by showing how the five different variations of the theory then competing with one another actually all belonged under one umbrella. In this interview, Witten talks about how the big bang could have coughed up a string so large that it might still survive in the universe and be visible with telescopes—and other ideas to make your head spin. Note: For a definition of unfamiliar terms, see our glossary. Spreading out particles NOVA: What is string theory? Witten: String theory is an attempt at a deeper description of nature by thinking of an elementary particle not as a little point but as a little loop of vibrating string. One of the basic things about a string is that it can vibrate in many different shapes or forms, which gives music its beauty. If we listen to a tuning fork, it sounds harsh to the human ear. And that's because you hear a pure tone rather than the higher overtones that you get from a piano or violin that give music its richness and beauty. So in the case of one of these strings it can oscillate in many different forms—analogously to the overtones of a piano string. And those different forms of vibration are interpreted as different elementary particles: quarks, electrons, photons. All are different forms of vibration of the same basic string. Unity of the different forces and particles is achieved because they all come from different kinds of vibrations of the same basic string. In the case of string theory, with our present understanding, there would be nothing more basic than the string. NOVA: Why does something as simple as replacing points with strings make such a huge difference? Witten: It's indeed surprising that replacing the elementary particle with a string leads to such a big change in things. I'm tempted to say that it has to do with the fuzziness it introduces. So the particle is spread out. But it turns out that everything about spacetime is a little bit spread out; it's blurred. You have to start doing some calculations to really see it. It's hard to explain it just in words or by drawing pictures. “You enter a completely new world where things aren’t at all what you’re used to.” Spreading out the particle into a string is a step in the direction of making everything we're familiar with fuzzy. You enter a completely new world where things aren't at all what you're used to. It's as surprising in its own way as the fuzziness that much of physics acquired in light of quantum mechanics and the Heisenberg uncertainty principle. NOVA: In reading about string theory and in talking to people about physics in general, we hear a lot about string theory being beautiful, but what does that mean? What's beautiful about it? Witten: Even before string theory, especially as physics developed in the 20th century, it turned out that the equations that really work in describing nature with the most generality and the greatest simplicity are very elegant and subtle. It's the kind of beauty that might be hard to explain to a person from a different walk of life who doesn't deal with science or math professionally. But the beauty of Einstein's equations, for example, is just as real to anyone who's experienced it as the beauty of music. We've learned in the 20th century that the equations that work have inner harmony. Now there must be skeptics out there who will tell you that these beautiful equations might have nothing to do with nature. That's possible, but it's uncanny that they are so graceful and that they capture so much of what we already know about physics while shedding so much light on theories that we already have. Enter fuzziness NOVA: Can you give us an example of something concrete that string theory does for physics that goes beyond previous theories? Witten: In Einstein's general relativity the structure of space can change but not its topology. Topology is the property of something that doesn't change when you bend it or stretch it as long as you don't break anything. You can imagine a bowling ball and you can imagine a coffee cup that has a handle—the coffee cup is different topologically because there's a handle. Even if you could bend it or stretch it, as long as you don't break it, it's still got that handle, which makes it topologically different. There was a long history of speculation that in quantum gravity, unlike Einstein's classical theory, it might be possible for the topology of spacetime to change. And it turned out in string theory in the late '80s and early '90s we actually were able to calculate examples where you could really see changes in the topology of spacetime. That was fun because it was very concrete—you could understand it pretty well. And it illustrated how the theory went beyond Einstein's general relativity as understood before in a very nice and down-to-earth way. NOVA: How does string theory allow you to change the topology of spacetime? Witten: Quantum mechanics brought an unexpected fuzziness into physics because of quantum uncertainty, the Heisenberg uncertainty principle. String theory does so again because a point particle is replaced by a string, which is more spread out. And even though it's a naïve statement, it leads in the right direction: when we study it more deeply, we find that in string theory, spacetime becomes fuzzy. So imagine now we have this coffee cup. If the handle is big enough, you can see it's there. But if you had a very small handle, because of the fuzziness of spacetime, you couldn't tell if it was there or not. Then it could disappear. That fuzziness of spacetime leads to the possibility that the topology can change. Extra dimensions required NOVA: String theory requires 10 dimensions of space. Does that complicate the theory or does it solve problems? Witten: Technically you need the extra dimensions. At first people didn't like them too much, but they've got a big benefit, which is that the ability of string theory to describe all the elementary particles and their forces along with gravity depends on using the extra dimensions. You have that one basic string, but it can vibrate in many ways. But we're trying to get a lot of particles because experimental physicists have discovered a lot of particles. The electron and its heavy cousins the neutrinos, the quarks, photons, gravitons, and so on. There is really a big zoo of elementary particles that you're trying to explain. Having those extra dimensions and therefore many ways the string can vibrate in many different directions turns out to be the key to being able to describe all the particles that we see. NOVA: But why do we think that they might actually exist? We certainly can't see them. Witten: We see light waves with our eyes, but most of the other particles take 20th-century equipment to discover them. As for the forces, electromagnetism and gravity we experience in everyday life. But the weak and strong forces are beyond our ordinary experience. So in physics, lots of the basic building blocks take 20th- or perhaps 21st-century equipment to explore. “I would conclude that extra dimensions really exist. They’re part of nature.” As far as extra dimensions are concerned, very tiny extra dimensions wouldn't be perceived in everyday life, just as atoms aren't: we see many atoms together but we don't see atoms individually. In a somewhat similar fashion, our experiences and our observations would average over the extra dimensions if those were small enough. NOVA: So just because we don't see them doesn't mean the theory is wrong? Witten: The theory has to be interpreted that extra dimensions beyond the ordinary four dimensions the three spatial dimensions plus time are sufficiently small that they haven't been observed yet. So we would hope to test the theory, conceivably directly at accelerators. I suspect that's a long shot. More likely we'll do it indirectly by making more precise calculations about elementary particles based on the existence of extra dimensions. NOVA: Do you think extra dimensions actually exist, or are they a mathematical device? Witten: If I take the theory as we have it now, literally, I would conclude that extra dimensions really exist. They're part of nature. We don't really know how big they are yet, but we hope to explore that in various ways. They're beyond our ordinary experience just like atomic nuclei are. On the other hand, we don't understand the theory too completely, and because of this fuzziness of spacetime, the very concept of spacetime and spacetime dimensions isn't precisely defined. I suspect that the fuzziness of spacetime will play more of a role in the eventual answer than we understand now. [To try to picture a fourth spatial dimension, see Imagining Other Dimensions.] NOVA: If these extra dimensions exist, does string theory offer any explanation of why there are apparently three space dimensions larger than the rest? Witten: That's a big problem that has to be explained. As of now, string theorists have no explanation of why there are three large dimensions as well as time, and the other dimensions are microscopic. Proposals about that have been all over the map. Verifying string theory NOVA: It seems like the standard criticism of string theory is that it isn't testable. How do you respond to that criticism? Witten: One very important aspect of string theory is definitely testable. That was the prediction of supersymmetry, which emerged from string theory in the early '70s. Experimentalists are still trying to test it. It hasn't been proved that supersymmetry is right. But there is a very precise relationship among the interaction rates of different kinds of particles which follows from supersymmetry and which has been tested successfully. Because of that and a variety of other clues, many physicists do suspect that our present decade is the decade when supersymmetry will be discovered. Supersymmetry is a very big prediction; it would be interesting to delve into history and try to see any theory that ever made as big a prediction as that. NOVA: What are some of the other ways that string theory could be confirmed experimentally? Witten: There are a lot of conceivable ways we could get experimental information that would help with string theory. Explorations of cosmology, studying the cosmic background microwave radiation and hopefully finding gravitational waves left over from the big bang and studying their properties are very plausible avenues for eventually testing string theory, although there isn't yet to my thinking a satisfactory theoretical understanding of what to expect. But it's conceivable that the big bang could have produced a string so large that it would be present in today's universe and visible in telescopes, perhaps discoverable by the satellites that are now mapping out the microwave sky. If that were discovered, it would be a dramatic confirmation of the existence of strings. Still, that's a story that will develop over the next decade or two as the experiments progress and conceivably as the theory progresses. NOVA: How likely do you think it is that string theory will be proven correct? Witten: Well, I don't have a crystal ball. You know, the theory of neutron stars was tested and the same is true of the theory of black holes and the theory of gravitational waves. A lot of the theories that were there in the '20s and '30s that looked like they were way beyond reach were eventually tested. They were tested because there were new technologies, there were new instruments, there were newer things found in the sky. Things happened that you couldn't foresee. That's what happens in science. “I think that nature will turn out to be kind to us and that there will be some nice surprises, as there have been so many times in the past.” So when you ask me how string theory might be tested, I can tell you what's likely to happen at accelerators or some parts of the theory that are likely to be tested. But I also have to point out that part of the answer is the unknown. Just as the theory of neutron stars, black holes, gravity waves, and so many other things were tested because of things that nobody foresaw, there are just so many ways that nice surprises could happen that would lead to new advances in string theory. There are all kinds of possibilities, like literally seeing a string in a telescope if nature has chosen to be kind to us in that particular way. I think that nature will turn out to be kind to us and that there will be some nice surprises, as there have been so many times in the past in science. But if I could tell you what they were, they wouldn't be surprises. NOVA: Do you think it's possible that string theory will turn out to be wrong, or at least some branch of knowledge that just isn't connected to nature? Witten: I guess it's possible that string theory could be wrong. But if it is in fact wrong, it's amazing that it's been so rich and has survived so many brushes with catastrophe and has linked up with the established physical theories in so many ways, providing so many new insights about them. I wouldn't have thought that a wrong theory should lead us to understand better the ordinary quantum field theories or to have new insights about the quantum states of black holes. The question reminds me a little bit of the question about interpreting fossils. When fossils were first explored 100 or 200 years ago, some people thought they were traces of past life that had survived in the rocks and others thought that they had been placed there at the creation of the universe by the creator in order to test our faith. So I guess string theory might be wrong, but it would seem like a kind of cosmic conspiracy. NOVA: It's been said that string theory really belongs to the 21st century. Do you agree? Witten: Back in the early '70s, the Italian physicist, Daniele Amati reportedly said that string theory was part of 21st-century physics that fell by chance into the 20th century. I think it was a very wise remark. How wise it was is so clear from the fact that 30 years later we're still trying to understand what string theory really is. What Amati meant was that usually the physical theory isn't developed until there are more or less the concepts and ideas in hand for making sense out of it. By the time Einstein developed general relativity, he actually knew what he was doing. But string theory wasn't like that. The first traces appeared in 1968 with the Venetziano model. Nobody at the time had the conception that could have led to string theory in a clear way or understood what it was. It was something incredibly beautiful, a trail that people followed without understanding what it was. We've come through 30 years of remarkable discoveries, and we can see a lot of puzzles still ahead. NOVA: Where does inspiration in this field come from for you? Witten: You have to be open-minded because ideas come from different places. You can think about something in one way for a long time and it seems like the only way to think about it, but it really isn't. Somebody could make a suggestion that really sounds naïve. It might even be naïve, but it could have an important element of the truth in it. And it could be truth that one's overlooking. So it's really hard to state a general rule. If one could say the general rule about where to find inspiration, we would just teach it to our students and then science would be much more straightforward.

Math Fest, 2006

This is a summary from my obeservations from the Mathematical Association of America's Math Fest 2006.

Everything here has been included in my personal blog, blueollie. Here I've attempted to strip off irrelevant materal. I am starting with a little bit about my adventure getting to the conference; those who want to read only about mathematics should scroll down to the next section.

Day One; getting there, getting situated.

Today, I work up a bit later than normal due to being very tired. Hence, I limited my workout to a 3 mile run on the treadmill followed by 15 minutes of yoga.

This hotel (Crown Plaza) has an excellent workout facility and a reasonably well stocked weight room; I'll use it a bit more extensively tomorrow morning, I hope.

The trip itself: interesting. First of all, I noticed that in Chicago (the day prior to this trip) and at times on this trip, more people than normal have given me nods of approval.

I wonder why; it isn't as if I've trimmed down that much; then I realized: I now have a very close cropped crew cut. My guess is that many think that I am military or perhaps police.

Go figure; if only they knew that I was a Kos reading lefty.

As far as the road trip, here is the good: my Prius got 45.8 miles to the gallon, and that included the stretch through big hills. The bad: I left my credit card in a Peoria drugstore (CVS at Campus town). Fortunately, they still had my card, but I didn't know that at the time. I had to call after hours to get this card cancelled and that was a trip. I highly suggest carrying the numbers of your credit card companies with you for after hours emergencies such as this one.

Numbers: VISA (800) 847-2911
Master Card: 1-800-MC-ASSIST (1-800-622-7747)
Discover: 1-800-DISCOVER (1-800-347-2683)
American Express: 1-888-412-6945

Anyway, having these numbers would have helped me some.

But, between playing with Vickie, hitting an early rain storm and discovering that my check card was lost, and having to stop almost every hour to relieve my bladder, I didn't get in until after 9 pm central time, or after 10 pm local time.

I was very tired.

Day One: The Mathematics

The good news is that the talks so far have been outstanding.

Dorothy Buck lead off with an excellent talk about knot theory and its role in understanding the chemical reactions in DNA. Basically, Dr. Buck and Erica Flapan have shown that, subject to three biological assumptions, that only 6 types of links need to be considered (when one wants to understand the cross changing operations that DNA undergoes) and, if one uses tangle theory, one can pass to the theory of knots and lens spaces by passing to the double branched cover of a tangle (which is a solid torus) and studying Dehn surgery.

Next, Fields Medalist W. T. Gowers gave a nice talk about some "easy to state, hard to solve" combinatorial problems. Exampe: let R be a commutative ring and A a subset. The sum set is the set of all elements of R of the form x + y where x and y are both in A (note: x + x is permitted). The multiplicative set is defined in a similar way.

So, suppose that you know that the additive and multiplicative sets are of a certain size: what can you say about the given set to begin with?

Next, Jesus De Loera gave a nice lecture on convex polytopes and gave some simple to state, yet still unsolved conjectures. Here is one:

in dimension three, it is known that if you are given a triple of integers x, y, z such that

x -y + z = 2
2y >= 3z
2y >= 3x

Then there is a unique convex polytope with x vertices, y edges and z faces. For example, for a tetrahedron, x = 4, y = 6, z = 4,for a cube: x = 8, y = 12, z = 6, for a pyramid: x = 5, y = 8, z = 5.

Such a formula, if one exists, is completely unknown for 4 dimensional polytopes.

Of course, these were some of the major invited addresses; I expected these to be good. But the contributed talks went well as well.

Among those: I head a nice talk about Hill's cipher algorithm (given by Bill Wardlaw, who was my first abstract mathematics professor at Annapolis), check digits of codes which detect errors in digits as well as order of digits (using the action of permutation groups on dihedral groups) , using magic squares (and their relatives) as examples of vector spaces and ideals in rings, enumeration of sudoku puzzles. I also heard a nice talk by Ed Burger on how to (not) teach a class that introduces the idea of proofs to undergraduates. Ed was a friend in graduate school; he now has a list of honors 10 miles long; included in these was the Chauvenet Prize for expository writing. Sigh...I knew him when...

This is Ed Burger, answering questions after his well received talk. If someone thinks that stellar teaching and good research can't go hand in hand, think again! Ed is proof that it can.


Though this shot leaves something to be desired, this is Bill Wardlaw of the U. S. Naval Academy, who taught me my first course on Modern Algebra back in the spring of 1979.

Day Two: I give my talk (ok, it is a 10 minute talk)

Personal Journal: I started the day with a short walk on the treadmill (3 miles, 37 minutes; 24 minutes for the last two miles) followed by weights and yoga.

Today I gave my talk; it seemed to go ok. Here is what I talked about: I presented a paper which said, in effect, that if one attempted to evaluate the limit of a two variable function by evaluating it over all curves that had continuous first derivative and ran through a given point and if one obtained the same limit over all such curves, then the limit of the function exists there.

Note that this is false if one replaces "all curves with continuous first derivatives" by "all lines" or even by "all real analytic functions" (functions which have a power series expansion at every point in their domain which is valid on some open set).

In addition to giving my talk, I also dropped some dollars on math books. This is to be expected.

Notable talks, in addition to the Gowers talk, were the series on Physical Knot Theory.

First, Ken Millett talked about using the computer to make random knots (physical) and to see what knots were produced. Note that he wasn't merely interested in knot type, but also the various physical qualities that a given knot representation had. He was especially interested in "equilateral polygonal knots" (knots formed by line segments, each of the same length).

Eric Rawdon informed me that, up to now, equilateral polygonal knots were realized at their (known) minimum stick number presentation, though the knot 8-19 might not have this happen.

Next Lou Kauffman talked about knots and rational tangles. He described work with Sofia Lambropoulou. In particular, he used the theory of rational tangles to find a way to recognize the unknot. Note that recognizing the unknot isn't always easy on computational grounds; a Theorem of Hass and Lagarias shows that one can always unknot a diagram by using 2^cn Reidermeister moves where c is some fixed constant and n is the number of crossings of the diagram. Unfortunately, the lowest known value of c is 10¹¹.

Tom Banchoff then talked about Piecewise Circular Space Curves; that is, curves that are formed out of pieces of honest to goodness circles. He talked about various properties (e. g., rigidity) and about various kinds of "dual polygons" associated with such curves.

Eric Rawdon gave an interesting talk about tight knots. Here, the definition of "tight knot" means the following: Suppose you have a regular neighborhood of a smooth knot, where one, say, ensures that the regular neighborhood (a knotted solid torus) has some fixed radius as a diameter. Call this diameter "1". Then, using this scale, what is the shortest "length" that one needs to obtain a given knot?

Eric's papers can be found here: http://www.mathcs.duq.edu/~rawdon/Preprints/
There is enough here to keep you busy for a very long time.

Day Three: Another good session.

This post will start with the personal and then get to the mathematics. Again, those who wish can scroll down a bit.

Athletically: 4 miles of "running" on the treadmill; the first mile was a 10:10 warm up (5:30 first half mile). I admit that I was a bit distracted when writing this post as I am in a hotel lobby and a tall lady in a pretty, loose but sort of clingy dress just got on the elevator.

Hey, if I didn't like women, I wouldn't like my wife.

Now back to the post: my run was both good (got some exercise) and bad (couldn't hold a 8:30 pace for 2.5 miles; that used to be my marathon pace). The road back is very long; the good news is that my hip/back/IT band is slowly getting better.

While in Knoxville, I found a nice place to eat called The Tomato Head. It is in the Market Square. They serve good pizza as well as other types of food (salads, sandwiches, etc.) The servers and workers are from the "tye dye" set and are downright pleasant. One even asked me about my ultramarathon t-shirt (no, she wasn't my server so she wasn't sucking up for a tip). This is the kind of place you go to after yoga class; it is the type of place you go to in order to forget that George W. Bush is our President.


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The Mathematics from Day Three

The morning invited talks were outstanding. In the second talk of the day, Gowers finished his three part lecture on analytic combinatorics by showing the utility of the Discrete Fourier Transform, among other things.

In the first talk of the day, Trachette Jackson, who is one of the hottest young researchers, gave an excellent talk on the mathematics of cancer cell modeling.

The following is a very incomplete sketch of what she talked about: first, she gave the basics of cancer cells by giving a basic definition: what makes a cell cancerous? In a nutshell, a cancer cell is one which gets around dying naturally (evading apoptosis), doesn't take growth/death cues from its environment, can initiate and sustain angiogenesis (grows blood vessels to take blood from the host body), is invasive and has nearly limitless reproductive potential.

She said that the cells first start out as a somewhat spherical cluster which gets nutrition via its boundary. Eventually, the inner core starts to die off (gets starved; this is stage I); at that point, the cell starts growing capillaries to get a blood supply (this is stage II). At stage three, it is growing and has established its own blood supply.

There are ways to attacking cancer cells; some deal with trying to shrink the cell itself, some deal with trying to kill off the blood vessels supplying the cell (this was Judah Folkmin's 1998 breakthrough in mice that failed to be replicated in humans).

Her research deals with using partial differential equations to model the situation at various stages, and to see what happens to growth when various parameters are changed (reducing the ability to obtain oxygen, shrinking the newly made blood vessels, limiting the production of certain secreted chemicals, for example).

The partial differential equations that are obtained are not for the faint of heart; one such is
dN/dtt = phiv(V)N(1-N/N0) where N is the number of cells in the larger cancer cell, V is a measure of the blood vessel density, and phi is a function:

(r1c^2)/(c1^2 + c^2) - r2(1 - sigmac^2/(c2^2 + C^2)) where C is the oxygen concentration which is itself a function of v:
c(V) = CmV/(k+V) and Cm, sigma, k, c1, r1 and r2 are parameters which must be determined from experiment.

Of course, these equations must be solved numerically.

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The next invited talk was one about math history and the current teaching of calculus by David Bressoud. His talk centered around some of the basic theorems of real analysis and calculus and their historical context. What made this talk a bit different is that he used the history to explain why some of these theorems are difficult for the average undergraduate to understand or even appreciate.

The talk he gave can be accessed here: http://www.macalester.edu/~bressoud/talks/Knoxville.ppt

Some examples: take how we teach the Fundamental Theorem of Calculus. It wasn't even published in its current form until 1907; prior to that, various forms of "integral as some sort of an anti-derivative" definition was used rather than the traditional Riemann sums. The power of the fundamental theorem is that it shows that the Riemann sum and the anti-derivative are equivalent concepts, under certain special conditions. This association is not at all natural.

Yes, I know the proof.

He talked about Cauchy's attempt to provide some rigor to calculus, and how even good contemporary mathematicians found some of his stuff difficult to understand. And, he shows one of Cauchy's big mistakes; he thought that he proved that the infinite sum of a convergent series of continuous functions converges to a continuous function; Abel showed that this was false (think: Fourier Series) unless one had the notion of uniform convergence.

He then talked about the so-called Heinie-Borel theorem (ironically, Shoenflies was the first to refer to this theorem by this name) and how this theorem is really only needed at the start of the discussion of measure theory. My take: that might well be true, but H-B sure makes life easier.

For those who don't know what the H-B theorem says, it says that closed bounded sets in the real line are compact under the standard metric topology. That is, if one covers any closed interval by a collection of open intervals, then a finite subset of the collection of open intervals will also cover that given closed interval.

By the way, Heinie had very little to do with this theorem, though he did talk about uniform convergence.
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I attended two afternoon sessions of contributed talks. One was a session on the teaching of numerical methods. There were a few talks of "here is what I did and this is what happened" variety, one talk was about an ambitious plan to talk about discrete methods of solutions to partial differential equations, one was about automatic differentiation, and one talk was about the use of some cool applets for exploring various curve approximation schemes (splines, Lagrange, etc.)

The automatic differentiation talk by Richard Neidinger of Davidson was especially interesting. It basically views differential functions as 2-dimensional vectors: the first coordinate is the value of the function at a point, and the second coordinate is the value of the derivative. Then one does a weird sort of algebra with these vectors; one scalar multiplies and adds in the usual way (component wise). But multiplication of these vectors becomes interesting; one multiplies the first two components in the usual way, but one uses a bizarre dot product to multiply the second component (to respect the chain rule): (a , b) * (c, d) = (ac, ad + bc) under this "algebra".

The value is that differentiation can be done purely numerically once one knows the formula for the functions; for example, one doesn't need to enter the formula for the derivative of a function into, say, a Newton's root finding program.

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I then ended the conference by catching the tail end of the last general contributed paper session. There were a few good talks here as well.

Elena Constantin gave a nice talk about how one can generalize the Lagrange multiplier method/second derivative test even when these methods don't work (e. g., the Hessian matrix is singular). She showed how her method worked, though she didn't have time to give an outline of the proof. My guess is that she used the multi-variable power series method, but that is just a guess.

David Austin gave a nice talk about how to visualize the Cauchy-Riemann equations in complex analysis.

Basically, the theorem is this: if f(x + i y) = u(x,y) + i v(x,y) is a complex function, f is differentiable on an open set if and only if the so-called Cauchy-Riemann equations are satisfied:

u_x = v_y and u_y = - v_x

The usual proof is to use the limit definition of derivative and approach the point in question from the real axis direction and also from the y-axis direction, realize that one must get the same limit, and equate real and imaginary parts.

Austin gave a nice visual way of seeing this that uses the concept of how the complex derivative acts on sets in the domain (stretches by the magnitude of the derivative and rotates by the argument of the derivative evaluated at the given point).

His talk can be found here: http://merganser.math.gvsu.edu/david/mathfest06/

Jay Shiffman gave a talk about Collatz k-tuples. The Collatz conjecture says that any positive integer can be reduced to 1 by the following rules: if the integer is odd, multiply by three and add 1; if it is even, divide by two.

For example: start with 7. 7 * 3 + 1 = 22
22/2 = 11
11* 3 + 1 = 34
34/2 = 17
17*3 + 1 = 52
52 / 2 = 26
26 / 2 = 13
13*3 +1 + 40
40/2 = 20
20/2 = 10
10/2 = 5
5*3 + 1 = 16

16 is a power of 2: 16/ 2 = 8, 8 /2 = 4, 4/2 = 2, 2/2 = 1.
So, we say that 7 has "Collatz length" 16. (my note: to me, it was over when we reached 16 since 16 is a power of 2, so to me a better number to focus on would be 12, but I digress)

Now if, say, there are numbers n, n+ 1 that have the same Collatz length, these are called a Collatz pair. If there are numbers n, n+1, n+2 that have the same Collatz length, these are called a Collatz triple, and so on.

Anyway, Schiffman's talk was about this. Note: it is still unknown if any positive integer can be reduced to 1 by this process, though this has been verified for all numbers <= 3 x 2^(53). Then Mahmoud Almanassra closed the session with a nice talk about finding estimators for parameters of "quality adjusted lifetime hazard functions.

My reliability engineering course is a long way in my past, so here is my best recollection of the issues: a hazard function is a probability distribution which models the lifetime of something (human, electronic component, etc.) Typically, the density function follows a bathtub curve; models have initial defects, then go through ok, then start dying off near the end of their lives. If one wants to measure only for a certain length of time and then stop counting, one is "censoring" the data.

There are estimators for the parameters of such functions, but unfortunately two of the most common ones are not monotonic. This is bad because a non-monotonic estimator might well provide for "births after deaths." Almanassra showed how to correct for this.

Then, there is something called the quality adjusted hazard function; one takes into account the "quality of life" (obvious meaning in humans, say, usefulness in components?) He then develops consistent, monotone estimators for this setting.