College Math Teaching

May 29, 2013

Thoughts about Formal Laurent series and non-standard equivalence classes

I admit that I haven’t looked this up in the literature; I don’t know how much of this has been studied.

The objects of my concern: Laurent Series, which can be written like this: \sum^{\infty}_{j = -\infty} a_j t^j ; examples might be:
...-2t^{-2} + -1t^{-1} + 0 + t + 2t^2 ... = \sum^{\infty}_{j = -\infty} j t^j . I’ll denote these series by p(t) .

Note: in this note, I am not at all concerned about convergence; I am thinking formally.

The following terminology is non-standard: we’ll call a Laurent series p(t) of “bounded power” if there exists some integer M such that a_m = 0 for all m \ge M ; that is, p(t) = \sum^{k}_{j = -\infty} j t^j for some k \le M .

Equivalence classes: two Laurent series p(t), q(t) will be called equivalent if there exists an integer (possibly negative or zero) k such that t^k p(t) = q(t) . The multiplication here is understood to be formal “term by term” multiplication.

Addition and subtraction of the Laurent series is the usual term by term operation.

Let p_1(t), p_2(t), p_3(t)....p_k(t).... be a sequence of equivalent Laurent series. We say that the sequence p_n(t) converges to a Laurent series p(t) if for every positive integer M we can find an integer n such that for all k \ge n, p(t) - p_k = t^M \sum^{\infty}_{j=1} a_j t^j ; that is, the difference is a non-Laurent series whose smallest power becomes arbitrarily large as the sequence of Laurent series gets large.

Example: p_k(t) = \sum^{k}_{j = -\infty} t^j converges to p(t) = \sum^{\infty}_{j = -\infty} t^j .

The question: given a Laurent series to be used as a limit, is there a sequence of equivalent “bounded power” Laurent series that converges to it?
If I can answer this question “yes”, I can prove a theorem in topology. 🙂

But I don’t know if this is even plausible or not.

May 24, 2013

Beware of the limiting process….

Filed under: advanced mathematics, research — collegemathteaching @ 12:49 pm

If you’ve done mathematical research, you are probably aware of the following minefield (especially if you study non-compact spaces):

1. Establish f_1 = f_2 = ....f_n for all n \in \{1, 2, 3, ...\} . Note: I am abusing the = sign here; I mean “equivalence class equality”.

2. Then try to conclude that lim_{n \rightarrow \infty}f_n = f only to find out….that the limit fails to exist, though it might exist if you put restrictive conditions on either the f_n or on HOW the equivalence is obtained. 😦

This really shouldn’t surprise me at all; after all one of the things we teach our advanced calculus students is this example:

let f_0(x) = 1, 0 \le x \le 1 . Index the rational numbers by q_i
Let f_n(x) = 1 if x \notin \{q_1, q_2, ... q_n\} , f_n(x) = 0 if x \in \{q_1, q_2, ...q_n \} . Then, while it is true that \int^1_0 f_n(x) dx = 1 for all n , the limit lim_{n \rightarrow \infty}f_n fails to be Riemann integrable (though it is Lebesgue integrable).

How quickly I sometimes forget the basics. 🙂

May 22, 2013

In the news….and THINK before you reply to an article. :-)

Ok, a mathematician who is known to be brilliant self-publishes (on the internet) a dense, 512 page proof of a famous conjecture. So what happens?

The Internet exploded. Within days, even the mainstream media had picked up on the story. “World’s Most Complex Mathematical Theory Cracked,” announced the Telegraph. “Possible Breakthrough in ABC Conjecture,” reported the New York Times, more demurely.

On MathOverflow, an online math forum, mathematicians around the world began to debate and discuss Mochizuki’s claim. The question which quickly bubbled to the top of the forum, encouraged by the community’s “upvotes,” was simple: “Can someone briefly explain the philosophy behind his work and comment on why it might be expected to shed light on questions like the ABC conjecture?” asked Andy Putman, assistant professor at Rice University. Or, in plainer words: I don’t get it. Does anyone?

The problem, as many mathematicians were discovering when they flocked to Mochizuki’s website, was that the proof was impossible to read. The first paper, entitled “Inter-universal Teichmuller Theory I: Construction of Hodge Theaters,” starts out by stating that the goal is “to establish an arithmetic version of Teichmuller theory for number fields equipped with an elliptic curve…by applying the theory of semi-graphs of anabelioids, Frobenioids, the etale theta function, and log-shells.”

This is not just gibberish to the average layman. It was gibberish to the math community as well.

[…]

Here is the deal: reading a mid level mathematics research paper is hard work. Refereeing it is even harder work (really checking the proofs) and it is hard work that is not really going to result in anything positive for the person doing the work.

Of course, if you referee for a journal, you do your best because you want YOUR papers to get good refereeing. You want them fairly evaluated and if there is a mistake in your work, it is much better for the referee to catch it than to look like an idiot in front of your community.

But this work was not submitted to a journal. Interesting, no?

Of course, were I to do this, it would be ok to dismiss me as a crank since I haven’t given the mathematical community any reason to grant me the benefit of the doubt.

And speaking of idiots; I made a rather foolish remark in the comments section of this article by Edward Frenkel in Scientific American. The article itself is fine: it is about the Abel prize and the work by Pierre Deligne which won this prize. The work deals with what one might call the geometry of number theory. The idea: if one wants to look for solutions to an equation, say, x^2 + y^2 = 1 one gets different associated geometric objects which depend on “what kind of numbers” we allow for x, y . For example, if x, y are integers, we get a 4 point set. If x, y are real numbers, we get a circle in the plane. Then Frenkel remarked:

such as x2 + y2 = 1, we can look for its solutions in different domains: in the familiar numerical systems, such as real or complex numbers, or in less familiar ones, like natural numbers modulo N. For example, solutions of the above equation in real numbers form a circle, but solutions in complex numbers form a sphere.

The comment that I bolded didn’t make sense to me; I did a quick look up and reviewed that |z_1|^2 + |z_2|^2 = 1 actually forms a 3-sphere which lives in R^4 . Note: I added in the “absolute value” signs which were not there in the article.

This is easy to see: if z_1 = x_1 + y_1 i, z_2 = x_2 + y_2i then |z_1|^2 + |z_2|^2 = 1 implies that x_1^2 + y_1^2 + x_2^2 + y_2^2 = 1 . But that isn’t what was in the article.

Frenkel made a patient, kind response …and as soon as I read “equate real and imaginary parts” I winced with self-embarrassment.

Of course, he admits that the complex version of this equation really yields a PUNCTURED sphere; basically a copy of R^2 in R^4 .

Just for fun, let’s look at this beast.

Real part of the equation: x_1^2 + x_2^2 - (y_1^2 + y_2^2) = 1
Imaginary part: x_1y_1 + x_2y_2 = 0 (for you experts: this is a real algebraic variety in 4-space).

Now let’s look at the intersection of this surface in 4 space with some coordinate planes:
Clearly this surface misses the x_1=x_2 = 0 plane (look at the real part of the equation).
Intersection with the y_1 = y_2 = 0 plane yields x_1^2+ x_2^2 = 1 which is just the unit circle.
Intersection with the y_1 = x_2 = 0 plane yields the hyperbola x_1^2 - y_2^2 = 1
Intersection with the y_2 = x_1 = 0 plane yields the hyperbola x_2^2 - y_1^2 = 1
Intersection with the x_1 = y_1 = 0 plane yields two isolated points: x_2 = \pm 1
Intersection with the x_2 = y_2 = 0 plane yields two isolated points: x_1 = \pm 1
(so we know that this object is non-compact; this is one reason the “sphere” remark puzzled me)

Science and the media
This Guardian article points out that it is hard to do good science reporting that goes beyond information entertainment. Of course, one of the reasons is that many “groundbreaking” science findings turn out to be false, even if the scientists in question did their work carefully. If this sounds strange, consider the following “thought experiment”: suppose that there are, say, 1000 factors that one can study and only 1 of them is relevant to the issue at hand (say, one place on the genome might indicate a genuine risk factor for a given disease, and it makes sense to study 1000 different places). So you take one at random, run a statistical test at p = .05 and find statistical significance at p = .05 . So, if we get a “positive” result from an experiment, what is the chance that it is a true positive? (assume 95 percent accuracy)

So let P represent a positive outcome of a test, N a negative outcome, T means that this is a genuine factor, and F that it isn’t.
Note: P(T) = .001, P(F) = .999, P(P|T) = .95, P(N|T) = .05, P(P|F) = .05, P(N|F) = .95 . It follows P(P) = P(T)P(P \cap T)P(T) + P(F)P(P \cap F) = (.001)(.95) + (.999)(.05) = .0509

So we seek: the probability that a result is true given that a positive test occurred: we seek P(T|P) =\frac{P(P|T)P(T)}{P(P)} = \frac{(.95)(.001)}{.0509} = .018664. That is, given a test is 95 percent accurate, if one is testing for something very rare, there is only about a 2 percent chance that a positive test is from a true factor, even if the test is done correctly!

May 17, 2013

College Misery: Poem about Residue Integrals

Filed under: academia, advanced mathematics, calculus, complex variables, integrals — Tags: , — collegemathteaching @ 12:40 am

Seriously. Check it out.

Screen shot 2013-05-16 at 7.39.11 PM

May 16, 2013

Big Breakthrough in Number Theory: progress toward the twin primes conjecture.

Filed under: advanced mathematics, number theory — Tags: , , — collegemathteaching @ 7:26 pm

It is a long standing conjecture in number theory that there exists an infinite number of twin primes: twin primes are prime integers that differ by 2.

Example: 3 and 5, 11 and 13, 17 and 19 are examples of twin prime pairs.

Very large twin primes have been found: (2,003,663,613 \times 2^{2195,000}) - 1 and (2,003,663,613 \times 2^{2195,000}) + 1 .
But, up to now: We don’t know if this pairing “stops” at some point (is there a largest pair?)

In fact, up to recently, we had no statement of the following form: given a finite integer M there exists an infinite number or pairs of primes p, q such that p - q \le M (assuming that p is the greater of the pair).

Well, now we do. The Annals of Mathematics (the top ranked mathematics journal in the world) has accepted a paper that shows the infinite pairs statement is true, if M = 70,000,000 :

The twin prime conjecture says that there is an infinite number of such twin pairs. Some attribute the conjecture to the Greek mathematician Euclid of Alexandria, which would make it one of the oldest open problems in mathematics.

The problem has eluded all attempts to find a solution so far. A major milestone was reached in 2005 when Goldston and two colleagues showed that there is an infinite number of prime pairs that differ by no more than 16. But there was a catch. “They were assuming a conjecture that no one knows how to prove,” says Dorian Goldfeld, a number theorist at Columbia University in New York.

The new result, from Yitang Zhang of the University of New Hampshire in Durham, finds that there are infinitely many pairs of primes that are less than 70 million units apart without relying on unproven conjectures. Although 70 million seems like a very large number, the existence of any finite bound, no matter how large, means that that the gaps between consecutive numbers don’t keep growing forever. The jump from 2 to 70 million is nothing compared with the jump from 70 million to infinity. “If this is right, I’m absolutely astounded,” says Goldfeld.

Zhang presented his research on 13 May to an audience of a few dozen at Harvard University in Cambridge, Massachusetts, and the fact that the work seems to use standard mathematical techniques led some to question whether Zhang could really have succeeded where others failed.

But a referee report from the Annals of Mathematics, to which Zhang submitted his paper, suggests he has. “The main results are of the first rank,” states the report, a copy of which Zhang provided to Nature. “The author has succeeded to prove a landmark theorem in the distribution of prime numbers. … We are very happy to strongly recommend acceptance of the paper for publication in the Annals.”

Hey, 70 million is a LOT less than “infinity”. 🙂

May 1, 2013

Job Candidates in Today’s Math Professor Job Market: take a second post-doc?

Filed under: academia, calculus, editorial, research — collegemathteaching @ 9:02 pm

This American Mathematical Society blog post is interesting:

This is a post that requests comments and advice from the readers. The issue is the following: Student X finished her PhD in mathematics at a highly-ranked university. Upon graduating, she was able to get a three-year postdoc position in a math department that emphasizes research, so that everything seemed to go well so far. As she was finishing her postdoc and searching for jobs, she had a couple of interviews but nothing very promising and then it was March and X had no tenure-track job offers. However, she had an offer to do a second postdoc at a different university for 2 more years. Question #1: Is it a good idea to do a second postdoc if her plan is to get a tenure-track faculty position? In the absence of another option, obviously X took the second postdoc offer. She thinks that, at the end of her second postdoc, getting a faculty position at a highly-ranked research university will be very difficult. She also likes teaching and has done some teaching as a postdoc; however, she thinks she should take action and get involved in activities that will appeal to undergraduate institutions and liberal arts colleges where teaching is emphasized more than the research (although research is also important). Question #2: Is this a good plan and, if so, what type of activities should she get involved in as a postdoc so that her file looks attractive to undergraduate institutions the next time she applies for jobs?

I teach at a 11-12 hour load “teaching institution” that has a modest…but real..research requirement. You are expected to publish but obviously not in Annals of Mathematics. 🙂

Two thoughts:
1. We teach a LOT of calculus (mostly for engineering or business) and we’d expect a solid record of teaching success in calculus. We don’t want to hire new problems. Stuff in “course development” probably isn’t necessary; there is time to grow into that when they join our department.

2. We also have that modest research requirements; we expect new faculty to be excited about mathematics and be willing to tell us what they are doing. And if someone doesn’t publish (or submit stuff) in a post-doc situation, given the lighter load and division I resources and seminars, etc., then they don’t have a chance of publishing at place like ours.

So I’d say: take the post-doc, get some research done, and have experienced faculty watch you teach calculus so they can give a nice reference letter.

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