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.

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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 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.

April 30, 2013

My research

Filed under: advanced mathematics, research, topology — Tags: — oldgote @ 9:18 pm

If you are interested, I’ll be posting on topics that deal with my current area of research here.

I am reproducing this post below but will NOT publish further installments on this blog:

A knot is an embedding of the circle \{(x,y)|x^2 + y^2 =1 \} into 3-space. By “3-space” we usually mean R^3 or S^3 , which is the 3-sphere, which can be thought of as R^3 with a point added at infinity. S^3 is sometimes preferred because it is a compact space.

Note: sometimes we focus on the image of the embedding itself (i. e., the geometric object) and sometimes we focus on the map, which includes information about orientation.

Example: If one has t \in [0, 2\pi), then f(t) = (x(t), y(t), z(t)),  x(t) = (2+cos(3t))cos(2t), y(t) = (2+cos(3t))sin(3t), z(t) = sin(3t) is a knot. Here are two different MATLAB plots of the image:

trefoil

trefoilproject

The second is a projection of the image of the trefoil onto the x, y plane. If we endow such a projection with “crossing information”, we call the image a diagram for the knot.

trefoilDiagram

Here, the broken line indicates that the strand passes under another strand.

It is custom to insist on “regular” projections, which means that:

1. All “singularities” (points on the diagram which correspond to more than one point of the knot) are double points (there are no points where 3 or more strands of the knot’s projection meet)
2. All crossings are “honest” crossings; that is there are no “tangents” (places where the projection “kisses” another strand).

Note: one can think of a diagram as a “shadow” of the knot on a plane, provided one adds crossing information at all double points.

Now not all knots possess a diagram, but it is a known fact that all smooth knots (knots that arise from differentiable embeddings) and all picewise linear knots (knots whose image consists of a finite number of straight line segments glued end to end) have a projection.

Most of knot theory research deals with smooth or piecewise linear embeddings of the circle into S^3 or R^3 . There is knot theory of similar embeddings into other 3-manifolds, embeddings of S^2 into S^4 (higher dimensional knot theory) or embeddings of graphs into S^3 .

Also, link theory deals with multiple knots together.

BorromeanRings.svg

The above shows the Borromean Rings, which are three linked knots, no two of which are linked to each other. This is a famous 3-component link.

This blog will mostly focus on the following:
1. non-smooth (and non-piecewise linear) embeddings of the circle into S^3 .

twowildknots

These two diagrams are of non-smooth (and non-p. l.) knots; we call these wild knots. Notice how the stitches and arcs get smaller and converge to a point? That point is called a wild point. I will give a precise definition later; for right now we’ll tell you that it is impossible to assign a tangent vector to those points in some well defined way.

2. An arc is the image of [0,1] into 3 space. The mathematics of smooth (or p. l.) arcs in 3-space is pretty boring. Every smooth or p. l. arc “can be straightened in space” into a straight, boring arc.
On the other hand, the mathematics of wild arcs (think: non-smooth/p. l. ) is every interesting.

foxartinarc

The above arc has two wild points (the end points) and can NOT be straightened out in space into a straight arc. We’ll make this concept clear a bit later in another post).

3. Straight lines (a copy of the real line) into open 3-manifolds; we will insist that the “two infinities” of the line go to the “infinities” in the manifold.

someproperknots

In the above, the reader is invited to think of the “line” being embedded in the space D^2 \times R where D^2 is the standard 2-disk. Think of an infinitely long solid tube or cylinder (like a long pipe).
I will call this Proper Knot Theory; the term “proper” is a technical term, which I will explain here: a continuous map f:X \rightarrow Y is said to be proper if for all compact sets C \subset Y, f^{-1}(C) is compact. Here is an example of a non-proper embedding: consider f: R \rightarrow R given by f(x) = arctan(x) . The inverse image of [0, \frac{\pi}{2}] is not compact.

Equivalence Classes for Knots
In most of knot theory, what is studied is NOT the knots themselves but their “equivalence classes”. For example: the first example of the knot we have had a very specific function to define it. However, if we were to say, take a strand of the knot and move it a little, we’d get a different embedding, but mathematically we’d want to think of it as being “the same as” the original embedding. This makes the subject much more doable. Besides, knot theory is studied mostly because it impacts the study of the topology of 3-manifolds: such spaces are modified by doing operations (called “surgery”) which are often defined as being done along some embedded circle: a knot. In many cases, the objected obtained doesn’t differ “topologically” if the surgery knot is changed by some “motion of space”.

The same principle often applies if a scientist is, say, studying a knotted molecule or DNA strand.

knotteddna

So we need to state the equivalence classes.

Classical Knot Theory (the kind most often done)
Note: sometimes oriented knots are studied (the diagrams have arrows) and sometimes the unoriented knots are studied (no arrows). Sometimes this makes a difference as we shall see later.

project4

The above is an example of an oriented knot diagram.

The most common equivalence class used:
Given two knots (or links) in three space, say, K_1, K_2 ; we say that K_1 is equivalent to $K_2$ if there is a map called an “ambient isotopy” that connects the two. More particularly there exists F: S^3 \times [0,1] \rightarrow S^3 where:
1. F(-,t) is a homemomorphism of S^3 for all t \in [0,1] .
2. F(K,0) = K_1 and F(K,1) = K_2 for some K \subset S^3 , K homeomorphic to the circle.
The above is just a fancy way of saying that we can “deform space” to turn K_1 into K_2; almost never do we worry about finding, say, a formula for F.

It turns out that this definition is equivalent to the following simpler definition: K_1, K_2 are equivalent knots if there is some orientation preserving homeomorphism f: S^3 \rightarrow S^3 such that f(K_1) = K_2 . Needless to say, this is easier to state, but one loses the sense of taking a knotted piece of string and playing with it (which is what you are doing in the first definition).

There is also another type of equivalence that is used: two knots K_1, K_2 can be declared to be equivalent if there is a homeomorphism (possibly non-orientation preserving) f: S^3 \rightarrow S^3 such that f(K_1) = K_2 .

If f is orientation reversing and f(K_1) = K_2 then K_1 and K_2 are called mirror images.

So, classical knot theory (the kind most often studied) boils down to four different kinds:
1. oriented knots; mirror images considered equivalent.
2. oriented knots; mirror images NOT automatically considered equivalent.
3. non-oriented knots, mirror images considered equivalent.
4. non-oriented knots, mirror images not automatically considered equivalent.

A knot that is different from a knot with the same image but with a different orientation (arrow direction) is said to be non-invertible.
A knot that is different from its mirror image is said to be chiral.

The trefoil knot: is chiral but invertible (you can reverse the arrows by an orientation preserving homeomorphism)
The figure 8 knot: is NOT chiral and is invertible.

Blue_Figure-Eight_Knot

Non-invertible knots exist; here is an example: (8_{17} )

8_17_Knot.svg

The astute reader might wonder: “hey, you didn’t say anything about your isotopy or homeomorphism being smooth, piecewise linear or merely topological”. It turns out that in classical knot theory, this is a settled foundational question and therefore unimportant (here and here).

However this issue does appear in other kinds of knot theory, including those we will be discussing.

Wild knots
A knot (link or arc) is said to be tame if it is equivalent to a smooth (or p. l.) knot (equivalence class of choice). If it isn’t, it is called wild.
Note: it isn’t always immediately obvious if an arc is wild or tame; for example, the arc in the upper left hand corner is wild (wild point is the left end point) whereas the the lower right arc (which has separate trefoil knots converging to an endpoint) is actually tame!

wildkarcs

We will discuss this later; note that the “infinite trefoil” arc is just on the edge of being wild; were we to add on, say, a straight segment at the left hand endpoint and extend it any finite distance at all, the arc would become wild. That appears to make no sense at all (at first glance) but in a later post I will provide a proof.

We will study wild knots of various kinds; note: it is possible for a knot to be wild at ALL of its points. We’ll get to this in a later post; if you can’t wait, here is an example: consider the following picture, which is supposed to represent a nested series of solid tori, (think: a bagel or doughnut) which are nested inside one another. If we intersect all of these knotted up tori, we end up with a very ill behaved wild knot in 3-space; this knot is wild at all of its points:

bingsling

I am running out of steam; so in our next installment I’ll talk about different types of equivalence classes for knots in 3 space and for lines (proper knots) in open 3-manifolds. (note: I’ll post this on my research blog, not here).

April 27, 2013

Unsolicited advise to young professors at heavy teaching load universities: Go to Research Conferences anyway!

This is coming to you from Ames, Iowa at the Spring American Mathematical Society Meeting. I am here to attend the sessions on the Topology of 3-dimensional manifolds.

Note: I try to go to conferences regularly; I have averaged about 1 conference a year. Sometimes, the conference is a MAA Mathfest conference. These ARE fun and refreshing. But sometimes (this year), I go to a research oriented conference.

I’ll speak for myself only.

Sometimes, these can be intimidating. Though many of the attendees are nice, cordial and polite, the fact is that many (ok, almost all of them) are either the best graduate students or among the finest researchers in the world. The big names who have proved the big theorems are here. They earn their living by doing cutting edge research and by guiding graduate students through their research; they are not spending hours and hours convincing students that \sqrt{x^2 + y^2} \ne x + y .

So, the talks can be tough. Sure, they do a good job, but remember that most of the audience is immersed in this stuff; they don’t have to review things like “normal surface theory” or “Haken manifold”.

Therefore, it is VERY easy to start lamenting (internally) “oh no, I am by far the dumbest one here”. That, in my case, IS true, but it is unimportant.
What I found is that, if I pay attention to what I can absorb, I can pick up a technique here and there, which I can then later use in my own research. In fact, just today, I picked up something that might help me with a problem that I am pondering.

Also, the atmosphere can be invigorating!

I happen to enjoy the conferences that are held on university campuses. There is nothing that gets my intellectual mood pumped up more than to hang around the campus of a division I research university. For me, there is nothing like it.

This conference
A few general remarks:
1. I didn’t realize how pretty Iowa State University is. I’d rank it along with the University of Tennessee as among the prettiest campuses that I’ve ever seen.

2. As far as the talks: one “big picture” technique that I’ve seen used again and again is the technique of: take an abstract set of objects (say, the Seifert Surfaces of a knot; say of minimal genus. Then to each, say, ambient isotopy class of Seifert Surface, assign a vertex of a graph or simplicial complex. Then group the vertices together either by a segment (in some settings) or a simplex (if, in one setting, the Seifert Surfaces admit disjoint representatives). Then one studies the complex or the graph.

In one of the talks (talking about essential closed surfaces in the complement of a knot), one assigned such things to the vertex of a graph (dendron actually) and set up an algorithm to search along such a graph; it turns out that is one starts near the top of this dendron, one gains the opportunity to prune lower branches of the group by doing the calculation near the top.

Sidenote
The weather couldn’t be better; I found time over lunch to do a 5.7 mile run near my hotel. The run was almost all on bike paths (albeit a “harder” surface than I’d like).

February 28, 2013

Math is not easy!

Filed under: advanced mathematics, research, topology — collegemathteaching @ 7:05 pm

penetrationindex

The above is an example of a “wild arc”: it is an arc in space (a continuous image of the unit interval [0,1] ) that is so pathologically embedded that it cannot be “straightened out” by a deformation of 3-space. Or if you have had some calculus, it is impossible to define a tangent vector to the arc at two points; in this case, the end points.

Now do you see the red circles around one of the end points? Those represent the equator of a round sphere whose “center” is at the end point. As you can see, the arc hits those spheres at 3 points. So, take any old sphere that has that endpoint “inside of it” (technically, inside the ball bounded by the sphere, and for you experts, we insist that the sphere be a “smooth” sphere) . Now, what is the fewest number of points that this arc meets such a sphere? That is called the “penetration index” of the endpoint.

It sure looks like the penetration index is 3, but all this picture does is to show that the penetration index is at most three. How do you know that there isn’t some sphere that you haven’t thought of that contains the endpoint inside of it (ok, inside of the region bounded by the sphere) that hits this arc in two points, or even one point?

It turns out that no such sphere exists, but that requires proof and the proofs aren’t always that easy.

I bring this up because I was doing a calculation similar to this and wondered why it was getting so complex; then I realized that my calculation was akin to calculating a “penetration index”.

February 13, 2013

Writing Math: being comprehensible and accurate at the same time….ARGGGHHH!!!

Filed under: advanced mathematics, calculus, editorial, research, topology — collegemathteaching @ 7:47 pm

One of the irritating things about writing mathematics is that one has to be accurate in what one says, but one also WANTS to speak comprehensibly. Here is what I am trying to say: “for a certain subclass X of wild knots, to each equivalence class of knots of class X there corresponds a sequence of equivalences classes of tamely embedded solid tori.” This is accurate but obscures what I am trying to say: sequences of solid tori are a knot invariant for knots of type X.”

(If you are wondering what I am talking about: a tame solid torus can be thought of as a possibly knotted solid bagel in space. A smooth knot is an image of a differentiable, one to one map of the unit circle into 3 space, and a wild knot is an image of a continuous map of the unit circle into 3 space which cannot be deformed (by a continuous, one to one function of 3-space to itself) into a smooth knot.

wildKnot

This is an example of a wild knot; one can define a tangent vector to every point of this knot EXECPT for one exceptional point. It is that point that makes the knot “wild”.

The knots I am studying are wild at ALL of their points.

Note: if you wondered “why is he being so stilted in his definition of “wild knot””, here is why: consider the following image of a circle: join the graph of f(x) = xsin(\frac{1}{x}) to the semi-circle from the graph: g(x) = -\sqrt{\frac{1}{4} - (x - \frac{1}{2})^2}. This forms the continuous, non-differentiable, one to one image of a circle. But it is NOT wild; it can be shown that this knot is actually equivalent to a differentiable knot (the “unknot” actually).

This is an excellent example of precision conflicting with ease of understanding.

February 5, 2013

The Rabbit Hole of the Sabbatical Paper

Filed under: advanced mathematics, research — Tags: — collegemathteaching @ 8:12 pm

First of all, let me be clear: I LOVE being on sabbatical (one semester plus a summer). And this time, I did something different: I started my paper at the beginning of the sabbatical rather than doing the “relearning stuff I once knew but forgot or never learned that well to begin with” stuff.

And yes, the idea for the paper formed quickly and started writing….and have now entered the rabbit hole.

It works something like this: “wow, I know how to prove Theorem X for a class of objects Y” and you start to write the proof. As you go you realize “oh wait, I really don’t need this hypothesis, and if I use this other technique, that I only “sort of” understand, I’ll be able to prove something much stronger. There is a temptation to just put the idea off, but no one wants a referee’s report that says that the author of the submitted paper is, well, a lazy idiot. So there I go…making sure “does this REALLY follow? It seems counter intuitive”…and I disappear.

I’ve got to give myself a time limit though.

January 21, 2013

Research Paper Writing: into the cursing stage….

Filed under: academia, advanced mathematics, research, topology — Tags: — collegemathteaching @ 9:14 pm

I am attempting to end a publication drought; I still have one paper at a referee and I am currently writing a more research level paper (it deals with the subject of “wild knots” in 3-space: these are topological homeomorphic images of the circle, but these embeddings are NOT “equivalent” to any differential embedding of the circle in 3-space; it is “no fair” doing calculus in this setting! 🙂 )

I *think* that I have a good idea on how to procede and the first bits have been typeset with no difficulties. But now comes the “cursing” part: I am trying to organize an infinite number of “maps” in a way that allows one to compose these maps in a well defined way.

The frustrating part is that I have to be precise in my language, and at times, precision tends to bury and obscure the idea. People who teach undergraduate mathematics for a living sure understand that! 😉

But the precision is necessary; I remember spending 2 years trying to prove something that was “obviously true”; it turns out that it was false. But I was able to publish the counter-example, and even the reviewer thought that this example was counter-intuitive.

But that was many years ago….so……bring on the cursing!!!!!!

October 19, 2012

If you have to referee an article…

Filed under: academia, advanced mathematics, research — oldgote @ 6:20 pm

Please, if you accept the task of refereeing an article, please do your job seriously.

We don’t need this.

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