The above is an example of a “wild arc”: it is an arc in space (a continuous image of the unit interval ) 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”.

I remember looking at the course schedule when I was near the end of my first semester, senior year. I noticed that a “topology” course was being offered; I also remember reading a bit about “topology” in the Time-Life book on mathematics. I remembered a donut shaped object (called torus), Klein Bottles, Mobius bands and the like. I wondered if the course would be a sequence of parlor tricks.

Then when I got to the course; well….it wasn’t what I expected.

Ok, what is going on? I heard terms like “open”, “closed”, “basis”, “Hausdorf”, “Regular”, “Normal”, “second countable”, “first countable” etc. It reminded me of “analysis on steroids”. I began to wonder if the Time-Life book was just making it all up.

But it wasn’t; toward the back of our text (Munkres) there was even a drawing of a double torus.

So, what in the world is going on?

Well, I’ll just write informally; if you can catch my omissions you already know this stuff. 🙂

Probably the most basic question topology asks is: “when are two spaces “the same””. So, what do you mean by “the same”?

In topology the answer is almost always: given spaces is there a continuous bijection (one to one and onto function) which as a continuous inverse ? If there is such a function, the spaces are said to be “topologically equivalent” or “homeomorphic”. You might notice the term “continuous” and wonder what it means in an abstract context like this.

The usual calculus “epsilon-delta” definition works if are real n-spaces with the usual open intervals/disks/balls, etc.
So here is an elementary example: the unit interval is homeomorphic to any other closed interval . Here is the proof: the map (called a “homeomorphism”) is given by . Notice that is a bijection and has a continuous inverse. On the other hand, is NOT homeomorphic to . The standard way to see the latter is to use the tools developed in either analysis courses or a beginning topology course that I am describing; the quick answer is that the closed interval is “compact” whereas the half-open interval is not. Or, another way: has all of its limit points; is missing a limit point.

You go on to talk about what an open set is. The calculus notion is that an open set is one that is built up as the collection of open intervals (in the real line) or open disks, open balls, etc. In topology, you take a collection of sets and if this collection meets the following properties: the whole space and the empty sets belong to and if an arbitrary union of subsets of belong to and any FINITE intersection of elements of belong to , then is said to be a topology for the space.

Clearly, the standard “open” intervals form a topology for the real line; we do calculus with these. But one can form a topology generated by, say, half open intervals (these things are bizarre) or other stranger collection of sets.

Yes, the complement of an open set is a “closed set”, and yes, in some topological spaces, there are sets that are both open and closed at the same time. (grrr…) Example: given a set , declare EVERY point in to be an open set.

It is the study of these things that often constitute the first part of a first topology course. And yes, you DO need to know this stuff.

So what about the geometric stuff?
Well, let’s start small. You know that there is a bijection between and the unit circle (the bijection is . You also know that and the unit circle are both compact sets (think: “closed and bounded” if you are new). But they are NOT homeomorphic sets. Learning why starts you in the more geometric direction. One easy way: if you look at the intervals, removal of any point except the end points separates the intervals into two pieces, where no one point separates the unit circle into two pieces. That is a more “global” property.

So in this more global view, you’ll learn not only geometric type arguments but also how do use algebra (yes, at advanced levels, math is not as compartmentalized as it appears to many undergraduates). That is part of algebraic topology.

And, of course, to simplify the types of objects studied, one might want to but a differential structure on a space (assign the notion of a derivative and “tangents”) by attaching something called the “tangent bundle” to a space. That is the subject of differential topology. Here the homeomorphisms are often required to be infinitely differentiable as well.

So, yes, there IS a connection between the “rubber sheet” geometry stuff that you read about in the popular media and the abstract sounding stuff that you sometimes get at the start of an undergraduate topology class. It just takes a bit of time and effort to get there.

Now the type of topology that really never gets to the geometric stuff is called “point set” topology; there is something called “general topology” too. (don’t ask). 🙂

Ok, you are taking, maybe an analysis class or perhaps your first abstract algebra class. You are learning how a proof works. Of course, you might be studying a proof of, say, one of the Sylow Theorems in group theory, or perhaps a convergence proof in analysis.

That proof is elegant and to the point, isn’t it? But here are some things to remember:

1. You are seeing “what worked”; you aren’t seeing the scores of attempts that failed.

Example: one of my papers contains a counterexample to something I thought “for sure” was true; in fact I spent 2 years trying to “prove” the conjecture that I ended up publishing the counterexample for!

2. You are seeing a polished proof.

Example: right now, I am finishing up a paper on wild knots (simple closed curves in 3 space that are not deformable to smooth simple closed curves). I spend 3-4 days on one step of a construction, only to realize that not only were my steps not convincing, they WEREN’T at all necessary!

Here is what lead me to realize I was headed toward a dead end: I was proving something that directly depended on specific properties of the type of knots that I was studying, yet my construction was not using those properties. I was doomed to fail if I kept on this path!

For the record, here is the mistake that I was making:

Suppose you have an annulus in the plane; example: . Now suppose you take another annulus in the region above the plane and attach it to along its two boundary circles. You get a torus . But is necessarily unknotted in 3 space? Hint: we knot that in , bounds at least ONE solid torus, but does it bound two of them?

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.

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 to the semi-circle from the graph: . 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.

Imagine a world in which it is possible for an elite group of hackers to install a “backdoor” not on a personal computer but on the entire U.S. economy. Imagine that they can use it to cryptically raise taxes and slash social benefits at will. Such a scenario may sound far-fetched, but replace “backdoor” with the Consumer Price Index (CPI), and you get a pretty accurate picture of how this arcane economics statistic has been used.
Tax brackets, Social Security, Medicare, and various indexed payments, together affecting tens of millions of Americans, are pegged to the CPI as a measure of inflation. The fiscal cliff deal that the White House and Congress reached a month ago was almost derailed by a proposal to change the formula for the CPI, which Matthew Yglesias characterized as “a sneaky plan to cut Social Security and raise taxes by changing how inflation is calculated.” That plan was scrapped at the last minute. But what most people don’t realize is that something similar had already happened in the past. A new book, The Physics of Wall Street by James Weatherall, tells that story: In 1996, five economists, known as the Boskin Commission, were tasked with saving the government $1 trillion. They observed that if the CPI were lowered by 1.1 percent, then a $1 trillion could indeed be saved over the coming decade. So what did they do? They proposed a way to alter the formula that would lower the CPI by exactly that amount!
This raises a question: Is economics being used as science or as after-the-fact justification, much like economic statistics were manipulated in the Soviet Union? More importantly, is anyone paying attention? Are we willing to give government agents a free hand to keep changing this all-important formula whenever it suits their political needs, simply because they think we won’t get the math?

Well, most probably won’t get the math and even more won’t be able to if some have their way:

Ironically, in a recent op-ed in the New York Times, social scientist Andrew Hacker suggested eliminating algebra from the school curriculum as an “onerous stumbling block,” and instead teaching students “how the Consumer Price Index is computed.” What seems to be completely lost on Hacker and authors of similar proposals is that the calculation of the CPI, as well as other evidence-based statistics, is in fact a difficult mathematical problem, which requires deep knowledge of all major branches of mathematics including … advanced algebra.
Whether we like it or not, calculating CPI necessarily involves some abstract, arcane body of math. If there were only one item being consumed, then we could easily measure inflation by dividing the unit price of this item today by the unit price a year ago. But if there are two or more items, then knowing their prices is not sufficient.

The article continues on; it is well worth reading.

This debate matters. Making mathematics mandatory prevents us from discovering and developing young talent. In the interest of maintaining rigor, we’re actually depleting our pool of brainpower. I say this as a writer and social scientist whose work relies heavily on the use of numbers. My aim is not to spare students from a difficult subject, but to call attention to the real problems we are causing by misdirecting precious resources.

The toll mathematics takes begins early. To our nation’s shame, one in four ninth graders fail to finish high school. In South Carolina, 34 percent fell away in 2008-9, according to national data released last year; for Nevada, it was 45 percent. Most of the educators I’ve talked with cite algebra as the major academic reason.

Shirley Bagwell, a longtime Tennessee teacher, warns that “to expect all students to master algebra will cause more students to drop out.” For those who stay in school, there are often “exit exams,” almost all of which contain an algebra component. In Oklahoma, 33 percent failed to pass last year, as did 35 percent in West Virginia.

Algebra is an onerous stumbling block for all kinds of students: disadvantaged and affluent, black and white. In New Mexico, 43 percent of white students fell below “proficient,” along with 39 percent in Tennessee. Even well-endowed schools have otherwise talented students who are impeded by algebra, to say nothing of calculus and trigonometry.

California’s two university systems, for instance, consider applications only from students who have taken three years of mathematics and in that way exclude many applicants who might excel in fields like art or history. Community college students face an equally prohibitive mathematics wall. A study of two-year schools found that fewer than a quarter of their entrants passed the algebra classes they were required to take.

“There are students taking these courses three, four, five times,” says Barbara Bonham of Appalachian State University. While some ultimately pass, she adds, “many drop out.”

Another dropout statistic should cause equal chagrin. Of all who embark on higher education, only 58 percent end up with bachelor’s degrees. The main impediment to graduation: freshman math. […]

In other words: math is too hard! 🙂

Well, “gee, I won’t need it!” Well, actually, math literacy is a prerequisite to understanding many seemingly unrelated things. For example, I am reading The Better Angels of our Nature by Steven Pinker. Though the book’s purpose is to demonstrate that human violence is trending downward and has been trending downward for some time, much of the argument is statistical; being mathematically illiterate would make this book inaccessible.

We some basic mathematics when in discussions on our economy. For example: how does one determine if, say, government spending is up or not? It isn’t as simple as counting dollars spent; after all, our population is growing and we’d expect a country with a larger population to spend more than a country with a smaller one. Then there is gross domestic product; spending is usually correlated with that; hence “government spending graphs” are usually presented in terms of “percent of GDP”. But then what if absolute spending hits a flat stretch and GDP falls, as it does during a recession? That’s right: a smaller denominator makes for a bigger number! You see this concept presented here.

But if you are mathematically illiterate, all of this is invisible to you.

Ever see the “jobs graph” that the current Presidential Administration touts?

Politics? If you can’t read a poll or understand what the polls are saying, you are basically sunk (as were many of our pundits in 2012). Of course, if you can’t understand a collection of polls, you can be a journalist or a pundit, but there is limited opportunity for that.

Science? Example: is evolution too improbable to have occurred? Uh, no. But you need some mathematical literacy to see why.

First of all, I’d like to make it clear that I am unqualified to talk about teaching mathematics at the junior high and high school level. I am qualified to make comments on what sorts of skills the students bring with them to college.

But I am interested in issues affecting mathematics education and so will mention a couple of them.

1. California is moving away from having all 8’th graders take “algebra 1”. Note: I was in 8’th grade from 1972-1973. Our school was undergoing an experiment to see if 8’th graders could learn algebra 1. Being new to the school, I was put into the regular math class, but was quickly switched into the lone section of algebra 1. The point: it wasn’t considered “standard for everyone.”

My “off the cuff” remarks: I know that students mature at different rates and wonder if most are ready for the challenge by the 8’th grade. I also wonder about “regression to the mean” effects of having everyone take algebra 1; does that force the teacher to water down the course?

I read Epstein School head Stan Beiner’s guest column on what kids really need to know for college with great interest because one of the main goals of my 40-years as a college professor was to help my students make a successful transition from high school to college.

I taught thousands of freshmen in Introductory Psychology classes and Freshman Learning Communities, and I was constantly amazed by how many of them suffered from a severe case of “culture shock” when they moved from high school to college.

I used one of my assignments to identify these cultural differences by asking my students to create suggestions they would like to give their former high school teachers to help them better prepare their students for college. A content analysis of the results produced the following six suggestion summaries.

The underlying theme in all these suggestions is that my students firmly believed they would have been better prepared for college if their high school teachers had provided them with more opportunities to behave in the responsible ways that are required for success in higher education […]

You can surf to the article to read the suggestions. They are not surprising; they boil down to “be harder on us and hold us accountable.” (duh). But what is more interesting, to me, is some of the comments left by the high school teachers:

“I have tried to hold students accountable, give them an assignment with a due date and expect it turned in. When I gave them failing grades, I was told my teaching was flawed and needed professional development. The idea that the students were the problem is/was anathema to the administration.”

“hahahaha!! Hold the kids responsible and you will get into trouble! I worked at one school where we had to submit a written “game plan” of what WE were going to do to help failing students. Most teachers just passed them…it was easier. See what SGA teacher wrote earlier….that is the reality of most high school teachers.”

“Pressure on taechers from parents and administrators to “cut the kid a break” is intense! Go along to get along. That’s the philosophy of public education in Georgia.”

“It was the same when I was in college during the 80’s. Hindsight makes you wished you would have pushed yourself harder. Students and parents need to look at themselves for making excuses while in high school. One thing you forget. College is a choice, high school is not. the College mindset is do what is asked or find yourself another career path. High school, do it or not, there is a seat in the class for you tomorrow. It is harder to commit to anything, student or adult, if the rewards or consequences are superficial. Making you attend school has it advantages for society and it disadvantages.”

My two cents: it appears to me that too many of the high schools are adopting “the customer is always right” attitude with the student and their parents being “the customer”. I think that is the wrong approach. The “customer” is society, as a whole. After all, public schools are funded by everyone’s tax dollars, and not just the tax dollars of those who have kids attending the school. Sometimes, educating the student means telling them things that they don’t want to hear, making them do things that they don’t want to do, and standing up to the helicopter parents. But, who will stand up for the teachers when they do this?

Note: if you google “education then and now” (search for images) you’ll find the above cartoons translated into different languages. Evidently, the US isn’t alone.

This New England Journal of Medicine article takes a look at well known “factoids” about obesity, and how many of them don’t stand up to statistical scrutiny. (note: the article is behind a paywall, but if you are university faculty, you probably have access to the article via your library.

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.