The Unapologetic Mathematician

There has been some discussion on “not coming off as a crackpot”, inspired by the recent purported proof of the Riemann Hypothesis.

Let me be clear here: Li may or may not have handled his potential proof in the most elegant way possible, but I most definitely do not consider him to have come off as a crackpot. Why? Because this is what a crackpot sounds like:

from: shishkindima2@mail.ru
subject: The Quantum geometry is created

To the world scientific community of units

There can be no doubt about the world is standing on the doorstep of the new scientific system of views and ways of perception. In the history and philosophy such situation is called “the changing of the scientific paradigm” and it means in reality the scientific revolution. To all appearances this changing will have unbelievable range and can change the root and the branch of our views to the world, the nature and human being. In such way a gap between the ancient wisdom and modern science, between the eastern mysticism, western pragmatism and positivism, between knowledge and intuition, mind and splash of mind is disappearing. Such important discoveries nearly almost have been made by unauthorized people or scientist with unusual turn of mind. A. Einstein, N. Bohr, M. Born, were “amateurs” and called them selves in a such way. I am not shy, that I’m an amateur too. It’s important to have an ability to look at old ideas in a new way. Somebody said once “We need crazy ideas, totally crazy to be right”. A. Einstein died about 50 years ago, and hasn’t fulfilled his dream - to create the unified theory, describing the Universe in the whole. The last couple dozen years of his life he dedicated to the searching of such theory that could explain everything - from elementary particle and their connection to global structure of the Universe. In spite of great effort Einstein hasn’t been succeed in it, because that time it was early for it. I’m a usual man from province had more luck, and the discovery is made. “The unified theory of Universal”, which is based on Quantum geometry is created. This theory explains lots of things. But most important thing is that most of what we call mysticism and superstition in our days contain knowledge that exceed the results of science. Technologies that are discovered will pass ahead of modern for thousands years. Right now we can begin to create teleportation modules which will be able to teleportate objects and people to every place on the Earth, solar system and galaxy. It is possible to create the time machine, quantum computer, magnetic charge accelerator, to gain energy from “pure” matter and antimatter. All that will allow us to avoid the destruction of our planet from commercial production of minerals. A lot of things will come to light for people. It will bring to the situation when modern technologies will look like “anachronism”. Moreover this today’s “anachronism” is a catalyzer of people’s vainglory that leads to the changing of the climate that can also influence and speed up the global catastrophe and self-destruction. In our days there is urgent question asked by the Shakespeare in ancient time. “To be or not to be” the existence of humane race on the Earth. But we can choose another way, people that become proficient in “Unified theory of the field” could be gods. For this we only need a faith of the world scientific community and the will of humanity to more reasonable existence on the planet and relation to the world.

Alexander Shikunov

The letter is sent by Dmitry Shishkin the student of Alexander Shikunov. To connect with Alexander Shikunov you can email me or phone: 011 7 926 555 99 88.

This is the entire text of the email sent this evening to every member of the Tulane mathematics department. There is no preface or reason given. This, friends, is psychoceramics. Getting caught up in enthusiasm for what seems to be a great result isn’t.

Mark Dominus over at The Universe of Discourse has a neat post about Möbius tranformations on the circle, and how one of them reminded him about Sharkovskii’s theorem.

Psst, Mark.. I’d call that circle the “projective real line”, just like the Riemann sphere is the projective complex line…

As I wait for the iTunes store to be less busy so it can reauthorize my iPhone to work with the updated firmware, we can finally get back on track.

Let’s consider a system of linear equations. We’ll use the variables , , and so on up to ; and we’ll let there be equations. Let’s write these out:

…

Here the constant are the coefficient of in the th equation, and is the constant term on the right hand side of the th equation.

But this is all but writing out exactly our matrix notation! We can take the above system and rewrite it as

Picking values for the variables is the same as picking the components of a column vector . We can collect the right hand sides of all our equations into one column vector , and the coefficients give a (linear) formula for taking the values we choose for the variables and turning them into the values on the right of our equations. That is, they define a linear map . We can thus rewrite our system in a more abstract notation as:

Suddenly it looks a lot more like the first — and simplest — linear equation we wrote down. But now we can’t just “divide by ” to solve it. We need heavier tools to manage this task, or even just to show when it can be managed at all! In short: we need linear algebra.

Incidentally, now we see why we indexed the variables with superscripts: because that’s how we wrote the components of a vector, and the variables are the components of a single vector variable. And if you’re still on the fence, I’ll note that physicists use superscripts all the time to index variables (for similar purposes), and they even do it when the equations aren’t all linear. Just try it. You’ll get used to it.

Some people had trouble grabbing the whole 50MB file that I posted, so Scott Carter broke it into pieces. He also included these comments:

The red, blue, and purple curves on the large (distorted) spherical objects at the bottom of each page of the eversion are the preimages of the the folds (color coded of course) and the double decker sets. Since at each time the sphere is immersed it may have double and triple points. Each arc of double points lifts to a pair of arcs on the ambient sphere, and each triple point lifts to three points on the ambient sphere. These lifts are the “decker sets.”

They are obtained via Gauss-Morse codes. Pick a base point and orientation on each curve in a movie. These are chosen
consistently from one still to the next. Label the double points and the optima and read the labels as they are encountered upon a single journey around the curve. The labels, too, are chosen consistently from one still to the next. Write these down for each curve in a movie, and connect the letters in the words as the curves change according to the basic changes that occur in each of the movie scenes.

These curves then are instructions on how to immerse the ambient sphere to create the illustrations.

Sarah’s thesis computes that the fold set is an annulus, the double point set is the connected sum of three projective planes, and the double decker set is the connected orientation double cover: a genus 2 surface.

So here are the pieces:

1. Immersed spheres as movies (2.2 MB)
2. The basic movie moves (3.4 MB)
3. The eversion from the red side to the quadruple point (19 MB)
4. Half of the eversion from the quadruple point halfway to the blue side (24 MB)
5. The other half of the eversion from the quadruple point halfway to the blue side (17 MB)

There’s a glitch in part 4, so I’ll post that as soon as I can.

As I mentioned in the comments yesterday, I may well have to be teaching math courses below the calculus level this year. The closest I’ve come to it before is “tutoring” some of my parents’ friends’ kids, which amounted to reminding them to do their homework. Amazingly, when they actually did their homework (and extra similar problems from the book before exams), their grades improved immediately. So I’m a bit nervous as to how this will play out. But I’m not saying it’ll go badly, so please don’t fire me quite yet, Dr. Hamburger.

Anyhow, I’ve got the impression that there’s a subtle, yet distinct difference between calculus and pre-calculus (including “college algebra”) math teaching. There’s a reason that a lot of schools don’t teach anything below calculus, and I’ve been able to feel my classes teetering on the edge of a conceptual continental divide. My intuition is that the calculus is the leading edge of a sea change from “how” to “why” in mathematics teaching — though I think that elementary mathematics could do with a lot more “why”, especially abstract reasoning. But I could be wrong about this, having little hands-on experience.

But when it comes down to it, I understand the position of the pre-calculus college math course. Most students taking it are scraping through whatever minimal mathematics requirement the institution sets, and will stop once it’s completed. And I understand that this will hold for most of them no matter how inspiring a teacher I am. The University of Maryland had mathematics courses below calculus, but I never knew a mathematics or physics major who started there. There was the occasional computer science major, but that was the heyday of the internet bubble, so you couldn’t spit on campus without hitting one.

The upshot is that such courses exist with certain “The Student Will Be Able To” line-items, and it’s my job to do the best I can to help them meet those. What I don’t know is the current thoughts on their pedagogy. What is the role (and the balance) of “how” vs. “why” in these courses? Are they really that different from calculus courses? If so, in what ways?

Since I’m now here in Bowling Green and taking a bit of a break while I look for an apartment, I thought I’d open this thread up to public discussion. JVP and JackieB have already started the ball rolling a bit. Jackie’s a high school math teacher, which provides her bona fides here, and I’ve yet to find the subject JVP doesn’t have some insight on. But there’s a lot of people out there from a lot of backgrounds. What do you think? And call up your friends teaching (or who have taught) below the calculus level in high schools and colleges, or those who might end up doing it some day, so they can weigh in as well.

I’ll be taking another short hiatus this week. I hadn’t mentioned this before because there was one more person I wanted to tell in person rather than let her read it here, but I’ve managed to forestall the death of my career by one more year. So in the fall I take up at Western Kentucky University, and I have to go this week to hunt the wild apartment.

So I figured that it would be appropriate to use the Kentucky state song, “My Old Kentucky Home”. This version is sung by Paul Robeson, and it includes the pre-bowdlerization lyrics. So I suppose I should make clear that my quoting them in no way means that I endorse their use. Ahem.
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I’ve mentioned Outside In before. That video shows a way of turning a sphere inside out. It’s simpler than the first explicit eversions to be discovered, but the simplicity is connected to a high degree of symmetry. This leads to very congested parts of the movie, where it’s very difficult to see what’s going on. Further, many quadruple points — where four sections of the surface pass through the same point — occur simultaneously, and even higher degree points occur. We need a simpler version.

What would constitute “simple” for us, then? We want as few multiple points as possible, and as few at a time as possible. In fact, it would be really nice if we could write it down algebraically, in some sense? But what sense?

Go back to the diagrammatics of braided monoidal categories with duals. There we could draw knots and links to represent morphisms from the monoidal identity object to itself. And topologically deformed versions of the same knot encoded the same morphism. This is the basic idea of the category of tangles.

But if we shift our perspective a bit, we consider the 2-category of tangles. Instead of saying that deformations are “the same” tangle, we consider explicit 2-isomorphisms between tangles. We’ve got basic 2-isomorphisms for each of the Reidemeister moves, and a couple to create or cancel caps and cups in pairs (duality) and to pull crossings past caps or cups (naturality). Just like we can write out any link diagram in terms of a small finite collection of basic tangles, we can write out any link diagram isotopy in terms of a small finite collection of basic moves.

What does a link diagram isotopy describe? Links (in our picture) are described by collections of points moving around in the plane. As we stack up pictures of these planes the points trace out a link. So now we’ve got links moving around in space. As we stack up pictures of these spaces, the links trace out linked surfaces in four-dimensional space. And we can describe any such surface in terms of a small collection of basic 2-morphisms in the braided monoidal 2-category of 2-tangles. These are analogous to the basic cups, caps, and crossings for tangles.

Of course the natural next step is to consider how to deform 2-tangles into each other. And we again have a small collection of basic 3-morphisms that can be used to describe any morphisms of 2-tangles. These are analogous to the Reidemeister moves. Any deformation of a surface (which is written in terms of the basic 2-morphisms) can be written out in terms of these basic 3-morphisms.

We can simplify our picture a bit. Instead of knotting surfaces in four-dimensional space, let’s just let them intersect each other in three-dimensional space. To do this, we need to use a symmetric monoidal 3-category with duals, since there’s no distinction between two types of crossings.

And now we come back to eversions. We write the sphere as a 2-dimensional cup followed by a 2-dimensional cap. Since we have duals, we can consider one side to be “painted red” and one side “painted blue”. One way of writing the sphere has the outside painted red and the other side is painted blue. An eversion in our language will be an explicit list of 3-morphisms that run from one of these spheres to the other.

Scott Carter and Sarah Gelsinger have now created just such an explicit list of directions to evert a sphere. And, what’s more, they’ve rendered pictures of it! Here, for the first time in public, is a 50MB PDF file showing the Carter-Gelsinger eversion.

First they illustrate the basic pieces of a diagram of knotted surfaces (pp. 1-4). Then they illustrate the basic 2-morphisms that build up surfaces (pp. 5-6), and write out a torus as an example (p. 7). Then come a few more basic 2-morphisms that involve self-intersections (pp. 8-9) and a more complicated immersed sphere (pp. 10-11). Each of these is written out also as a “movie” of self-intersecting loops in the plane. Next come the “movie moves” — the 3-morphisms connecting the 2-morphism “movies” (pp. 12-17). These are the basic pieces that let us move from one immersed surface to another.

Finally, the eversion itself, consisting of the next 79 pages. Each one consists of an immersed sphere, rendered in a number of different ways. On the left is a movie of immersed plane curves. On the top are three views of the sphere as a whole — a “solid” view on the right, a sketch of the double-point curves in the middle, and a “see-through” view on the left. The largest picture on each page is a more schematic view I don’t want to say too much about.

The important thing to see here is that between each two frames of this movie is exactly one movie move. Everything here is rendered into pictures, but we could write out the movie on each page as a sequence of 2-morphisms form the top of the page to the bottom. Then moving from one page to the next we trace out a sequence of 3-morphisms, writing out the eversion explicitly in terms of the basic 3-morphisms. As an added bonus, there’s only ever one quadruple point — where we pass from Red 26 to Blue 53 — and no higher degree points.

I’d like to thank Scott for not only finishing off this rendering he’s been promising for ages, but for allowing me to host its premiere weblog appearance. I, for one, am looking forward to the book, although I’m not sure this one will be better than the movie.

[UPDATE] Some people have been having trouble with the whole 50MB PDF (and more people might as the Carnival comes to see this page. Scott Carter broke the file up into five pieces, and I’ve put them up here in a new post. There’s a glitch in part 4, but I’ll have that one up as soon as I can.

Okay, now I really should introduce one of the most popular applications of linear algebra, at least outside mathematics. Matrices can encode systems of linear equations, and matrix algebra can be used to solve them.

What is a linear equation? It’s simply an algebraic equation where each variable shows up at most to the first power. For example, we could consider the equation

and clearly we can solve this equation by dividing by on each side to find . Of course, there could be more than one variable.

Sometimes people might use different names like , , and , but since we want to be open-ended about things we’ll just say , , , and so on. Notice here that because variables can only show up to the first power, there is no ambiguity about writing our indices as superscripts — something we’ve done before. Anyhow, we might write an equation

Now we have many different possible solutions. We could set and , or we could set and , or all sorts of combinations in between. This one equation is not enough to specify a unique solution.

Things might change, though, if we add more equations. Consider the system

Now we can rewrite the second equation as , and then drop this into the first equation to find , or . This just involves the one variable, and we can easily solve it to find , which quickly yields . We now have a single solution for the pair of equations.

What if we add another equation? Consider

Now we know that the only values solving both of the first two equations are and . But then , so the third equation cannot possibly be satisfied. Too many equations can be impossible to solve.

In general, things work out when we’ve got one equation for each variable. As we move forwards we’ll see how to express this fact more concretely in terms of matrices and vector spaces.

Everybody is talking about this.

I agree with Isabel that it doesn’t set off obvious alarm bells, but that I’m not nearly qualified to read this paper. That admittance puts me head and shoulders above the clamor at Slashdot who always think they know everything “geeky” even if they white-knuckled their way through the required Calc II course for their half-finished computer science B.A.s. Kudos to Walt for pointing out the distinction between Slashdot commenters and “the well-informed”.

Still, as a blatherer I suppose I’m contractually obligated to say something. So here. I’m saying something. Now go home.

Yesterday we defined the column rank of a matrix to be the maximal number of linearly independent columns. Flipping over, we can consider the obviously analogous quantity for rows. The “row rank” is the maximal number of linearly independent rows in a matrix. But what’s a row? It’s a vector in a dual space.

Okay, let’s talk a bit more concretely. Consider a linear transformation , which is described by the matrix . Then for each index we can take the th row in this matrix:

And use it as a linear functional on the space of column vectors . Specifically, it sends the vector with components to the number . Thus we get linear functionals — elements of the dual space .

But what linear functionals are these? They must be something special, and indeed they are. Remember that we’ve got our canonical basis for the target space . We also immediately have the dual basis of . The linear functionals we got from the rows are then just the pullbacks of these basic linear functionals .

To see this, notice that the linear functional is given by the matrix with a in the th component and everywhere else. That is, its components come from the delta: . We get a linear functional on by first hitting a column vector with the matrix and then with this matrix. We use matrix multiplication to see that this is equivalent to just using the matrix — the element of — with components . And this is just the th row of the matrix!

So the rows span the image of the dual space under the dual map . This is, of course, a subspace of , and its dimension is exactly the row rank of . We’ll come back later and show that this must actually be the same as the column rank.