Oh, No! The first unit of class ended. Now What?

Originally Posted 02-14-2012

The first part of class went smoothly enough. But, suddenly, the first unit of each class was done, and roughly at about the same time: one to two weeks ago.

So I have been scrambling to make it through each day and meet my scheduled obligations. I said something to my wife about how the problem was that the measured time between me and the next expected due date had become negative. She made fun of me. You know, I might have used the phrase “event horizon” in there, too, so I bet she was justified.

Anyway, what has been going on?

Dynamics

This course is going along well enough. We are now just a bit behind where I hoped to be, and I have a little knot of students who are too quiet for their own good. I recenly gave the class a “not-a-midterm” list of questions that I think they should be able to handle at this point. Perhaps that will motivate some quiet students to come and discuss things with me. If not, I’ll plan another intervention.

I don’t mind about the schedule–it probably means I was too optimistic in my planning. We have definitely gotten through the idea of a dynamical system as a modeling tool, how to use basic terminology correctly (like dynamics, orbit, phase space), basic orbit types like fixed points and periodic points, and we’ve learned to draw good pictures like phase diagrams, cobweb plots and something more like a “folding diagram”. (Does anyone know if those have a more standard name? I don’t recall ever learning one, but I know I am not the only one to draw this kind of picture.) I think we have at least understood how the intermediate value theorem can guarantee a fixed point, and we are very close to nailing down how the mean value theorem can tell us when a fixed point is attracting and repelling.

The next phase of study will be to look at some classical families of systems and study bifurcations. We can then draw some different diagrams and ask deeper questions. I have three different families I can use for the rest of the course, each of which has chaotic behavior in it:

  • the family of tent maps
  • the family of quadratic maps x \mapsto x^2 + c
  • the logistic family x \mapsto a\cdot x\cdot (1-x).

It seems to me that we need the following things out of our next unit:

  • types of bifurcations, including the period doubling route to chaos
  • an actual discussion of what “chaos” means as a mathematical term
  • some symbolic dynamics to provide proofs (hiding here is the notion of conjugacy of two systems)
  • an appearance of the Cantor Middle Thirds Set.

More advanced things that I’ll want my graduate students to do include:

  • presenting Sharkovski’s Theorem
  • Discussing some numerical algorithms for finding periodic orbits.

The last phase of the course should be about Newton’s method, Julia sets for the complex dynamical systems z \mapsto z^2 +c and possibly the Mandlebrot set.

Math in Decision Making

This has been going really well. Basically my class came around on what it means to say two sets have the same size, and then we explored how weird that is for infintie sets. They managed to see that the natural numbers, the evens, the odds, the positive rationals, the integers, the rationals, the set of all `mathematical words’ on the standard English alphabet, and the rationals all have the same size. Then we saw that the real numbers are different by way of Cantor’s diagonal argument. For a kicker, I gave a lecture day about the Middle Thirds set and we saw that a set could be “huge” but “hard to see” at the same time. A few of them were suitably impressed.

I am still grading exams. They did not take me seriously when I talked about writing to explain clearly…so they will be rewriting their exam papers as soon as I get them returned. I think it is an appropriate time to talk about the process of writing as something that includes revisions.

This week we are starting our unit on surfaces. We just started, so yesterday’s class was a bit of a mind-bender. We successfully noted that a torus was different from a sphere because there are simple closed curves on a torus that are non-separating. And we managed to see that a donut can be deformed into a coffee cup. I had my regular coffee mug and a tire innertube which is just too big to be worn as a hat for visual aids, and I managed to hit and stick the chalk tray with a thrown piece of chalk three times running. That was clearly the best performance of my chalk-throwing carreer.

I plan on hitting many questions about what curves can live on a surface and what shapes are created by cutting along those curves until we feel comfortable enough to go in the opposite direction and make surfaces out of polygons with “sewing patterns.” I still want to talk a bit more about the idea of “stretch equivalence” and later differentiate it from “cut and paste” equivalence.“ Also, I have a week to do about Mobius bands to introduce the ideas of boundary components and orientability. When the ground is suitably prepared, we will talk about the Euler characteristic, and then the classification will have to happen. I am not completely decided what proof I will use. There is the ”standard scissors and glue“ proof I learned in graduate school which involves putting a cut-up surface into normal form, and then there is the Conway ”ZIP proof." These are equivalent, of course, and I’ll have to think about which one I find conceptually easier.

I still hope to have enough time to do a third unit on classifying wallpaper patterns.

Oh, at the conclusion of yesterday’s class, a student told my teaching assistant that this is the most interesting and fun math course he has ever had. This was unsolicited, and wasn’t directed at me. It made me feel awesome for a bit.

Linear Algebra

Here I am just surviving. Though I do think we finished an interesting week of work on Monday. We got to a spot where we had made a model of the internet as a weighted directed graph, and a model for the behavior of a “random web surfer” and then set up a Markov Chain to describe the evolution of the probability that the random surfer is currently at page X. Then we showed how the long term behavior should be goverened by an eigenvector with eigenvalue 1, and discussed the basics of the “power method” for finding that vector. In short, we took a week to explore the basic structure of a naive Google PageRank algorithm.

The next application will be least squares and other polynomial approximation problems. I hope to use that as a springboard for more abstract material, since the notion of an abstract vector space made of polynomials will just happen, and we will see things like column and row spaces.

Not quite Research

My student research projects are coming along at vastly different rates. The undergraduate I have worked with since last May has hit a bit of a roadblock. We are now casting about again for something else interesting to say.

The graduate student I am supervising for just this term has made serious progress on some programming and graphics functions, and I still have hope that he will draw the limit sets of some Kleinian groups by the end of the term. (My secret hope is that he will get far enough to make an animation of how the limit sets change as we change one of the generators of a group inside PSL(2,C).)

The graduate student who will be doing a thesis option with me has come up with a neat sounding question on his own. I have no idea how to do it, nor any idea what is known, so I have sent him to the library as a feasibility check. If he can find some relevant literature, I can help him carve out a project.

Research

I haven’t done enough of this lately, but a colleague and I are having conversations about a problem of mutual interest. We are taking baby steps.

One thing I would like to explore is the Birman-Williams result about the kinds of knots that can appear in the classical Lorenz system, and Ghys’ stuff about how that is the same as the knots that can appear in the geodesic flow of the modular surface. It just looks so cool. Maybe I’ll need to write some expository blog posts to make myself really work the details.

I am going to not make a list of all the other projects in various stages of “incomplete”. But it is long.

Other Stuff

I have a problem solving contest to take students to in about ten days. Math club is starting to get rolling properly again. And now that hiring season is over, it will be time to start convening meetings about writing an REU grant.

More Research, and the Next LinAlg Activity

Originally Posted 01-24-2012

I had a pleasant and still day in the office. I had no formal teaching duties, but I did have two research oriented meetings, and a nice informal chat with a former student.

Meeting Number One: Undergraduate Project

I am working on a combinatorics/number theory project with an undergraduate student. Our meeting today had a bit of a “wow” moment, too. We have been studying some families of polynomials that “arise naturally” from the problem. The student has found a ton of identities involving these polynomials, all experimentally. We have not succeeded in finding a proof. So, as an alternative to feeling stuck, I thought we might change gears a bit and instead of looking at the coefficients (which is what brought us to this point), we could look at the roots. We finally got some Sage/Python code up and running to compute and plot these roots in a meaningful way, and, ‘pop’, they appeared to lie on a logistic S-shaped curve.
I can’t explain it, but it probably means something deeper is going on.

Meeting Number Two: Professional Research

My colleague Bill Wood and I had an exploratory meeting this afternoon. We attended a workshop on Discrete and Computational Geometry at the Joint Mathematics Meetings earlier this month, and we wanted to find a problem to work on together. After two hours of just talking, I think we had three or four problems that we agree are interesting. Most of them seem a bit out of our reach, but one just might be a good way to get going. So, it looks like I’ll be trying out something new this term. I am excited.

Linear Algebra: The Indoctrination Educating Continues

Feeling good about the last two days of linear algebra, I have designed some more for them to handle tomorrow. The activity is meant as a way to introduce a lot of important language quickly, including the concepts of span, linearly (in)dependent sets, dimension, basis, and determinant. It seems impossible that this will work so well, but I have to try. The focus is again on solving simple sounding problems presented in several equivalent ways, and noting what it all really means.
Also, I have designed a Sage based homework assignment. If either of these goes at all well, I’ll share them later.

The Summer Research Program for Students

I spent some time organizing our summer research program for undergraduate students today, too. This was mostly simple grunt work to prepare to advertise the opportunity. The program has run for three consecutive summers, so much of the organization is on auto-pilot now. Right now, we only have money to support two students–which is the other project I am supposed to be thinking about. Oh, look at the time…

Making Ends Meet

Originally Posted 01-18-2012

Figuring Out Linear Algebra

One of my challenges this semester is implementing a linear algebra class with many new components. The considerations are these:

  • I last taught a linear algebra class five years ago at and a different institution. Therefore, the audience is new to me.
  • I am incorporating use of the open-source mathematical computing system Sage.
  • I am using WeBWorK for “routine” homework assignments.
  • I am using an open-source text which is still “in development.”

This is probably too much. But I am doing it anyway. Another thing which wouldn’t count for anyone else, but counts for me, is that I am trying to blend in an inquiry-based learning approach. This isn’t new to me, though for this material it is.

So far, I find that I haven’t carved out enough time to make the WeBWorK assignments work. This is a major goal of the next few days.

I ran a couple of Sage introductory workshops, one last night, that seemed to go pretty well. The first night went well, with lots of great questions, but was poorly attended. The second night I had a full room, but the crowd was not interested in conversation. One outcome is that I have at least made materials for running these in the future.

The biggest trouble is really that I find I haven’t hit the right stride in class, yet. I can’t seem to get an interactive environment and still include the computer. I am finding it difficult to make activities centered around the material, given that I am roughly following a text. By the way, the text takes the interesting approach that the students should learn all of the computational skills right away, so all of it is in chapter one. Applications come next, and then the theory starts in chapter three. I have already discussed matrix and vector algebra (minimally); the equivalence between solving matrix equations, solving systems of linear equations, looking for a realization of a vector as a linear combination of other vectors and the geometry of intersecting hyperplanes; Gaussian elimination. Next time we will cover rank and determinant, and maybe eigenvectors. Then we’ll have a quiz.

I’ll get there. I suspect that this will be a “muddling through” experience, and then next summer I will try to do something more serious. Maybe I should look at the AIBL grant cycle.

Other Courses

I have two other classes running this term, and they seem to be going well.

Dynamical Systems

My dynamics class is going gangbusters. Almost. I have several students who have had classes from me before, and lots of people who are willing to give presentations and ask questions. I have two or three that I worry about because they have been very quiet. I’ll have to do a personal check on each of them.

Math in Decision Making

(My liberal arts class.) I think that this is going well. I have succeeded in making them confused about things, and then unconfused about some of it. Check!
I was a little aghast that they had no reaction to the weirdness of infinite sets. I mean, they just managed to prove that there is a bijection (we call it a ‘matching’ in class) between the natural numbers and the even natural numbers. I jumped up and down about how weird that is… They couldn’t really muster any emotion. Perhaps this is a bad indicator. At the very least, it means that they haven’t thought deeply about this issue (surprise!), at worst, it means they are just uncritically accepting whatever happens in a math class. I need to work on both of those items.

Other stuff from today

I mentioned in a department meeting that exactly zero people had volunteered to lead a summer research experience in math after my last call for proposals. This was a bit depressing. But I got responses from three colleagues right after the meeting. So, that is looking up.

Tonight, I am looking forward to a soccer game and a night out with friends (for free pie!) to help soothe the cares away.

Master’s Project

Originally Posted 01-16-2012

This semester I am supervising an MA student. This is not a full-blown MA thesis; instead, the student has chosen the “non-thesis option” for his work, which requires him to do 2 cerdit hours worth of research work and write a paper. I talked to a few of my colleagues, and it seems that the requirements are a bit nebulous, but certainly don’t include new work.

I talked with the student and he is interested in geometry, has a background in art (some computer animation), and is curious about fractals. Also, he did a summer project with me a while back, and learned a lot about the geometry of the hyperbolic plane. So, I pulled my copy of Indra’s Pearls down off the shelf…

Now, I haven’t read the book before. (One of my vices is buying math books that I want to read “someday.”) Easy selling points here were that the book is full of pretty pictures, and one of the co-authors is my academic grandmother.

I figure that if a student is going to do some research, they have to make something. In this case it is not a new theorem, or a new argument for an old theorem, but instead a picture. Rather, I am going to ask him to make some pictures of limit sets of Kleinian groups.

If all goes well, he will actually write a bunch of routines in Python/Sage/Cython for making such pictures that can be used for a variety of groups. I don’t know if this has been done before (I suspect it has), but I don’t really care. The student will be doing something of academic value, and it will help me learn some stuff that I really should have finished absorbing by now.

So today I read several chapters of the book to get rolling. It starts off very gently. The book is written for a mathematical layman, really, so they start with chapters introducing complex numbers, symmetry, groups, and Moebius transformations of the Riemann sphere. There is pseudo-code weaved in, too, so that a reader can get going with some basic drawings.

It looks like the rest of tonight’s reading will be about Schottky groups. I haven’t seen anything new (to me), yet, but several of these basic concepts are things that I find myself explaning to students often, and I might adopt one or two turns of phrase.

The goal I set for the first week was to try to understand how the group PSL(2,\mathbb{C}) is essentially the set of isometries of hyperbolic 3-space. I am sure that will keep him occupied for this week. Next we can try to draw pictures of the action of individual Moebius mappings

Oh, if you want, here is a link to the book.