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.

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