## Turing’s topological proof that every written alphabet is finite

Recently one of my colleagues was reading Alan Turing‘s groundbreaking 1936 article “On Computable Numbers with an Application to the Entscheidungsproblem.” This is the article in which Turing introduced the turing machine, solved Hilbert’s Entscheidungsproblem (`decision problem’), and proved that the halting problem is undecidable. It is viewed by many as the foundation of computer…

## Movie day in topology class: the Poincaré conjecture

Today was the last day of the topology class I’ve been teaching. I decided to devote the day to the Poincaré conjecture. I started by telling the students a little about the history of the problem. Then I showed them three videos. The first video was an excellent 50-minute lecture by Fields medalist Curt McMullen…

## Topological maps or topographic maps?

While surfing the web the other day I read an article in which the author refers to a “topological map.” I think it is safe to say that he meant to write “topographic map.” This is an error I’ve seen many times before. A topographic map is a map of a region that shows changes…

## Perelman to be awarded the Clay Millennium Prize

The epilogue of my book is devoted to the Poincaré conjecture, the famously challenging 98-year old topological puzzler that was proved in 2002 by Grisha Perelman. Perelman was awarded the Fields Medal in 2006 for this accomplishment, but he declined to accept the award. Today the Clay Mathematics Institute issued a press release that begins:…

## A tale of why you (U, that is) needs a tail

What is this collection of symbols? No, it is not a wallpaper border pattern, a brain teaser, or ancient hieroglyphics. It is a set identity, of course! When I was in college I had a math major friend who said that all he learned in our topology class was to put tails on his U’s…

## What is the cardinality of the Euclidean topology?

I’m teaching topology this semester. The students are looking at different topologies on the real number line. For homework I asked them to think about which topologies are “the same” (if any) and which are “different,” and why they thought that was the case. We haven’t yet talked about continuous maps or homeomorphisms, so I…

## A new way to collaborate: DropBox

I have a long-time collaborator who lives in Georgia (I’m in Pennsylvania). I’ve had good luck collaborating with him via email, but it is a pain. As soon as one of us edits a file he sends it to the other person as an email attachment. We haven’t had any “forked” files, but we do…

## The maypole braid group

Over the weekend I attended a May Day party thrown by one of my colleagues. During the party they had a traditional maypole dance. An example of a maypole dance is shown at left. A maypole is a tall pole with colorful ribbons attached to the top that are fanned out in a cone shape….

## How to use KnotPlot

As I’ve mentioned before, I’m teaching a knot theory class this semester. I’ve been playing around with KnotPlot, a powerful piece of software for drawing and working with knots. I want my students use it, but it has a somewhat unintuitive interface. So I’m trying to write up a list of easy-to-use instructions for them. The…

## Materials for a knot theory class

This is a call for help—or for suggestions, at least. I’m teaching a knot theory class next semester. I’m looking for good props to use in the class to make knots. I would like to be able to make knots such as the following (and have my students do so as well). I suppose the…