Zip-Apart Möbius Bands

I’ve taught topology many times. One of the highlights for the students (and for me) is the investigation of the Möbius band—the one sided, one edged, non-orientable surface with boundary. On the day we introduce the Möbius band I bring many strips of paper, clear tape, and scissors and have the students make conjectures about what…

How I teach topology: an inquiry-based learning approach

Recently I’ve had a number of people ask for more information about how I teach topology. I’ve taught it five times using a “modified Moore method” or “inquiry-based learning” approach. I’ve modified it each time, trying to work out the bugs. I think it is pretty successful now. Context. At our college all math majors…

Cantor set applet

I made this Cantor set applet for my Real Analysis class. It is nothing fancy, but it saves me from drawing it on the board.

Beautiful theorems about dynamical systems on the plane

I was reading through some papers written by my Ph.D. advisor (John Franks) from the early 1990’s and was reminded of a few beautiful results about the dynamics of planar homeomorphisms. So I thought I’d share them here. For those of you who are not familiar with the terminology, a planar homeomorphism is a bijective…

Furstenberg’s topological proof of the infinitude of primes

I just returned from a 10-day trip to India. It was my first visit there. I gave a talk at the ICM Satellite Conference: Various Aspects of Dynamical Systems. The conference was hosted by the Department of Mathematics at the M. S. University, Baroda, which is in Vadodara (formerly Baroda) in the Indian state of…

Mathematical magic tricks for kids

My six-year-old son loves the website ActivityTV.com, especially their science, origami, cooking, and magic videos. I watched a few of the magic how-to videos with him and was pleasantly surprised to see that some of them had a distinctly mathematical feel to them. For example: Jumping rubber bands: topological properties of circles and linked circles…

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…