Tag Archives: topology

Michael Atiyah’s Birthday!

Heads up!  In  a couple of days (April 22nd) it is the 87th birthday of “Britain’s mathematical pope”, (not just Britain’s, either, IMO), otherwise known as my doctoral advisor, Professor Sir Michael Atiyah.   HAPPY BIRTHDAY MICHAEL!

To celebrate, his son David is assembling an online tribute – see http://www.atiyah.eu/mfa87/    Please consider sending a tribute message to david@atiyah.eu  Here’s what hes ays:

We are collecting messages of congratulations on the occasion of Michael Atiyah‘s 87th birthday Friday, April 22, 2016.

If you have the time, memory, and an inclination, please also include your favourite personal story about Britain’s Mathematical Pope*. I keep hearing every mathematician has one – it would be a shame not to collect and archive them for posterity.

Bonus points awarded for photographs, with prizes for the best MP4 video message we can share on the night.

Pls include:
– your name
– your current position, & location (if appropriate)
– when and where you first met Michael

We will keep it simple and hope to collate and publish submisssions in due course.

* = with thanks to Siobhan Roberts for the expression used in her recent biog of J H Conway – i have simply extended his Popedom from England to Britain.

If you haven’t seen it, here is a great article from Wired last week: Mathematical Matchmaker Atiyah Dreams of a Quantum Union.

 

Tychonoff’s theorem II

In my earlier post on Tychonoff’s theorem, I talked about the original proof, based on the following characterization of compactness which is due to Kuratowski.

Definition  Let \(S\) be a subset of a topological space \(X\).  A point \(x\in X\) is a point of perfection of \(S\) if, for every neighborhood \(U\) of \(x\), the set \(U\cap S\) has the same cardinality as \(S\).

Lemma (Kuratowski) A topological space \(X\) is compact if and only if every infinite subset has a point of perfection.

Now I will give the proof of this characterization (again following Wright).

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Tychonoff’s theorem

Tychonoff’s theorem (an arbitrary product of compact sets is compact) is one of the high points of any general topology course.  When I’ve taught this in recent years, I’ve usually given the proof using universal nets, which I think is due to Kelley.

Recently though I read a very nice paper by Wright  which reproduces, and then modifies, Tychonoff’s original proof (otherwise inaccessible to me because of my lack of German).  I thought the original proof was really elegant and though I would try to give an exposition.

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Topology, Moore or Less

I get to teach Math 429 this semester.

This is the introductory topology course for undergraduates at Penn State – “point set topology” as the old-fashioned name would be.  I used to teach some of this material at Oxford but I have not had a chance to teach it at PSU before now.  I have about 25 students.

I decided to try a variation of a Moore method approach in this class.

So I started by showing the students the two-minute video above, which shows Steph Davis free-soloing and then BASE jumping from a Utah desert tower. Then I asked them, “Now you have watched the video, could you do that?”

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“Winding Around” now going up

The website for my MASS course, “Winding Around” (Math 497C, Fall 2013) is now live.

Winding Around” is an introduction to topology using the winding number as a unifying theme. It’s intended to be different from most introductory topology courses because we’ll try to define the key concept (winding number) as economically as possible and then  apply it in many different ways.

One of the inspirations for this course is the classic expository paper

Atiyah, M. F. “Algebraic Topology and Elliptic Operators.” Communications on Pure and Applied Mathematics 20, no. 2 (1967): 237–249. doi:10.1002/cpa.3160200202.

and if things go according to plan I hope that we may get to discuss the Bott periodicity theorem at the end of the course, in the spirit of Atiyah’s article.