“Operator K-theory” has appeared on AMS Open Math Notes

My final Penn State course (Spring 2017) was about K-theory and operator algebras – the connection between these two has been central to my mathematical life.  I wrote up lecture notes for this course, as has become usual for me.  I’m pleased to report that these have now appeared on the AMS Open Math Notes page.

The American Mathematical Society hosts AMS Open Math Notes,  which is “a repository of freely downloadable mathematical works in progress hosted by the AMS as a service to researchers, teachers and students.”

The Open Math Notes homepage continues  “These draft works include course notes, textbooks, and research expositions in progress. They have not been published elsewhere, and, as works in progress, are subject to significant revision.  Visitors are encouraged to download and use these materials as teaching and research aids, and to send constructive comments and suggestions to the authors.”

Topology, Moore or Less – Concluded

We’ve now finished the Moore method topology class that I wrote about in Topology: Moore or Less It’s been an intense experience for everyone, I think.  Many students have surprised themselves by what they have achieved.  At the end of the course we printed off copies of the co-written course textbook (complete with frontispiece photo of the authors!) and everyone received one in time for the “open book” final exam.  I hope that many students will hold on to these as a reminder of our common achievement.  Here’s the cover.

The first part of the course was descended from Bing’s notes  (and thus, indirectly, from Moore himself) as they are reproduced in the Journal of Inquiry-Based Learning in Mathematics.  After mid-semester we digressed into product and quotient topologies and then, briefly, into function spaces and the compact-open topology – the last proof that the students were guided through was a special case of the exponential law for function spaces,

$X^{Y\times Z} \cong (X^Y)^Z.$

That seemed like a good place to stop.

Students prepared their in-class presentations in “teams” of 4 rather than individually – this was a modification that I made to the standard Moore method. The online platform Piazza was the main mode of team collaboration – I held a couple of in-class sessions too where teams worked together and I acted as a roving consultant, but I should probably have done more of that.

What did the students think?  Here are a few comments:

“One of the biggest takeaways from this class was seeing how mathematics is constructed firsthand. This semester we constructed a complicated and powerful machine that I am eager to build upon in my later mathematics courses, and now I have the tools necessary to do so.”

“Having to prepare for class with what theorem or example we had to prove for the class and writing our own book has given me an understanding of the material that I don’t think would have happened if the course was taught similar to a traditional course.”

“I felt very engaged in the course as a result of the unique “Moore Method” used to teach the course.”

“The class structure challenged me to think differently than I ever had, and I genuinely appreciated that.”

I love these quotes, but of course not everyone feels the same way. I paraphrase the next comment:

“The method used to teach this class, while I can see its benefits, was really not right for me… I wanted my struggles with the material to be private, and because the class was so collaborative there wasn’t an easy way to do this.”

I hear what this student is saying, and wish I could have helped him/her better. One thing I would have liked to be able to share is that we all struggle, in math as in life, and that I had wanted the class to be a place where we could struggle together, not feel we have to project a brittle confidence or else stay silent.  That will be something to work on for next time, if I do this again.

Thanks, students! I really enjoyed the class and I hope you did too.

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 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).

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 thought I would try to give an exposition.