Category Archives: Teaching

K-theory course online

I am putting the notes from my K-theory course online as they are available.   You can find them here.

I am starting with a purely algebraic development, as in the first few chapters of Milnor, but will soon dramatically change gear and talk about C*-algebras.

We had the interesting surprise of having a TV reporter in our class the other day.  You can view his report/interview here.

 

AMS Open Math Notes

I wonder if you know about the AMS Open Math Notes project? I only just heard of it, but I feel very positive about any projects that make mathematical content more freely available.  Here is the link to the AMS page about the project:

http://www.ams.org/open-math-notes/omn-about

As my students know, I’ve made a habit over the years of putting together TeXed notes for the courses I deliver – especially graduate courses – and now I have quite a number of them.  With some pressure on my time (read: cancer) there is no way that I could bring all of these to formal publication, even if that was the right route for them.  But as “MathNotes” I can see that they might be helpful.  So I’m going to start submitting them, perhaps after light revision, to the AMS site.  I made a start today by posting my notes from the Penn State complex analysis (graduate) course, which I’ve delivered three or four times, taking a slightly different tack each time.  Based on what I learn from that, I have a good queue of other notes to submit as well.  This is actually quite exciting for me.

Updated, December 20th: The notes have now appeared on the OpenMathNotes site, and may be found here.

In further exciting (to me) AMS news, my Winding Around made it to their 2016 bestseller list!  Because of this, the AMS is offering a special discount for orders placed between now and the end of January…

I hope to upload further packages of notes in the new year! Best wishes to all!

 

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.

covThe 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.

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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