Math 312 and “structured proving”

This coming semester I will be teaching a couple of sections of Math 312, which is the introductory real analysis course at Penn State.  The only prerequisite for this course is Calculus II (Math 141) and, in particular, students are not required to have taken an “introduction to proofs” course; though, in practice, many of them will have done so.

I have long thought that in teaching a first or second proof-based course, especially in analysis with lots of quantifiers floating about, one should try to emphasize the “block structured” nature of proofs, analogous to the block-structured nature of a programming language like C.  I had the impression that I came up with this idea for myself, but Dan Velleman wrote a whole beautiful book (How to Prove It) from this perspective, and I know I read the first edition of that book when I was in Oxford, so probably that is where I became aware of this point of view.

Anyhow, I spent a day writing some TeX macros to format “block structured” proofs.  Follow this link for a few examples.

One of the pleasing things about structuring proofs this way is that one can describe how a proof is constructed, by compressing the lower-level data.  Here for example are compressed versions of the three proofs above.

The symbol \( \require{AMSsymbols}\blacktriangle\quad\blacksquare\quad\blacktriangledown \) for the omitted material is supposed to remind students that there are three ways to look: “up” for the givens at this point in the proof, “down” for the goals for which this part of the proof is reaching, and “across” to construct some argument linking the local givens to the local goals.

Does anyone have experience using this sort of explicitly structured proof in Analysis I? How did it work out?

Paper on sheaves of C*-algebras and K-homology published

After a busy day giving final exams in the MASS program it was nice to learn today that my paper with Paul Siegel about sheaves of C*-algebras has appeared in the Journal of K-Theory.  The link for the published version is

http://journals.cambridge.org/repo_A91rlWyM

This paper arose from some discussions when Paul was writing his thesis.  We were talking about the “lifting and controlling” arguments for Paschke duals that are used in the construction of various forms of operator-algebraic assembly maps (an early example is the one that appears in my paper with Nigel on the coarse Baum-Connes conjecture, which asserts that the quotient \( D^*(X)/C^*(X) \) of the “controlled” pseudolocal by the “controlled” locally compact operators does not depend on the assumed “control”).  At some point in these discussions I casually remarked that, “of course”, what is really going on is that the Paschke dual is a sheaf.  Some time later I realized that what I had said was, in fact, true.  There aren’t any new results here but I hope that there is some conceptual clarification.   (There is an interesting spectral sequence that I’ll try to write about another time, though.)

“Finite part of operator K-theory” V

This is the final one of a series of posts about the manuscript “Finite Part of Operator K-theory for Groups Finitely Embeddable into Hilbert Space and the Degree of Non-rigidity of Manifolds” (ArXiv e-print 1308.4744. http://arxiv.org/abs/1308.4744) by Guoliang Yu and Shmuel Weinberger. In previous posts (most recently this one) I’ve described their main result about the assembly map, what I call the Finite Part Conjecture, and explained some of the methodology of the proof for the large class of groups that they call “finitely embeddable in Hilbert space”.  Now I want to explain some of the consequences of the Finite Part Conjecture. Continue reading

“Finite part of operator K-theory” IV

Continuing this series (earlier posts here, here and here) on the paper of Weinberger and Yu, I’m expecting to make two more posts: this one, which will say something about the class of groups for which they can prove their Finite Part Conjecture, and one more, which will say something about what can be done with the conjecture once one knows it. Continue reading

“Finite part of operator K-theory” III – repeat

(I posted this yesterday but it seems to have vanished into the ether – I am trying again.)

This series of posts addresses the preprint “Finite Part of Operator K-theory for Groups Finitely Embeddable into Hilbert Space and the Degree of Non-rigidity of Manifolds” (ArXiv e-print 1308.4744. http://arxiv.org/abs/1308.4744) by Guoliang Yu and Shmuel Weinberger.  In my previous post I gave the description of their main conjecture (let’s call it the Finite Part Conjecture) and showed how it would follow from the Baum-Connes conjecture (or, simply, from the statement that the Baum-Connes assembly map was an injection). Continue reading