# Higher index theory with change of fundamental group

I gave a talk in our seminar yesterday which arises from trying to understand the paper of Chang, Weinberger and Yu (Chang, Stanley, Shmuel Weinberger, and Guoliang Yu. “Positive Scalar Curvature and a New Index Theorem for Noncompact Manifolds,” 2013) where they use relative index theory in a non $$\pi-\pi$$ situation to produce examples of manifolds with strange positive-scalar-curvature properties (e.g., a  non-compact manifold which has an exhaustion by compact manifolds with boundary carrying nice positive-scalar-curvature metrics, but which itself carries no such metric).

I wanted to develop an approach to this kind of index theory that was more accessible (to me) and the talk was a report on my efforts in that direction.  Here are the slides from that talk.

# 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 though I would try to give an exposition.

# 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?”

# Jost Bürgi’s Method for Calculating Sines

Jost Bürgi (28.2.1552 -31.1.1632)
Astronom, Mathematiker, Instrumentenbauer, Entdecker der Logarithmen.
aus:7523 (Rar);Frontispiz

I just learned (via Facebook, no less) of a fascinating paper with the above title by Andreas Thom and coauthors.

Jost Burgi (1552-1632) was a Swiss mathematician, astronomer and clockmaker.  He worked with Johannes Kepler from 1604 and is thought to have arrived at the notion of logarithms independent of Napier.  He was also reputed to have constructed a table of sines by a brand new method, but until now the details of his Kunstweg (“artful method”) for computing sines were thought to have been lost.  The beautiful book of van Brummelen, The Mathematics of the Heavens and the Earth: The Early History of Trigonometry,  (Princeton University Press, 2009) documents how computing trigonometric ratios was  a central theoretical and practical preoccupation of ancient mathematics. We know of Burgi’s Kunstweg via a statement of his colleague and friend Nicolaus Ursus: “the calculation (of a table of sines)… can be done by a special way, by dividing a right angle into as many parts as one wants; and this is arithmetically. This has been found by Justus Burgi from Switzerland, the skilful technician of His Serene Highness, the Prince of Hesse.”  But the details are not clear and apparently nobody, starting with Kepler himself, was ever able to reconstruct Burgi’s method.