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

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.

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Matt Wiersma on exotic group C*-algebras

Recently Matt Wiersma from Waterloo spoke in our seminar about some of his work related to “exotic group C*-algebras”.  A more detailed account is on the arXiv.  I thought I would try to write up some of what I learned (probably, as usual, this is the most elementary points, but it was new to me).

What is an exotic group C*-algebra?  It is a completion of the group algebra which is different from the two standard examples (maximal and reduced) that we describe in C*-algebra courses.  Oversimplifying, we might make an analogy with compactifications of a locally compact Hausdorff space.   There is always a minimal one (one-point compactification) and a maximal (Stone-Cech), but there are also plenty of other things in between.  Analogously, in the case where a group \(\Gamma\) is non amenable, one might imagine that there should be many other C*-completions of \({\mathbb C}\Gamma\) lying between the maximal and the reduced C*-algebras.   (Whether, in fact, there exists any group for which \({\mathbb C}\Gamma\) has exactly two distinct completions appears to be an open question.)

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The Witness Relocation Program

I gave a talk last week in the Geometry, Analysis and Physics seminar with the title “The limit operator symbol”.  This was an attempt to distill some of the ideas from my series of posts on the Lindner-Seidel and Spakula-Willett papers, especially post IV of the series.  In particular, I wanted to explain the crucial move from having a series of inequalities witnessed to having a similar series of inequalities centrally witnessed.  As Nigel put it during the seminar, we are attempting to describe a “witness (re)location program”: our witnesses are scattered all over \(\Gamma\), and we are attempting to move them all to the “courthouse”, that is, to a neighborhood of the identity, at the same time. Continue reading

“Winding Around” is now available

JohnWAI was excited to receive a package from the American Mathematical Society today!  My author’s copies of “Winding Around” have arrived!

This is a book based on by MASS 2013 course of the same title, which looks at the winding number – the central notion in plane topology – from a variety of perspectives, topological, geometrical, analytic and combinatorial.  Having (I hope) made the case that the winding number concept is the “golden cord which guides the student through the labyrinth of classical mathematics”, I conclude by following a beautiful paper of Michael Atiyah to explain how, by asking one natural question about the winding number, we can be led to the Bott periodicity theorem, a central result in the flowering of topology in the 1960s.

If you are interested in the book, please visit the AMS bookstore page.

I will attempt to maintain a list of typos and other corrections here.  At present this list is empty, but I doubt if that happy state of affairs will last for long!  (edit: it didn’t) Please contact me with any information about corrections.  And, to quote the final sentence of the book, “I wish you much happy winding around in the future.”


Traces and commutators

The following is a true (and well-known) theorem: \(\newcommand{\Tr}{\mathop{\rm Tr}}\)

Suppose \(A\) and \(B\) are bounded operators on a Hilbert space, and \(AB\) and \(BA\) are trace class.  Then \( \Tr(AB)=\Tr(BA) \).

This is easy to prove if one of the operators \(A,B\) is itself of trace class, or if they both are Hilbert-Schmidt (the obvious calculation works).  In the general case it is a bit harder.  The “usual” argument proceeds via Lidskii’s trace theorem – the trace of any trace-class operator is the sum of the eigenvalues – together with the purely algebraic fact that the nonzero eigenvalues of \(AB\) and \(BA\) are the same (including multiplicities).  Continue reading