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Tag: geometry

“God given time”

Published February 20, 2008 by lieven

If you ever sat through a lecture by Alain Connes you will know about his insistence on the ‘canonical dynamic nature of noncommutative manifolds’. If you haven’t, he did write a blog post Heart bit 1 about it.

I’ll try to explain here that there is a definite “supplément d’âme” obtained in the transition from classical (commutative) spaces to the noncommutative ones. The main new feature is that “noncommutative spaces generate their own time” and moreover can undergo thermodynamical operations such as cooling, distillation etc…

Here a section from his paper A view of mathematics :

Indeed even at the coarsest level of understanding of a space provided by measure
theory, which in essence only cares about the “quantity of points” in a space, one
finds unexpected completely new features in the noncommutative case. While it
had been long known by operator algebraists that the theory of von-Neumann
algebras represents a far reaching extension of measure theory, the main surprise
which occurred at the beginning of the seventies is that such an algebra M
inherits from its noncommutativity a god-given time evolution:

$\delta~:~\mathbb{R} \rightarrow Out(M) $

where $Out M = Aut M/Int M $ is the quotient of the group of automorphisms of M
by the normal subgroup of inner automorphisms. This led in my thesis to the
reduction from type III to type II and their automorphisms and eventually to the
classification of injective factors.

Even a commutative manifold has a kind of dynamics associated to it. Take a suitable vectorfield, consider the flow determined by it and there’s your ‘dynamics’, or a one-parameter group of automorphisms on the functions. Further, other classes of noncommutative algebras have similar features. For example, Cuntz and Quillen showed that also formally smooth algebras (the noncommutative manifolds in the algebraic world) have natural Yang-Mills flows associated to them, giving a one-parameter subgroup of automorphisms.

Let us try to keep far from mysticism and let us agree that by ‘time’ (let alone ‘god given time’) we mean a one-parameter subgroup of algebra automorphisms of the noncommutative algebra. In nice cases, such as some von-Neumann algebras this canonical subgroup is canonical in the sense that it is unique upto inner automorphisms.

In the special case of the Bost-Connes algebra these automorphisms $\sigma_t $ are given by $\sigma_t(X_n) = n^{it} X_n $ and $\sigma_t(Y_{\lambda}) = Y_{\lambda} $.

This one-parameter subgroup is crucial in the definition of the so called KMS-states (for Kubo-Martin and Schwinger) which is our next goal.

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censured post : bloggers’ block

Published February 6, 2008 by lieven

Below an up-till-now hidden post, written november last year, trying to explain the long blog-silence at neverendingbooks during october-november 2007…

A couple of months ago a publisher approached me, out of the blue, to consider writing a book about mathematics for the general audience (in Dutch (?!)). Okay, I brought this on myself hinting at the possibility in this post

Recently, I’ve been playing with the idea of writing a book for the general public. Its title is still unclear to me (though an idea might be “The disposable science”, better suggestions are of course wellcome) but I’ve fixed the subtitle as “Mathematics’ puzzling fall from grace”. The book’s concept is simple : I would consider the mathematical puzzles creating an hype over the last three centuries : the 14-15 puzzle for the 19th century, Rubik’s cube for the 20th century and, of course, Sudoku for the present century.

For each puzzle, I would describe its origin, the mathematics involved and how it can be used to solve the puzzle and, finally, what the differing quality of these puzzles tells us about mathematics’ changing standing in society over the period. Needless to say, the subtitle already gives away my point of view. The final part of the book would then be more optimistic. What kind of puzzles should we promote for mathematical thinking to have a fighting chance to survive in the near future?

While I still like the idea and am considering the proposal, chances are low this book ever materializes : the blog-title says it all…

Then, about a month ago I got some incoming links from a variety of Flemish blogs. From their posts I learned that the leading Science-magazine for the low countries, Natuur, Wetenschap & Techniek (Nature, Science & Technology), featured an article on Flemish science-blogs and that this blog might be among the ones covered. It sure would explain the publisher’s sudden interest. Of course, by that time the relevant volume of NW&T was out of circulation so I had to order a backcopy to find out what was going on. Here’s the relevant section, written by their editor Erick Vermeulen (as well as an attempt to translate it)

Sliding puzzle For those who want more scientific depth (( their interpretation, not mine )), there is the English blog by Antwerp professor algebra & geometry Lieven Le Bruyn, MoonshineMath (( indicates when the article was written… )). Le Bruyn offers a number of mathematical descriptions, most of them relating to group theory and in particular the so called monster-group and monstrous moonshine. He mentions some puzzles in passing such as the well known sliding puzzle with 15 pieces sliding horizontally and vertically in a 4 by 4 matrix. Le Bruyn argues that this ’15-puzzle (( The 15-puzzle groupoid ))’ was the hype of the 19th century as was the Rubik cube for the 20th and is Sudoku for the 21st century.
Interesting is Le Bruyn’s mathematical description of the M(13)-puzzle (( Conway’s M(13)-puzzle )) developed by John Conway. It has 13 points on a circle, twelve of them carrying a numbered counter. Every point is connected via lines to all others (( a slight simplification )). Whenever a counter jumps to the empty spot, two others exchange places. Le Bruyn promises the blog-visitor new variants to come (( did I? )). We are curious.
Of course, the genuine puzzler can leave all this theory for what it is, use the Java-applet (( Egner’s M(13)-applet )) and painfully try to move the counters around the circle according to the rules of the game.

Some people crave for this kind of media-attention. On me it merely has a blocking-effect. Still, as the end of my first-semester courses comes within sight, I might try to shake it off…

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vacation reading (2)

Published February 5, 2008 by lieven

Vacation is always a good time to catch up on some reading. Besides, there’s very little else you can do at night in a ski-resort… This year, I’ve taken along The Archimedes Codex: Revealing The Secrets Of The World’s Greatest Palimpsest by Reviel Netz and William Noel telling the story of the Archimedes Palimpsest.

The most remarkable of the above works is The Method, of which the palimpsest contains the only known copy. In his other works, Archimedes often proves the equality of two areas or volumes with his method of double contradiction: assuming that the first is bigger than the second leads to a contradiction, as does the assumption that the first be smaller than the second; so the two must be equal. These proofs, still considered to be rigorous and correct, used what we might now consider secondary-school geometry with rare brilliance. Later writers often criticized Archimedes for not explaining how he arrived at his results in the first place. This explanation is contained in The Method.
Essentially, the method consists in dividing the two areas or volumes in infinitely many stripes of infinitesimal width, and “weighing” the stripes of the first figure against those of the second, evaluated in terms of a finite Egyptian fraction series. He considered this method as a useful heuristic but always made sure to prove the results found in this manner using the rigorous arithmetic methods mentioned above.
He was able to solve problems that would now be treated by integral calculus, which was formally invented in the 17th century by Isaac Newton and Gottfried Leibniz, working independently. Among those problems were that of calculating the center of gravity of a solid hemisphere, the center of gravity of a frustum of a circular paraboloid, and the area of a region bounded by a parabola and one of its secant lines. Contrary to exaggerations found in some 20th century calculus textbooks, he did not use anything like Riemann sums, either in the work embodied in this palimpsest or in any of his other works. (For explicit details of the method used, see Archimedes’ use of infinitesimals.)
A problem solved exclusively in the Method is the calculation of the volume of a cylindrical wedge, a result that reappears as theorem XVII (schema XIX) of Kepler’s Stereometria.
Some pages of the Method remained unused by the author of the Palimpsest and thus they are still lost. Between them, an announced result concerned the volume of the intersection of two cylinders, a figure that Apostol and Mnatsakian have renamed n = 4 Archimedean globe (and the half of it, n = 4 Archimedean dome), whose volume relates to the n-polygonal pyramid.
In Heiberg’s time, much attention was paid to Archimedes’ brilliant use of infinitesimals to solve problems about areas, volumes, and centers of gravity. Less attention was given to the Stomachion, a problem treated in the Palimpsest that appears to deal with a children’s puzzle. Reviel Netz of Stanford University has argued that Archimedes discussed the number of ways to solve the puzzle. Modern combinatorics leads to the result that this number is 17,152. Due to the fragmentary state of the palimpsest it is unknown whether or not Archimedes came to the same result. This may have been the most sophisticated work in the field of combinatorics in Greek antiquity.

Also I hope to finish the novel Interred with their bones by Jennifer Lee Carrell (though I prefer the Dutch title, “Het Shakespeare Geheim” that is, “The Shakespeare Secret”) on a lost play by Shakespeare, and have a re-read of The music of the primes as I’ll use this book for my course starting next week.

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