• John Baez, The Hipparchus operad, *The n-Category Café*, 1 April 2013.

(Not an April Fool’s joke!)

]]>in the form of a sensational free English tome:

http://perso.ens-lyon.fr/ghys/accueil/

It’s hard for me to describe but I would say that a prominent theme in the book is the combinatorics of singularities. The cubical parabola is discussed very briefly on page 107. Here are five quotes from the book:

“This is a preliminary version. Comments are most welcome …”

“Amazingly, this example of a cross with identified opposite sides has already been considered by Gauss under the name Doppelring. In his remarkable paper “Gauss als geometer”, Stackel relates a conversation between Gauss and Moebius. Gauss observes that the “Doppelring” has a connected boundary. More interestingly, he notes that one can find two disjoint arcs connecting two linked pairs of points on the boundary. I recall that the impossibility of such a configuration in a disc was the crucial point in his proof of the fundamental theorem of algebra.”

“So, Hipparchus was right: there are a(10) = 2 x 103,049 ways of combining 10 assertions, using OR or AND, in the sense just described. … Most mathematicians, including

myself, have a naive idea about Greek mathematics. We believe that it only consists of Geometry, in the spirit of Euclid. The example of the computation by Hipparchus of the tenth Schroeder number may be a hint that the Ancient Greeks had developed a fairly elaborate understanding of combinatorics …”

“We will see that the collection of all singularities, up to homeomorphisms, can be seen as a singular operad and this helps understanding the global picture.

]]>“In his review on the book by Markl, Shinder and Stasheff on operads, John Baez explains one of the motivations for operads.

‘Most homotopy theorists would gladly sell their souls for the ability to compute the homotopy groups of an arbitrary space.'”

(Click to enlarge.)

The algebraic equation is taken straight from Lyashko, a few lines above equation (1).

]]>I think he means “Lyashko gave a talk in my seminar, in which he proved X.”

Scherbak also credits a very similar / identical result to Lyashko in “Singularities of families of evolvents in the neighborhood of an inflection point”, and he actually cites that paper. So perhaps it is in there somewhere.

]]>Remember, in the good old days before the collapse of the Soviet Union and the exodus of mathematicians, Russian mathematics was seminar-based. Anyone who proved anything about singularity theory would need to explain it to Arnol’d before it was accepted. Publication was often an afterthought in this tradition. That’s why we’re having trouble getting details.

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