April Notices
April 10th, 2007 by WaltThe April Notices of the AMS features a retrospective on Serge Lang’s mathematical career. This month’s What is…, What is… a tropical curve. A tropical curve is a curve defined over the tropical semiring, which is the real numbers with addition defined as max or min, and multiplication defined as real addition. Mathematicians are currently working on finding the analogues of classical results on algebraic curves to the tropical setting.
April 10th, 2007 at 10:46 pm
Thanks for pointing this out!
Semirings are also called rigs: rings without negatives. The tropical semiring is important in classical mechanics: if we take the theory of path integrals and do it using the tropical semiring, we get the classical principal of least action! I explained this in week12 and week13 of the winter 2007 quantum gravity seminar at UCR. It’s cool stuff.
April 11th, 2007 at 9:11 am
I have to admit I don’t like the rig terminology (I don’t like rng either). Semigroup/semiring/semifield is nicely consistent, and the pun involved in turning “ring” into “rig” doesn’t translate. (What’s the French analogue for “annele”? “Aele”?)
April 11th, 2007 at 10:33 am
That said, the analogies between the reals and the tropical semiring are pretty exciting. The interpretation of the path integral/classical action analogy is new to me; I’m looking at your lecture notes right now.
April 11th, 2007 at 11:18 am
I like the connection between the Legendre transform and the Fourier transform which is closely related to the way the tropical path integral gives Hamilton’s path of least action. Although they can both be seen as essentially the same operations in different semirings, I found an interesting paper a little while back by a guy called Sean Walston who noticed and wrote about the similarity, and its connection to quantum and classical mechanics, without knowing anthing about semirings, which in some ways made it more interesting. Unfortunately the link appears to be down right now: Sean Walston’s h-bar and grill but maybe it’ll reappear at some point.
April 13th, 2007 at 12:19 pm
Ah, this sounds vaguely familiar.
April 13th, 2007 at 12:22 pm
Ah, this sounds vaguely familiar. There are some applications of
this in network calculus - there’s a book by Le Boudec and
Thiran available as pdf (here).
April 15th, 2007 at 7:53 am
Lack of translation for the pun seams to be a weak argument to me. Mathematical terminology seams to vary a lot between languages anyway, with terms in different languages not being direkt translations of the English term anyway. Eg, field in English for the algebraic concept, but “body” in the European languages I am familiar with. (Of course, in small languages like Swedish, it is far from certain there even is a translation for the English terms, given the dominance of English in research.)
April 15th, 2007 at 8:48 am
Johan: I was actually looking through the terminology recently and found that almosst universally the term is not only “body”, but has the specific connotation of “dead body”, or “corpse”.
There is one notable European exception. In Flanders: “field”.
April 16th, 2007 at 12:38 am
Really? In Swedish it only means body. I wouldn’t dare say what the connotation is in other languages.
April 16th, 2007 at 1:46 am
Yes, like Johan Richter says for Swedish, in Portuguese there is also no sense of a “corpse”, just “body” (”corpo”, in Portuguese) and I think that this was the original meaning of the word given by Dedekind, where it is said that things worked “like a human body”.
P.S.: Just checked the English article at Wikipedia and it seems that Dedekind indeed used the word “Körper”.
P.S.2: I didn’t know that these “tropical” mathematics were so practical. I guess that I will (try to) chat with Prof. Imre Simon about his studies.