## What countries are mathematical objects?

I know about *Japanese rings* (commutative rings *A* which are integral domains and such that the integral closure of *A* in any finite extension of the ring of fraction is finite over *A*), and about *Polish spaces* (separable complete metric spaces). Are there any other mathematical concepts named after locations on Earth (or elsewhere)?

The only vaguely similar cases I can think of are *K3 surfaces*, which A. Weil mentions somewhere being named partly as a reference to the K2 mountain; and the recent innovation of *esperantist graphs*, which are defined in a new preprint of J. Ellenberg, C. Hall and myself (I’ll write about the latter in more detail soonish; the point of the name is that it alliterates with “expanders”, and it is indeed a condition related to, but weaker, than being an expander graph…)

## 26 Responses to “What countries are mathematical objects?”

In set theory we have “Canadian trees” (also called “weak Kurepa trees”).

And, of course, there is the Chinese remainder theorem.

Polish notation: Writing the function on the right and eliminating parentheses.

And what are those canadian trees (if it is not too highly technical…)?

One “feature” of such names is of course that they are rather hard to Google for…

Sorry — my previous post should have said “reverse Polish notation”. Polish notation is righting functions on the left and omitting parentheses. Lisp uses Polish notation. Postscript uses reverse Polish notation.

I should have thought about that one! I’m a great fan of RPN calculators (or was, really, since I don’t really use handheld calculators so often today; but I installed an RPN calculator application on my phone…)

The Forth programming language also uses RPN; though in terms of actual programming languages, I do prefer the Lisp syntax…

Tropical semirings are named (indirectly) after Brazil, I believe.

The expression = (1+|x|^2)^{1/2} is known as the “Japanese bracket” in PDE.

There are Egyptian fractions which are the multiplicative inverses of positive integers. Ancient Egyptians used a notation which allowed them to write most rational numbers as sums of those fractions, they had tables to simplify general fractions into sums of Egyptian fractions.

In mathematical finance one has the Greeks. And of course American and European options.

I should also have remembered egyptian fractions. There are quite a few fiendishly difficult number-theoretic problems involving those. (For instance there are many resources on this page, and E. Croot in particular has a number of papers in this area.)

There is an object named Hanoi tower.

well, I guess it is not exactly a country, though…

It’s true, but cities count as well.

There is the Nottingham group. In theoretical physics there was the Moscow zero (aka the Landau pole).

The “Hungarian method” is an optimization algorithm.

Does Russian-style seminar counts?

St Petersburg Paradox,

Monte Carlo Method,

Problem of Byzantine/Chinese generals,

Problem of Königsberg bridges,

Japanese Theorem,

Hawaiian eye-ring.

Unfortunately not the Catalan numbers nor Čech cohomology.

Jordan curve

Well, the Jordan curve is names after a mathematician

(Camille Jordan) rather than the country.

The Syracuse conjecture might well be an example of a “mathematical object” named after both a city and a country, as one of its (many) other names is the Czech conjecture. Still, a google search for “the Czech conjecture” gives pretty much nothing at all… That might seem surprising since the french translation “la conjecture tchèque” occurs much more often on the web.

A quick reminder about the statement of the conjecture: pick a positive integer, if it is even divide it by two, otherwise multiply by three and add one. Apply recursively the same procedure to the number obtained until the cycle (4,2,1) is (conjecturally) reached.

There’s the “Theorem of St. Etienne”, or at least that’s what Hida calls it.

The theorem is that q algebraic => j = 1/q + 744 + … is transcendental. The theorem also applies p-adically, and it follows that log_p(q) doesn’t vanish, and hence certain p-adic L-functions do not vanish accidentally.

Along these lines, Serre sometimes mentions a “theorem of Eugene”, which was not proved by someone named Eugene, but by a group of people (group theorists, I think) in Eugene, Oregon.

Re: Canadian trees.

A tree is a partially ordered set such that the predecessors of any point are well-ordered. We can then assign a height to each point (the order type of its predecessors). For each ordinal alpha for which this is non-empty, the alpha-th level of the tree is the collection of points of height alpha. The height of a tree is the supremum of the heights of its elements.

A cofinal branch through a tree is a well-ordered subset of the tree that meets each level.

A Canadian tree is a tree of height and size omega_1 with at least omega_2 cofinal branches.

In Hamiltonian N-body dynamics, periodic orbits of period T which satisfy f(t+T/2)=-f(t) for all times t are said to have “italian symmetry” (a term originally coined by Chenciner).

Roman numerals (or Arabic, or Chinese…)

A variant of a Monte Carlo algorithm is a Las Vegas algorithm.

Chinese restaurant (a way to build random partitions)

Hungarian construction (a way to strongly approximate stochastic processes by means of simpler objects)

In finance, there are also Israeli options, Asian options, Russian options, …