Olympiad vs. research mathematics

Recently, some interesting discussions on math education and mathematical philosophy have taken place on, of all places, my Facebook wall. Since Facebook is a rather restrictive medium, I feel it’s time to widen the scope of my blog to include such topics.

I am currently sitting in the main common area in our residence halls at the Math Olympiad Summer program (abbreviated MOP) in the middle of Lincoln, Nebraska. It is rather quiet, as the students are all taking their 5th “MOP test” in the last two weeks, and my fellow instructors and graders are either proctoring the tests or preparing more material to teach these brilliant kids.

It got me thinking about a topic that is often discussed among mathematicians and among those in math contest circles: How correlated is mathematical contest ability with mathematical research ability? Does one help or hinder the other? Is the social environment that math competitions create hostile to noncompetitive students, especially women?

Geometry of the real projective plane

I’ll be flying to Nebraska tomorrow to teach at the Math Olympiad Summer Program. One of my more advanced classes will be on projective geometry and how it can help us understand Euclidean geometry. I thought I would share the introduction from my handout, since it is an attempt to demonstrate one of the many reasons projective geometry is such a gemstone:

General projective planes

Let’s start with some general theory. A projective plane is any set $P$, which we call the set of points, and a nonempty collection $L$ of subsets of $P$ called lines, such that the following four axioms hold:

P1: Every pair of distinct points are contained in exactly one line.

P2: Every pair of distinct lines intersect in exactly one point.

P3: There are four distinct points $a,b,c,d\in P$ no three of which lie on a line.

P4: Every line contains at least three points.

Axiom P2 raises some alarms. In Euclidean geometry, we assume that there are parallel lines that never meet, but here, we’re requiring that every pair of lines intersect. So the usual Euclidean plane is not a projective plane.

Teeter-totter transpositions

In attempting to write problems for the USAMO, I came across an interesting variant of Knuth equivalence whose structure is rather intricate.

Suppose we start with the numbers $1,2,3,\ldots,n$ written in a row. At each step, we may choose three adjacent numbers in the sequence, say $a,b,c$, and if either $a\lt b \lt c$ or $c\lt b\lt a$ then we can switch $a$ and $c$. So for instance, we might start with 12345 and change it to 32145. At this point we cannot use the subsequence 2,1,4 to change the 2 and 4, but we can use 1,4,5 and end up with 32541. The question is, how many possible sequences can we end up with after a sequence of such moves? (I call them `teeter-totter moves’, since you can either change three consecutive elements from increasing to decreasing or vice versa.)

Tropical polynomials and your federal tax return

It’s Tax Day here in the United States, and I spent a larger portion of the past weekend than I would have liked filling out the appropriate forms. But even among tax forms and legal documents you can find a mathematical gemstone or two!

The instructions for filling out, say, IRS form 1040, are rather peculiar in that they try give the reader very simple mathematical operations to carry out, one at a time, to end up with a complicated function of many variables. As far as I could tell, the valid operations are:

• Entering a value for a variable (e.g. income, standard deduction, etc.)
• $+$: Addition of two entries
• $\times$: Multiplication
• $\min$: Taking the minimum of two entries
• $\overset{\cdot}{-}$: A modified version of subtraction, where $a \overset{\cdot}{-} b$ is defined to be $a-b$ if $a\ge b$ and $0$ otherwise.

I found it interesting that every complicated tax computing formula can be broken down into steps of this form. In fact, it says something about the tax formulae: they can all be written as a composition of addition, multiplication, scalar multiplication, min, and $\overset{\cdot}{-}$.

The q-factorial in terms of the major index

EDIT: As pointed out in the comments, the following proof is found in Knuth’s book, The Art of Computer Programming, Vol. 3, and is attributed to MacMahon.

There are several nice series expansions involving basic permutation statistics which I’ve been reviewing lately. One of these involves the number of inversions of a permutation.

An inversion of a permutation of the numbers $1,2,\ldots,n$ is a pair of numbers which appear ``out of order’’ in the sequence, that is, with the larger number appearing first. For instance, for $\DeclareMathOperator{\maj}{maj} \DeclareMathOperator{\inv}{inv} n=4$, the permutation $3,1,4,2$ has three inversions: $(3,1)$, $(3,2)$, and $(4,2)$. We write \[\inv(3142)=3.\]