Technical Mathematics and Its Exposition
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451
Many parts of mathematics are increasingly technical yet often progress into these technical domains come with important intuitive insights or with important applications. One can appreciate the value what has been discovered even if one can not follow the technical details. A good example of the pairing of a technical report and an explanation of its implications is given in the current issue of Nature (Volume 450; November 1, 2007; p. 40-41, exposition; p. 77-79 technical report), in a pair of articles: First encounters and First-passage times in complex scale-invariant media.
This particular pair of articles deals with random walks. Depending on the "rules" there are many mathematical issues here. One can imagine a random walk in the plane one starts at (0, 0) and one moves one unit at random either north, south, east or west. Now, one repeats the process from where one is located. A different idea would be to again start at (0, 0) but move in any direction at all, one unit, where the direction is chosen at random. Perhaps two dimensions is too hard, so one should follow the problem solving idea of starting at the 0 point on a line and moving left or right one unit. Now, there are a great variety of questions one might be interested in. For example: a. What is the probability that one will be less then k units from where one started after t time steps? or b. What is the probability that one will (first model) reach the point (3, 4) after exactly 7 time steps?
One can see that models of this kind might be useful in understanding the motion of very small particles in the air or the way rumors or diseases spread.
We should be encouraging more pairs of articles of this kind that open our minds to new ideas and applications. Anyone with intellectual curiosity can profit from the expository article and those with more technical skill can read the details, and perhaps see a way to improve them or carry they forward in some other way.