Note: First blog post in a while, been a crazy few months.
There are many analogies mathematicians use to try to explain the abstract realm they work in to laymen of various levels. For example, Freeman Dyson had tried to categorize mathematicians as “frogs” or “birds” and Tim Gowers tried to categorize them as “problem solvers” or “theory crafters”. In a post on MathOverflow, I tried to categorize certain theorems as ‘workhorses’.
Another useful way I recently thought of to think about various theorems and technique in mathematics is to think of them as “fuel” or “engines”. Fuel and engines are both parts of “machines”, which comprise theories and programs in mathematics. Fuel is very versatile; an improvement in fuel will almost certainly mean that all engines which can use said fuel can operate better. Engines are more specific; they are designed to perform a specific function. A car engine for example is useless if you try to make it chop vegetables, unless you design an entire system which chops vegetables while being powered by a car engine.
This is a good way to explain just how the approaches Yitang Zhang and James Maynard differ. One can think of the original sieve mechanism devised by Goldston, Pintz, and Yildirim, now known as the GPY-sieve. In a sense they constructed a ‘machine’ which allows us to turn churn out small gaps between primes. The engine of the machine is the GPY sieve mechanism; while the fuel (or the raw input) is the Bombieri-Vinogradov theorem. The machine works quite efficiently in that it produces gaps that are arbitrarily small compared to the average gap; but the engine just can’t put out enough power to establish bounded gaps between primes with the current quality of the fuel. This situation was thought to be as far as one can go back in 2013.
Yitang Zhang came along and showed that while it is not possible to develop an improved ‘all purpose’ fuel, i.e. an improvement on the level of distribution present in the Bombieri-Vinogradov theorem, one can devise a specialized and more efficient fuel that works specifically for the GPY engine. This extra boost in power was sufficient to establish bounded gaps.
Maynard, however, showed that one can rip out the GPY engine and replace it with a much more efficient and powerful engine that requires much less fuel and can output much more power. This is Maynard’s multi-dimensional sieve; something thought to be impossible previously.
The problem is that Maynard’s engine doesn’t run on Zhang’s fuel; because the latter is specialized for the GYP engine. Modifying the Maynard engine so that it runs on the Zhang fuel would be the next step forward towards the bounded gap problem.