Two months and countless draft embryos after initially promising it, here is the first part of an unfathomably long rant describing my field of research. I honestly don’t expect anybody to subject themselves to that read, but at least now I have a place to send those who foolishly ask me about it at cocktail parties.

The short answer is that I do research in Bioinformatics, which is where Mathematics (along with Computer Science and a dozen other disciplines) meet with Biology and Genetics in a dark back-alley, and do all sorts of indescribable things to each other in the hope of: creating a better world, curing cancer, breeding the next race of eugenic übermenschen or making a few bucks for Big Pharma… whichever comes first.

But that sort of answer, while technically correct, does not really tell you why such an unnatural coupling of disciplines was warranted in the first place. Allow me to start at the beginning. Way at the beginning.

[open long semi-relevant digression that can be advantageously replaced by a thorough read on Complexity Theory, if you feel up for the more sciencey and truthy version of things]

The scientific problems of this world tend to fall in either of two categories: those you might eventually solve with a good computer and some time… and those you will never solve exactly, no matter how much crazy sci-fi supercomputing power you throw at them.

This “solvable” vs. “not solvable” demarcation might sound like a tautology, until you understand the full meaning of “never” in the above statement: these, are not problems that might be solved one day, when science progresses far enough or computers get ten, a hundred or a million times faster. These are problems whose solutions require calculation of a complexity that is proven to be beyond the reach of any conventional means of computation in any foreseeable future (“unconventional means” would begin with the discovery of heretofore unknown laws of Physics: in other words, unlikely in your lifetime. at best¹Yea, yea, I know… “Quantum Computing”. Let’s wait and see when we get there, ok?).

By and large, the mathematical complexity of a problem, is the order of time (or computing power) it will take to solve it, relative to its size.

Without calculating the result of a certain task, it is often possible to predict whether producing this result could or could not be done in a reasonable amount of time (where “reasonable” usually means “in less than the age of the universe, assuming the use of every single computer on earth”, or somesuch).

There are countless examples of tasks falling in the first category, “easy” tasks that can be solved quickly, regardless of how big they are. For example, anybody past kindergarten age can presumably add two numbers of practically any size with a piece of paper and a pen. You just add each digit one by one (and, yes, carry the one) and adding two 100-digit numbers will take barely more time than adding two 3-digit numbers.

Now consider a different task: say you are a traveling salesman who needs to plan their next sales route. You have a map of the region, with the towns you must visit and all the distances between them, given in kilometers. How do you find the absolute shortest route that will take you to each city at least once without wasting gas or time?

More to the point: how difficult do you think finding that route will be?

Sure, it sounds easy enough: pick a starting point, follow every roads that go from that city to another one, then onto the next etc. Keep the shortest distance you’ve found. Can’t be that tough, right?

Let’s say there are five cities: you pick a city to start from, then check all remaining four, and from each four, go onto one of the remaining three etc. etc. In total, that’s 5x4x3x2x1 = 120 different paths to compare (that product can also be written using the factorial function: n! = n x (n-1) x … x 3 x 2 x 1. e.g. 5! = 5x4x3x2x1). Not so bad.

What if there are a few more cities… for instance, two times more: 10 cities. That’s 10! = 10x9x8x7x6x5x4x3x2x1 = 3,628,800 paths to look at. Huh, that might take a bit longer to do by hand. No worries: somebody will write a computer program that gives you the answer in a couple seconds.

Except that, you guessed it, each time I double the number of cities, the difficulty does way more than just double.

For 20 cities, the number of paths to look at is: 20! = 2,432,902,008,176,640,000.

For 70, cities, there are 70! (that’s factorial of 70: 70x69x68x…x3x2x1) possible paths to check one by one. That number has exactly 100 digits. This is (very) roughly the number of particles in the entire universe. Assuming you were to put every single computer in the world to work on this, you likely would not be done by the time the Sun explodes.

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