Another classic recently-solved is the 4-colour problem.
Take a plane. Divide it into countries (no oceans, countries all in one section). How many different colour pens might be needed to colour the countries so adjacent (across a line, not a corner) countries are different colours?
You might need four colours: consider a small country at the junction of three larger nieghbours. Each touches every other, so after no two can have the same colour. It was shown a while ago that any map can be coloured with 5 colours. Recently a team proved that 4 colours are always enough.
Interestingly, they proved that the only possible maps that needed 5 colours all were basically equivalent to a large but finite collection, and showed all these were 4-colourable with a computer.
Mr. Cecil Adams wrote about the four-color map problem here.
The proof wasn’t very elegant; basically they just tried every possible graph within a finite space to see if they could color it with four colors, and they could.
I’m a graduate student in mathematics going for a PhD right now. This is my answer to the OP: Mathematicians answer questions that
(a) can be stated in sufficiently abstract and precise language so that the answer depends only on the words used in the question and not on any facts about the physical world; and
(b) are hard enough that people who have to do something else to make a living don’t have the time to solve them.
Sometimes these questions arise in or are motivated by efforts to solve problems in the physical world. Often, however, the problems mathematicians are interested in depart from any obvious physical applications, only to turn out to be useful latter.
For example, many of the theorems in Euclid’s geometry might have been motivated by such practical fields as architecture. If you are designing a building and you want to put in square windows, you need to know how to recognize a right angle. So Euclid will tell you the handy theorem which states that a triangle with side lengths a, b, and c is a right triangle if, and only if, a[sup]2[/sup] + b[sup]2[/sup] = c[sup]2[/sup].
But when the mathematicians went about solving this problem, they found that, to make the language sufficiently abstract (see condition (a) above), they needed to assume some axioms of geometry from which they could deduce this result. One of these axioms is called the Parallel Postulate. Without getting into details, let it suffice to say that mathematicians became very interested in knowing whether the Parallel Postulate could be proved instead of being assumed as an axiom. This question was no longer motivated by the architectural design of buildings; the abstract problem itself was interesting apart from any physical applications. The quest to solve this problem eventually led to non-Euclidean geometry.
Now, as is often the case when abstract mathematical problems are pursued for their intrinsic interest, a real-world use for non-Euclidean theories did eventually turn up when they found applications in Einstein’s theory of general relativity.
That said, what modern mathematicians do day-to-day is produce papers like those that you can download free from the Mathematics ArXiv.
> There are usually two distinct departments of Applied
> Mathematics and Mathematics at most respected universities.
Well, there are two distinct departments at many large universities. I think that to say that it’s true at most respected universities is to slightly exaggerate.
C K Dexter Haven writes:
> Mathematicians employed by universities are usually required
> to publish research results. There are several monthly journals
> that accept such publications.
I think it should be noted that there are hundreds of mathematical journals.
I’ve read an estimate recently that about 250,000 new mathematical theorems are published each year.
We have hired a few mathematicians to help us prove that markets are not random. We are starting a Quantitative Analysis Hedge Fund on October 1 (yay!). Our mathematicians know very little about financial markets, which is cool for us. We just feed them TONS of data and sorta explain what we are looking for, and they attempt to come up with models that can predict market movement on a 12 to 24 hour basis. We’re trying to upgrade our computers because some of the models they come up with take 5 days to run on a 3gig processor!
Most mathematicians that I know are computer programmers in disguise. To do most of the high-level math these days one needs to be able to test things. There are a few software packages out there that help (We use MatLab), but they still require VBA, C++, Fortran and other languages as well.
