<homer>Mmmm…trig identity proofs!</homer>
<font color=#ECECEC>----------------
rocks</font>
Group theory, in particular the study of finite simple groups, is still very popular because it is so incredibly useful. A lot of modern mathematics revolves around taking a hard problem in one branch and figuring out a mapping to a different system or branch that is better understood. This is how Fermat’s Last Theorem was solved.
Finite simple groups are often used since group theory offers a lot of powerful insights into their properties.
Group theory is by no means closed, but you may be thinking of the classification of all finite simple groups was finished up in 1980. It took some time for the mathematical community to accept that the classification was complete.
What’s the classification of all finite simple groups, you ask?
Theorem: Any finite simple group is either
1.cyclic of prime order; or
2.an alternating group on at least five letters; or
3.a group of Lie type; or
4.one of the 26 sporadic groups
You think geometry proofs are hard? The above theorem took 15,000 pages to prove which explains why it took some time convince everyone that it was done. Mathematicians are working to pare that down to 3000 pages.
Understanding the sporadic groups, especially the Monster group, seems to be the rage these days.
Andrew Warinner
Group Theory is absolutely central in modern chemistry, particularly spectroscopy.
Offhand I would guess that there are about a thousand mathematicians specializing in group theory in the world. The nonsensical notion that there are only twenty people in the subject sounds like the urban legend that there are only thirteen people in the world who understand the theory of relativity, which is also nonsense.
When I took Trig in high school I loathed the identities, but now that I am into second semester Calculus, I wished that I had payed a little more attention.
The facts, although interesting, are irrelevent.
Thanks, Warinner, that was probably my confusion. Back in the 1970s, I was an algebraic topologist, which also relies heavily on group theory.
Aside from the sheer joy of finding such relationships, the trig identities teach logical manipulation of abstract concepts. When you are presented with massive amounts of confusing data, the ability to manipulate it (logically and therefore consistently) is a fine-honed skill. This is a commonly needed job skill, in many occupations.