practical applications for algebraic topology?

Question

I am a high schooler in a high school math class, and i know that this stuff is way over my head, but i have to do a small five-minute presentation in which i simply give a quick introduction to topology. I am not getting detailed at all. I know about point-set topology, but i would like to be able to talk briefly about algebraic topology. I will show the mobius strip tricks. That's interesting, but i want to know some real-life things that topology has been used to help with.

Answer 1

Another very important--and very accessible--branch of algebraic topology is knot theory. (See link below.)

A small anecdote: The mother of an algebraic topologist I knew worked in a biochem lab. It turns out that one way to identify proteins is to expose them to certain enzymes and see how the proteins "twist up"--in other words, to see what knot they form. But, at the time (I think this was the 50's or 60's), they were still doing this by taking x-rays (or some similar image) of the knots, modeling them with string, and manipulating them until they could figure out what the knot was! This is some 50 years after mathematicians could do this topologically...

Answer 2

You're not going to find much in the way of engineering applications. The difference between geometry and topology is that in geometry there's a concept of distance and in topology there isn't. So topology isn't going to help many people figure out the dimensions of what they're building.

That said, you can get a LOT of applications if you expand your talk to include graph theory, which is formally part of combinatorics but has a whole lot of topological concepts in it. Check out the book Linked, for example. There's been a huge amount of popularization of that kind of network analysis over the past decade, much of it inspired by that enormous graph that is the Internet.

Knot theory in biochemistry is a cool example of scientific applications. I'm not sure I'd agree that string theory or other cosmology are about topology so much as they're about differential geometry, but clearly topological concepts are in play at least helping with physicists' intuitions.

Answer 3

while Fourier stated that some discontinuous applications are the sum of infinite sequence it replaced into no longer just about useful awareness. Fourier is likewise credited with the invention in 1824 that gases interior the ambience might desire to advance the exterior temperature of the Earth lower back no longer just about useful(on the time). Now we use derivatives of Fourier's paintings for each little thing from voice acceptance to genetic learn. in basic terms on the grounds which you do no longer see a pragmatic use does not advise that somebody else does not have a topic that the applying of those works might desire to tutor useful in fixing. This is going decrease back to the previous arguments that have been raging between scientist and engineer for generations. I study a narrative as quickly as some scientist and an engineer speaking over a lager in a close-by pub. The Engineer replaced into explaining to the scientist that he builds machines that separate different length pebbles. The engineer factors out that in case you vibrate the textile at this cost one length is separated and at that cost yet another length. The scientist says "nicely why does that ensue?" The engineer says "i do no longer understand is it important?" different mindsets, the engineer seen his subject solved the scientist replaced into presented with a topic to clean up. i'm an engineer via commerce yet (If I had the money) i might gladly return to varsity to soak up a study of technology, i've got not got self belief the two mindsets are on an identical time unique

Answer 4

string theory is very active area with topological applications. Try wikipedia: http://en.wikipedia.org/wiki/String_theory

and google searches on applications of topology to string theory (also called superstring theory). The maths is VERY tough though but it is cutting edge!

Answer 5

In algebraic topology, there is homology and cohomology Betti numbers, Euler characteristic, Brouwer fixed-point theorem, Hairy ball theorem, Borsuk-Ulam theorem, Ham sandwich theorem, Chern classes, Stiefel-Whitney classes, Pontryagin classes.
Maybe you may like to search for it in gahooyoogle.com?