How To Present Algebraic Topology To Non-Mathematicians?
Solution 1:
Here are two examples of algebraic topology (actually homology theory) being applied to "real world" problems.
1) Ghrist's paper "Homological sensor networks." (pdf)
2) Topological data analysis, as pioneered by Gunnar Carlsson.
Solution 2:
Explain that in a society that values learning, the pursuit of abstract knowledge is an end unto itself. This is how pure mathematicians have answered that question for millennia and is, as far as I know, the only truly irrefutable answer.
Solution 3:
It may not be "useful", but you can explain the subject to a nontechnical audience by relating it to "wire and string" puzzles. The solvability of wire-and-string puzzles can basically be reduced to questions about the fundamental group of the complementary ambient space surrounding the metal wire.
For example, look at the following "astroknot puzzle", where the goal is to remove the long flexible string from the solid metal,
This also, incidentally, leads to one of the best ways to solve such puzzles as a human, which is to imagine that the inflexible metal can bend into a form where the puzzle is easy, and then imagine how the nearby air would deform under the reverse transformation.
Solution 4:
I think it's very doable, I actually gave a short talk on topology to high school students once. Forget about the algebra and treat it very geometrically instead. Most importantly, draw lots of pictures! After all, it's really a visually intuitive subject, and even topologists spend a lot of time drawing pictures and convincing themselves about the geometry.
Showcase some of the more immediately understandable results of the field - classification of surfaces, Brouwer fixed point theorem, hairy ball theorem, etc. and the cool examples - tori, Möbius band, Klein bottle, projective space, etc. Whenever possible, appeal to physical, visual, and geometric intuition.
For applications, tell your audience that topology is a great reduction tool in many problems that at first seem complicated, examples being persistent homology, Morse theory, and topological degree theory. The IMA is having a special focus year on applications of algebraic topology (http://www.ima.umn.edu/2013-2014/) - check out the lectures! And make special mention of the computability of the theory - homology is a great tool because we can make the computer do it.
Edit: I was speaking primarily about explaining topology to a general audience. I don't know much about homotopy groups, but if I were to say anything about fundamental groups I would make sure to mention the Poincare conjecture. It wouldn't hurt to mention how it had a USD 1 million bounty attached to it, and besides attempts at Poincare led us to some of the most fruitful mathematics of the past century, such as Morse theory.
Solution 5:
You might want to watch the talk of Tim Gowers "The Importance of Mathematics".
In an nutshell, his answer to the politician was that some areas of mathematics have more practical applications than others but:
- It is very hard to predict which areas of mathematics will have applications in the future.
- The different areas of mathematics are very interconnected. Cutting the funding of a "non-profitable" area could harm the development of "profitable" ones.
Of course he also talked about beauty and the quest for knowledge.