How To Present Algebraic Topology To Non-Mathematicians? I am writing my master thesis in algebraic topology (fundamental groups), and, as a system in my school, students must write one page about their theses explaining for non-mathematicians the purpose of the study (in general) and why it is important in the real life. I don't know actually how shall I start, because I'm not allowed to use some mathematical terminologies like "Topological Space", for instance. My teacher told me it's important to convince the "politicians" to support financially the research in the department of mathematics mainly in algebraic topology to show them that we are doing something useful. Furthermore; it is to make the new students like this kind of mathematics.
So any suggestion or idea will be very appreciated! 
 A: 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.
A: I belive that you can find really interesting this stuff posted by Prof. Porter. It's not about the foundamental group but nonetheless present some real world application in algebraic topology (and there are some links to some documents too).
That is an answer to a post of Tom Leinster addressing a similar question: he was looking for application of abstract math to motivate applied mathematicians into studying abstract maths and the opposite, find interesting application that motivates abstract maths for theoretical mathematicians.
here the complete link.  
Anyway if you try to google a little bit you can find a lot of application of algebraic topology in applied field, in particular in data analysis and machine learning. If I'm not mistaken there are lots of work in persistent homology that have real world application. Unfortunately I don't have papers under my hand but try to look you'll most certain find lots of stuff.
Now you could argue that these stuff doesn't directly regard fundamental group
none the less to study such object it comes in handy to know a lot about fundamental groups. That's more lots of this applied tools were born in order to solve theoretical problems and later it was discovered they could be used into real world application. They are a sort of practice proof that theoretical research is as much important as applied one.
Hope this very short and poor answer could help.
Have good luck with you work :)
Edit: After a while a was able to retrieve this old post on MathOverflow which I believe can be useful. There are some real world application of algebraic topology with some reference too.
A: I worked in the classification of 3-manifolds.  Briefly, here's what I came up with to explain this to people who were really interested.  This doesn't so much justify the work politically, but it does give the general idea.


*

*Imagine you're a two-dimensional creature.  What could your world look like?  It could be a plane. It could also be a sphere, which would make your life different because you always come back where you started.  Or it could be the surface of a doughnut, which would be different again: there are two ways to loop around, and you know that they're different.
The size of the sphere or doughnut doesn't matter, just the overall shape.  The universe in the video game Asteroids is also a doughnut.  This is "topology".

*In fact there's an infinite list of universes: the plane, the sphere, the doughnut, doughnuts with any number of holes, some others which are twisted like a Moebius strip...

*People solved this problem in the 19th century: we have a complete list of the two-dimensional universes, and if you give a two-dimensional universe, I can easily tell you which one it is.  We have a "classification".  The main tool for this classification is to use numbers to describe the shapes.  That's "algebraic topology".

*We seem to be living in a three-dimensional universe.  What shape is it?  What shapes could it be?  Cosmologists, for example, would need to know this.  But it turns out to be a much harder classification problem, and only partly solved.  I worked on tools of algebraic topology to help with this problem.
A: You might consider looking into topological mixing. One studies the optimal way you can mix, for instance, a viscous fluid with n-rods. This applies to such things as taffy pullers. Here are slides from a professor at my institution.
A: Here is something that may be close to the politically minded hearts - the Human Heart! I have heard that methods of algebraic topology have been used in medical research of the human heart.


*

*When I was in grad school (late 80s), there was some documentary (may be just a news clip?) about them meds diagnosing imminent heart problems from the topology of the electric currents on the heart tissue. On the chalkboard behind the docs the calculations definitely included homotopy groups - don't remember whether it was the fundamental group. This was probably related to the knots formed by those currents, but obviously they didn't give the details. But something like: "the 2-torsion of the $n$th homotopy group of your heart currents changes => you will soon develop arrythmia (I'm making this up) would sound kinda convincing.

*When the technology of 3D-imaging of the human body was emerging, the people at the general hospital across the street from the sciences building asked a colleague the following question (he had helped them with compression algorithms and other handling of medical images before, so became the obvious target): the computer has stored a 3D-image of the heart of the patient as a 3D-array of small cubes, and can reliably tell whether each cube contains heart tissue or is empty (like part of an atrium or a ventricle). If that data is presented in a useful 3D-image, a trained human can quickly spot whether there are extra holes in the walls separating the chambers. But can this process be automated? We talked it over a lunch, and the best suggestion we came up with was to calculate the Euler characteristic of the 2D-surface formed by the boundary of the chambers (connectedness also). Surely the same solution had been proposed earlier elsewhere, but I think it was adopted (though there may be difficult to analyze border effects due to the granularity of the 3D-image). Anyway: letting the computer calculate the homology groups of the heart => the docs know whether heart surgery is needed doesn't sound too useless an application.


Keeping this CW, because this is largely vague recollections, second (or third) hand information and such. Welcoming edits/links to references.
A: Have a look at the Knot Exhibition on this website and the applications there, including the history of knots, which seems to date back at least 300 millenia. One scientific application is to Knots and DNA, you can also do a web search on this. The design of the exhibition is to show how the work of understanding knots and their mathematics comes before you can make the applications.  Tim Porter and I also did an article on Making a mathematical exhibition. See also Out of Line. 
The following picture shows a loop of string tied around part of a pentoil knot according to the formula 
$$xyxyxy^{-1} x^{-1} y^{-1} x^{-1} y^{-1} $$

Then the loop of string comes off the knot!  Explanation is in this seminar at Liverpool, for a demonstration  this with a well travelled pentoil knot made of copper tubing and a length of thin rope. 
The idea for this trick comes  from the determination of the fundamental group of the complement of the pentoil knot in $3$-space. See also pages 350-351 of  Topology and Groupoids. 
A: I like the pictures from chapter 0 of Hatcher for that purpose.

And also Louis Kauffman made some nice pictures for the Fox calculus.

And Thurston's 3-manifolds book has good pictures as well.

"Normal maths works with squares, triangles, perfect circles; it counts beans. In topology we're freer to work with more interesting shapes." (cue picture from Dr Seuss)

This is basically what I did in http://tmblr.co/ZdCxIydt_Owi, drawing on the familiar philosophical distinction between reductivism vs holism to work in a cohomology element (as an argument for holism: some things can't be decomposed into atoms without losing some essential property that only shows up in how they're put together).
A romantic notion I felt when I was first reading Hatcher is that the "counting rocks" and "straight-sight cannonball" problems that seem more typical of a more boring mathematics, get replaced by "feminine" "weaving-based" mathematics. (Women were the ones tasked with purling, picking, and wrapping in most of history.) So this is a nice tie-in for non-mathematicians, although I'm not sure AT actually is more "feminine" (whatever that means) it's an attention-grabber.

You can also talk about topological spaces vs metric spaces by using highways

versus airplane connections

taking a ring road versus trying to not back-track, versus taking small streets with different speed limits, could be thought of as metrics -- whereas if you're at the airport, then it's a question more of does the city you're in actually connect to where you're going or not, rather than 'physical' distance. You might end up flying past your destination (or at least extra distance) before taking a second flight/network hop, so the "network distance" is two (2 nbds), compare that to "ground distance". (Maybe for example Colorado Springs does not receive flights from L.A., so one has to go further east to Denver first ... if my geography is wrong then some pattern like what I'm suggesting does exist somewhere.)

As far as applications, I like the suggestions of Ghrist & Carlsson's work. I would add Ghrist's robots-not-bumping-into-each-other in warehouses, anyone can easily envisage as a braid the robots not bumping into each other over time. (See Ghrist's EAT book ch 1 for the relevant picture.)

But I would make a further point, which is that the goal is not to rehash the applications already developed, but rather why the novel approaches might lead to something new. In other words, "How has this already been applied?" and "How might this generate applications in the future?" are two separate questions.
To that end I think it's fine, as long as you use clear non-technical language, to talk in generalities about how the "style" of topological thinking differs from maths they may be more familiar with. You could give some examples of "wiggly" or "non-rigid" situations and say that the ambition is that with enough development of AT, these situations could be handled.
Justin Curry made a pitch like this in http://arxiv.org/abs/1303.3255, saying essentially "Linear algebra was amazing for applications; I think my thing could be similarly productive, from a different starting point."
On the other hand, we probably all know examples of scientists who thought their research would be used for totally different purposes than what it ended up in. My favourite example is the inventor of NMR (worth a Nobel prize) thought NMR would be used for "calibrating industrial magnets". Instead it was used to look inside people's brains. (But that was someone else's work, to apply it to that.)
On the other other hand, probably no politicians slapped Purcell & Bloch on the wrist when they said they were working on technology to calibrate industrial magnets. That can be envisaged by a politician and sounds science-y and maybe-important (just not as important as what it ended up being). Some mathematicians have also been known to name their ideas for marketing purposes, for example "dynamic programming" and "information entropy". (There are funny stories going along with the naming of each of these.)

If I had a piece of paper, my 10-second elevator pitch for algebraic topology would be:


*

*Most data-analysis methods assume data is "Square" (meaning Euclidean--independent dimensions, flat, etc)

*But how do we know that? What if the data is shaped like a peace sign? 

*We don't want to be caught off-guard by assumptions we weren't necessarily trying to make, so do AT to be aware of other possibilities the world might hit you with.



A: 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.
A: 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.
A: The following examples come to mind. Show that the real projective plane, i.e., a Möbius strip glued with a disc along their boundaries, cannot be embedded in ${\mathbb R}^3$. In its turn, you can additionally show that the Möbius strip (without boundary) is the space of all non-ordered pairs of points on a circle. There are many ways to prove these two statements, and some are quite elementary. I believe some proofs can be performed almost without words, just using pictures (well, elaborating pictures requires some serious work). You might take one and try to adapt it for "politicians". Good luck!
A: 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.
A: 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.
A: Roger Penrose in some of his lectures challenged students to find an everyday accessible example best described by second or higher cohomology.
I would construct an functional or relational description of a dynamic system and make some predictions about it using algebraic topology. Then you can use ordinary language. Try modelling two administrative systems using only relations of parts and show their topological equivalence, if it has to convince politicians...
I have some examples, but they are much too complicated to fit on one or two pages if stated verbally. And they might make you fail, depending on what politics your professor has, so be careful to choose examples which people don't have strong feelings about, if doing this.
