German learners' society

ah I see this makes sense.
I haven't even noticed the warning above. It's only lucent red though.
Am I guessing right that the 'warning Level:1'-hint on the right relates to this problem?

Was deutsche Bücher betrifft, habe ich wirklich keine Ahnung. Bisher habe ich nur Damals war es Friedrich und ein paar andere Bücher gelesen und na ja, ich suche jetzt etwas anders. Ich suche ein Buch, das mich herausfordern wird aber auch das nicht so kompliziert ist, dass ich gar nichts verstehen kann.

Wenn man mal von der Sprache absieht, was für Bücher liest du denn gerne? Und sag jetzt bitte nicht "eigentlich alles" - kein Mensch liest wirklich alle Arten von Büchern gleich gerne.

Hallo Leute. Wenn ihr mir helfen könntet, würde es mich freuen.

Meine Prüfungen sind endlich zu Ende gekommen und ich suche ein paar gute deutsche Bücher, die ich während der Sommerferien lesen kann - und ich brauche Empfehlungen!

Was deutsche Bücher betrifft, habe ich wirklich keine Ahnung. Bisher habe ich nur Damals war es Friedrich und ein paar andere Bücher gelesen und na ja, ich suche jetzt etwas anders. Ich suche ein Buch, das mich herausfordern wird aber auch das nicht so kompliziert ist, dass ich gar nichts verstehen kann.

Tut mir Leid, dass mein Post so unklar ist aber.. eure Hilfe ist trotzdem sehr willkommen!

OK, bis hierhin verstehe ich es so halbwegs (glaube ich zumindest) - aber was mir immer noch nicht ganz klar ist: was genau kann man denn dann mit Hilfe dieser Vereinfachung über so einen Raum herausfinden?

Oh, sorry - I completely missed your post. I'll answer in English, because it's 4:24am and my German maths vocab isn't up to it.

Well, certain spaces have properties called completeness (you don't get sequences trying to converge to points that don't exist or naughty things like that), compactness (if you cover the space up with infinitely many "open sets" (just think of them as bits of cloth or something ), someone can come along and remove loads of superfluous ones and leave you with a finite cover), and so on. You might want to know if the space is disconnected, i.e. whether it would break apart or not if you tried to pick it up. (A square would be connected, but if you removed an infinitely thin line down the middle, like a crack, it'd be disconnected. A sphere and a doughnut are both connected.) You could, of course, just want to know what a space looks like (does it have a hole in it like a doughnut, or is it completely continuous like a sphere?).

The Poincaré conjecture, a problem very much like this, was recently solved by Perelman. Imagine a 2-dimensional sphere (i.e. a normal one - I say 2-dimensional because I'm referring to the surface of the sphere, not of a whole solid sphere) and wrap a loop of string around it. No matter where you put your loop of string, you can always tighten it and make it smaller and smaller (picture) until it's eventually just a point. Now, of course, this also works for anything that's vaguely like a sphere. It could be a cylinder - you'd have to tighten it "around" sharp corners, but that doesn't matter. As long as you're not breaking the string or breaking the cylinder, it's fine. It could be a cube or a pyramid or something even more irregular, as long as it's equivalent to a sphere topologically, i.e. it has no holes in it. Now think of a doughnut, or anything equivalent to it (like the coffee cup). If you make a loop in certain places on the doughnut, you can pull it tight, but if you make a loop in other places (e.g. around the "handle" of the coffee cup) you can't pull it tight along the surface.

It was known for a long time that the ability to pull any loop on the surface of a "2-manifold" (the name for this type of space) tight was something that was unique to a sphere - the only manifolds out there are manifolds that look like spheres, ones that look like doughnuts, ones that look like 'doughnuts' except with two holes (try gluing two doughnuts together), and so on. The sphere was the only one with this property. This is important because sometimes you can't tell whether a space is equivalent to a sphere or not (which is obviously a nice property to have because we know loads about spheres), so you can stick an arbitrary loop on it and try to pull it tight, and if it always works then your space is a sphere. Similar properties were also known about the sphere in 5 dimensions and above, and were soon proved for the sphere in 4 dimensions too. But the 3-sphere proved really difficult to get a grip on, because in dimensions 3 and 4 you get spaces "knotting" into themselves (which is exactly what you'd expect it to mean - forming a knot and being awkward about pulling the loop tight), whereas in 2 dimensions you don't have enough room for anything to knot together and in 5 or above you have enough room to trivially unknot anything that knots itself together. Perelman proved the result true for 3 dimensions, and was offered the Fields medal - often called the maths equivalent of a Nobel prize - and turned it down.

Sounds fascinating. Thanks for the explanation.

Did we read the same thing? Whatever floats your boat I guess!

Oi! Nothing wrong with maths.

Did we read the same thing? Whatever floats your boat I guess!

Speaking of knot theory, have you ever come across this book? It's meant to appeal to a wider audience, obviously, but they did actually include all the mathsy bits as well, so you might enjoy it (even if you don't wear ties on a daily basis).

Sei froh. Es ist noch viel schlimmer wenn du nach deinen allerletzten Prüfungen auf einmal keine Arbeit mehr hast. Man stellt sich das nur vorher immer so toll vor, aber in der Praxis sieht das dann leider oft ganz anders aus...

PS: Auch wenn das eine etwas dämliche Frage ist: Was sind eigentlich metrische und topologische Räume (irgendwas mathematisches vermutlich, aber was genau)?

Far too advanced for me to understand mathematically, but it's very well explained.

Funny, you say things like that yet you do Maths at Cambridge. I automatically expect someone doing that to be able to do anything in Maths with ease...misconseptions eh?