[Here’s a big dump of maths stuff from my previous web-site:]
Local invariants of real hypersurfaces (my final year undergrad thesis) PDF | PostScript
Constructions in Category Theory PDF | postscript
A Theory of Road-Crossing PDF | PostScript
Why D(f.g)=Df x Dg PDF | PostScript (A correct construction, but poorly explained – for a one-paragraph-of-words-and-a-diagram explanation of this, see Milnor’s book on differential topology).
A Fundamental Theorem of Calculus PDF | PostScript
Algebraic Topology sketches pdf
Intrinsic Geometry Formula Sheet pdf
Two sketches in Group Reps pdf
8 Comments
Hi!!
I’m doing Mathematics in Stellenbosch, South Africa.
I don’t really know in which field of Mathematics you’re
specializing, but I certainly hope that you will be able to help me and my friends with a
project that we have to do for Mathematics (3rd year).
The project is in General Topology and we have to give a whole discussion about Quotient
Mappings (ie Mobius Strips, Klein Bottles and the like…).
There are so many web sites that we don’t really know how to get organised, so it would be really
great if you could give us some advice.
By the way… like your room… ;)
Rinske,
I’m not sure that I’m the best person to ask about this; I’m very comfortable & familiar with all of the concepts but over the past half hour thinking about them I’m not sure what is the best way to introduce them at your level. I’m just going to throw out a few things of various sophistications, some might be useful, but maybe not (I hope so though).
There are quite a few different ways of viewing quotient mappings; looking at them as being identification maps (the standard “glueing” explanation) is the most geometrical, but that doesn’t explain explicitly what the open sets are.
for that, you’ll need to approach the who thing pretty topologically; it’s worth trying to figure it out yourself but in practicality people deal with quotient maps (that maps the space before glueing onto the space after it’s been glued it’s a many-to-one mapping if glueing’s taken place) , and say that the quotient space has the topology induced by the quotient map. Definitions can be got here.
Also, the description of the quotient topology as a universal property is definitely worth thinking about and trying to figure out what it means, because the description of things as universal properties becomes more and more common as you go up the strata in maths.
More than Klein Bottles, there are lots of other cool spaces you can construct: cross-caps, projective spaces, and other things. If you’re willing to skip over a lot of the technicalities (or ask your lecturers some problems) there are a lot of cool examples given in the book here:
just search for instances of the word “quotient” and you should soon enough come across a couple of examples: it’s worth knowing what the suspension of a space is (the suspension of a space T is got by looking at Tx[0,1], and then gluing all the (x,0) together and all the (x,1) together, and all the (x,0) together – see his book for a picture, the generic image is that of two cones either side of the space).
The issue of orientability is probably the main cool thing about moebius strips; but I don’t know if there’s much conceptual insight beyond the very basic stuff you can find out online about it (here, for example), but I can’t think of any really conceptually cool way of conneting it together with quotient spaces.
Beyond all of that (and up a little in complexity), you could maybe say something about covering spaces (universal coverings, maybe even fibre products and other things), but I don’t know what much you can say about them.
With regards to resources: wikipedia, mathworld, and planet math are all good places.
Perhaps you might consider posting this question on the newsgroups (on sci.math via here, say); a lot more people will read your request there and you’ll get a more diverse range of replies (there’s a lot of bullshit on sci.math, so ignore all the people who seem even slightly weird, or check with your lecturers if anything seems a bit odd).
I hope you plan on getting some of your math’s back online… I really liked reading through them.
Moon-guardian: What maths is offline?
By offline, he means that there are a lot of broken links on the maths page http://www.maths.tcd.ie/~icecube/maths
Regards,
David
Oh…that *is* weird…you guys were right…seems that a lot of pages can’t be accessed from outside the maths department. Pretty weird….will have to do something about fixing that…
Yes please, I would like to look at some of them.
(Apropos one of the randomly selected “x says” items (here –> http://www.maths.tcd.ie/~icecube/rules.php — the one I’m talking about is accredited to Cardiff University) I do wonder what Niall Crowley would think about squeezing an inequality too hard.)
>Yes please, I would like to look at some of them.
Sorry, I’m totally in some weird spacetime warp here, could someone please please giving me just one URL that isn’t accessible so I can work on it?
I have no idea what the fuck is up with that page…I mean, I have
suspicions, but yeah, there’s some site work I’ve been putting off for a
bit, so.
Thanks.
Stephen
[[UPDATE:I’ve not been able to see what the trouble was, but 2/3rds of the pages here have been moved into the wordpress database, so should, simply by virtue of being different, probably work where they hadn’t before]]