Realization of Cyclic Spaces
In what feels like a previous life I was a mathematician, and I just recently heard that appearantly my Master’s thesis is graded now. So here it is, for all you people who are interested in the realization of cyclic spaces!
This is what you’ll find in this 41-page document that took me many, many months to craft:
If you are not a mathematician: Gibberish formulas, arrows and some diagrams that don’t seem too impressive. Also 17 occurrences of the word “obvious”, and 12 occurrences of “it is clear”. Obviously, some of these things I wrote down half a year ago aren’t clear to me any more either, so don’t fret.
Mathematicians, especially those specializing in algebraic topology, might find that this text covers a number of interesting fundamental aspects of the theory of simplicial and cyclic spaces with a rather extensive level of detail. Feel free to use this text if it helps you or others.
The citation I prepended to the thesis, written a hundred years ago by my favourite author, so accurately describes the material at hand and my experience that it warrants a translation here:
It is only when one looks not toward the outside at their utility, but within mathematics itself at the relationships among the unused parts, that one sees the other, real face of this science. It is not goal-oriented, but uneconomical and passionate. – The average person doesn’t need much more mathematics than he learns in elementary school; the engineer only enough to find his way around in the collection of tabulations in his technical handbook, which isn’t a lot; even the physicist ordinarily works with quite simple mathematical tools. If they should need something different, they are mostly left to figure it out for themselves, since the mathematician has very little interest in such applied tasks. And this is why specialists in many practically important branches of mathematics are not mathematicians. But not far away are immeasurable realms that exist only for the mathematician: an enourmous nerve center has coalesced around the point of origin of a few lesser muscles. Somewhere inside, the individual mathematician is working, and his windows do not open to the outside, but into adjoining rooms.
There is an interesting story to how this thesis happened: I wrote it while travelling. To me this seemed perfectly normal at the time, but I’ve since heard that it astonishes people, so let me share the story.
My argument went somewhat like this: If you don’t like the cold and desolate winter months (and I don’t like them) and you have a bit of money in the bank (and I had some) and your professor agrees to consult with you using Skype (and I thank him for that) and you happen to live in times where one owns devices that can display PDFs – then it only seems natural that you should travel towards the tropics, following the sun, thinking about mathematics wherever it seems adequate. So that’s what I did.
This was my route: I started off in Spain to visit a friend and see if the nomadic life suits me (two weeks; early drafts of five introductory pages); it did, so I went back to Germany for three days to pack, and went off to Lebanon (I stayed for one month, writing 10 pages). Then I visited a friend in Dubai (two weeks; five pages and a few diagrams; grappled with fundamental problems of my formalism). On to Oman, where at first things went well and I produced a few pages; however I discovered a fundamental flaw in my understanding which my primary sources didn’t deem necessary to address: I painfully remember trying to understand a single, central diagram for eight days in a row, backtracking my way 14 hours a day, becoming increasingly desparate until I gave up and my friend John came to visit me over New Year’s. (Time spent in Oman: roughly a month with a short bus trip back to Dubai because of Visa issues. Unclear how much work I put to paper.) Next was Sri Lanka, where I arrived the day before presidential elections, which luckily went down without civil unrest breaking out. In Sri Lanka I mostly wrote in parks and the jungle, after these sandy countries everything seemed so exotic! And by painstakingly going through all I had written, blowing up seemingly innocent one-line statements to one-page proofs just to make absolutely sure I was correct, I finally found my way out of the trap that I had been in while in Oman. After a month in Sri Lanka, I went to Manila in the Philippines for two weeks. What an awful place! I didn’t get much done there, being busy with other stuff. From there I went to Sydney for a week (I went for job interview; incidentally, this is also where I live and work now…). I had three weeks of holidays on the island of Palawan in the Philippines, where my friend Felix visited me; no work was done there. In Singapore I resumed work and was pleasantly surprised that a considerable amount of work was done already (I only stayed there for a week due to budget constraints). I subsequently traded my windowless 12m² room for an equally priced 40m² flat in the heart of Bukit Bintang, Kuala Lumpur, where I stayed for a month and wrote most of the remainder of the thesis (15 pages). A short 3-day stint in Venice reunited me with my family, and I travelled back to Hamburg with them, where I did a final pass of the text, corrected numerous details and attended to day-stretching last-minute panics induced by seemingly poor choices of category-theoretic models in the very beginning. Then I travelled to Berlin and handed my thesis in. – All done, and not a single day was spent in the desolate winter!