Germs & Sheaves

Hyperbolic triangle, via Wikipedia
Hyperbolic triangle, via Wikipedia


I spent most of the day reading up on differentiable manifolds, Riemann surfaces, germs, and sheaves. Some of the concepts are extremely interesting since they tie into category theory. This led me to differential geometry. I supposed that differential geometry had more to do with Euclidean geometry, an undergrad class that I didn’t enjoy all that much³, but it’s got a lot more to do with the geometry and structure of differentiable manifolds, which interest me¹.

Since the late nineteenth century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds.

The study of calculus on differentiable manifolds is known as differential geometry.

The tangent bundle of a circle, via Wikipedia
The tangent bundle of a circle, via Wikipedia


There are a few professors at NTNU that study differential geometry. I’ll try to meet them sometime next week to find out more about their research and RA positions. There’s also a PDE prof that I need to contact.

I’ve started my research notebook². Right now, it’s concentrating on the topics mentioned in (I). Currently, I’m just working through a bunch of definitions. I haven’t started the Lebesgue outer measure and the analysis concepts that we’ve seen in class. I’ll use those notes in order to prepare for my real analysis exam in two weeks.

My mind is awash with sheaves, presheaves, stalks, and germs. I like the terminology and once again, when my wife read over my shoulder, she didn’t get a thing. She finds that funny. That being said, I could talk to a lot of people without them knowing what I was really talking about.


I worked into the night learning about topology, including paracompactness, mesocompactness, metacompactness and orthocompactness. Differentiable manifolds also led me to tangent and cotangent bundles.

I read up on filter, ultrafilters, and ideals again. This time, I noted stuff down. I think that, in total for today, I spent about 7 hours on research. That isn’t original research, just researching topics on my own. This is something that I really like doing. It also lets you find more interests. For example, before today, I wouldn’t have been able to say that I was interested in differential geometry. Now I am since it involved the study of differentiable manifolds and their geometric structures, or calculus upon differentiable manifolds.


My friend Charlotte had her PhD exams last week. By her comments, it looks like she’s going to do pretty well. That’s cool. I’m a bit sad that she’s heading to France for a few months. Initially, she was going for about 6 months, but she had to shorten her visit. She told me that the Taiwan scholarship that she got, which is about $1,000/month for 4 years is quite a hassle. It looks like the one from private or semi-private institutions are a lot less of a hassle. Charlotte has got a lot of paperwork to do for that.

I’m hoping to get a similar scholarship next year. Coupled with two RA salaries⁸, as well as my writing revenue and another base scholarship, I should be doing well⁷. I’d barely have to teach ESL anymore. I will though, it’s just good to use that teaching salary to live with, and save the rest or pay off bills with the other salary⁴.

Other than bicycles, getting a MacBook Pro 17″, an FX frame camera⁶, I won’t be spending much on other things. I might get an iMac and send it over to the US, but that’s pretty much it⁵ . The camera and the Mac can wait until the last year that I’m in Taiwan, which would be 2011 or 2012.

When the MacBook Pro 17″ came out, the urge to get one was strong, but it abated. I still want one, but I won’t need it until much later, as my 13.3″ Asus WFM is fine for now. One thing that I’m looking forward to is maxing out the specs on that little laptop. Getting a 500GB HD instead of the 250GB one, and adding more RAM and naturally, having Win 7 installed.


I’m still trying to stop myself from getting a motorcycle to replace my scooter on my daily commuting. I’d be getting a gray area motorcycle⁹, with something like 400-900cc. I wouldn’t spend more than $1,800 and it would be fun and a lot safer, since everything is safer on a real motorcycle. However, it would be a bit of a waste, as I would be leaving shortly after. I wouldn’t have it for more than a year or two. I’ll think it over. Old Nancy is getting quite used. Her 125cc aren’t enough and I’d never get a new scooter, I’d get a used motorcycle instead. I’d keep Old Nancy though. Scooters are great to do the shopping or errands on.

* * * * *

[¹]: Actually, quite a lot.
[²]: I decided to use my Canadian Blueline A91 200p hardcover notebook. These notebooks are usually used for lab notes. I bought one about a year ago and used two pages for essentially the same thing. Only, since then, my interests have become more focused.
[³]: Actually, hated wouldn’t be far from it. Well, maybe not that extreme. Clearly disliked or left me ambivalent would be more appropriate. Geometry, as I saw it, wasn’t very interesting. Differential geometry is quite interesting. I don’ know why I didn’t come upon it before.
[⁴]: Or buy some bikes!
[⁵]: The new 27″ iMac is the one to get, as it’s significantly cheaper as the 30″ Apple Cinema display. It’s especially bright and beautiful, which is perfect. I’ve got a 22″ Viewsonic screen here that I probably will sell before leaving. The iMac can be used as an auxiliary screen for notebooks.
[⁶]: A Nikon, of course. We are a Nikon family. At that time, it will either be a D700 or a D3S, or their updated models. Personally, I prefer the D700, as it does almost everything that its bigger brother does. Kind of like the D200 and the D2X at the time that I bought my D200. The D700 hasn’t received an update yet, and I’m waiting what it will be like.
[⁷]: That would be around $2,400 a month, not including the freelance writing + ESL teaching. I find that it would be important to work mostly in math than in teaching.
[⁸]: Hopefully they won’t be mutually exclusive.
[⁹]: Before 2001, big engined motorcycles, over 500cc, weren’t sold here. They were imported in pieces and assembled. In 2006, a new law was passed. The legal motorcycles with 500cc and more could go on highways. They have a red license plate. Any motorcycle that has a red plate is very expensive. Gray area motorcycles are those that were imported and assembled. They don’t cost that much and have a regular white license plate. You can’t ride on highways with those.


I spent

Author: range

I'm mathematician/IT strategist/blogger from Canada living in Taipei.

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