Posts Tagged 'differential geometry'

First Week of the Fall Term 2011 – Mathematics Graduate School

Gradient Flows by Luigi Ambrosio

It’s the second week of school, but with a bank holiday last Monday (09/12/11), graduate school started up slowly. It will take until next week until everyone is finally registered to all of their classes. It took me a while to do so as well, because they changed up the system from a program that you installed on your computer to an online version. The online version is better, but you need to know where to go. I finally registered to my classes last Tuesday. I have three this semester:

Topics in Geometric Analysis: This class is with my thesis advisor and we will be exploring gradient flows in metric spaces. We will be using Luigi Ambrosio’s book of the same title. It promises to be an interesting class. Although, it’s not exactly what I’d like to do in my thesis, it’s getting there. I’d like more measure theory, but luckily, my advisor is doing research in the field. There are six students in the class, but only 4 were registered on Monday. I don’t know if the other two will be registering, my guess is yes. Two of my classmates are the other graduate students of my advisor. We are all going to a workshop in Hsinchu in Differential Geometry  on Saturday. Since I work most Saturdays, it’s not really a problem getting up. It will be a break from the norm, and I have a keen interest in Differential Geometry.

Continue reading ‘First Week of the Fall Term 2011 – Mathematics Graduate School’

Paracompact Spaces

Toroid, via Wikivisual

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It’s strange that I was actually researching paracompact topological spaces on Sunday and that we are seeing those types of spaces in my complex analysis class. We just started the Berenstein & Gay Complex Variables¹ book and things are pretty interesting. I actually deduced that we were heading there because of some of the concepts that we are seeing.

A paracompact space is a topological space in which every open cover admits an open locally finite refinement.

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Germs & Sheaves

Hyperbolic triangle, via Wikipedia

Hyperbolic triangle, via Wikipedia

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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.

Continue reading ‘Germs & Sheaves’


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