I thought I might add a post today. I chose not to go to the farmer’s market with some of the rest of the crew this morning, ensuring that the day would be full of solitude. It’s been nice though. I read some of my notes, and organized a myriad of PDF documents on a new app called Goodreader. Good reader allows me to actively annotate my PDFs, which is an important feature now that I have a good collection of textbooks that I need to read.
My roommate, Jesse, just woke up. It’s 3:00.
Anyway, moving on. A group which includes some of my favorite people here left yesterday afternoon to see the arches in southern Utah. I hope they are having fun right now.
I feel pretty lame for not having much exiting to say, but a large part of the purpose of this blog is for me to catalogue my thoughts for my own mental health and posterity.
I should add that I’m getting a good sense of what I want to study next spring. I didn’t realize until I came here how fundamental “the big three” in math is. ” The big three” in math is what I call the three pillars of modern mathematics. They are Topology, Algebra, and Analysis. I didn’t realize that one can’t even begin to get a good insight into mathematics until one has had some decent level of exposure to all three. It is also worth noting that Linear Algebra is sort of an ambient topic that is useful in all three of the aforementioned areas of mathematics. Coming into this program in Utah, I had only a taste of each of these areas in the form of of semester-long undergraduate introductions in each. I am glad that I will be taking graduate level Topology and Algebra next semester, and I am already formulating the direction of further study. Jesse is close to my age and his mathematical interests are bizarrely aligned with my own. As a result, he has opened my eyes to a world of interesting directions that I have to pursue. As such, I plan to continue my graduate algebra course into the spring, and add a course in complex analysis. Depending on how confident I find myself in 4 months’ time, I might add to that an independent study in differential geometry. It turns out that physicists who study General Relativity are basically differential geometers, except that they use the Minkowski pseudo-metric, meaning that it’s not really differential geometry in the strictest sense. Differential Geometry is especially intriguing since it provides for an abstraction of calculus to smooth manifolds. Our universe behaves like a smooth 4-manifold, and the general consensus is that it is a hyperbolic manifold, meaning that every point in space-time has a so-called negative curvature. There are many types of curvature, but the basic meaning is that straight lines tend to diverge from each other in negative curvature, whereas straight lines converge with positive curvature. With flat (zero) curvature, straight lines go, well, straight. They neither converge nor diverge. They continue in the direction we feel they should continue without bending. Flat curvature, as you can see, is a special case, and it’s the special case that we all learn about in high school when we draw lines on paper.
So, the point is that there is a lot more I want to learn before I graduate next year. Seeing the amount of experience everyone else here has in mathematics, I can’t help but feel like something went wrong in my education. I love the fact that I am a part of a liberal arts program, and that I have taken a wide variety of enlightening and educational classes, but in terms of math and physics, something went terribly wrong. I look at Jesse, who I see to be a version of myself who had the right opportunities, and I become determined to change the way things work. I think that people who study physics need to first understand vector spaces, basic group theory, and complex analysis. This is something that can begin at the freshman level in college. I am grateful for all the physical concepts that I learned, but it was always frustrating that I never had the proper tools to understand complicated physics problems. I did extremely well in my classes, but it didn’t seem like I left with a greater insight into why equations are the way they are and how that related to the physical world. I might be being a little extreme, since that is not entirely true, but I definitely feel a lot more basic mathematical construction and logic could be imparted to students much earlier.
This being said, I will probably need to discuss my budding interests with Professor Hermiller, and perhaps break to her the bad news that I’m not very intrigued with discrete-y type things (which is what my project with her is about).