ChapterZero

I’m going to be reading a lot this summer. I got hold of a copy of

and realized how sadly underprepared I am. Of the (16!) topics he has listed: Linear Algebra, Real Analysis, Differentiating Vector-valued Functions, Point Set Topology, Classical Stokes’ Theorem, Differential Forms and Stokes’ Theorem, Curvature for Curves and Surfaces, Geometry, Complex Analysis, Countability and the Axiom of Choice, Algebra, Lebesgue Integration, Fourier Analysis, Differential Equations, Combinatorics and Probability Theory, and Algorithms… I am comfortably familiar with at most 5. So I’m going to read this book, which gives a broad strokes intuitive overview of all these topics, and then pick a few books from the bibliography and read a couple of chapters on those subjects that I think are actually useful in graduate school. I would be very surprised to find that Algorithms turned out to be a crucial subject in grad school. And I’m surprised number theory is not on the list.

To that end, I’m going to start an assigned reading list, with daily quotas to meet. I used to think that studying math is something that shouldn’t be rushed, in a sense; and I’ve heard that most professors only read/research for a couple of hours each day. This seemed fine— math is hard, after all, and you can’t work continuously at it, can you— but watching Grey’s Anatomy last night, I realized understanding is hard won in medicine, and even more crucial in a practical sense than in mathematics. So why is it that surgeons and doctors are expected to be at the top of their game, and spend so much time studying in med. school, and so much time working when they’re out of school? If they can do it, I can do it. So no more Mr. lazy pants; I’m going to study. We’ll see how long that lasts

I’m finding, from studying topology, that it’s a lot better to read one or two chapters from one book and then switch to another, and read the chapters on the corresponding material, using one book to coordinate the effort, than trying to read all the way through one book. Weird, but it keeps me from being bored with one style of exposition, and each different book gives me a different view of the subject.

Possibly relevant posts:

• Functional Analysis is What? (10/3/2005)
• Kvetches about Janich’s book (7/2/2007)
• Upcoming analysis test (3/27/2005)

One of my favorite aspects of studying math is picking up new definitions. Especially when they are in areas that I previously knew nothing about, so I have no previous connotations attached to them.

First, there’s always the question of motivation: what in the world made someone take the time to define e.g. a topology, homeomorphism, or quotient map. What possible uses are there for the concept of a fiber, or a saturated set? It seems to me that most definitions are made in hindsight— rarely is one clear-eyed enough to see that a particular concept is meaningful in itself until you’ve been dealing with it for a while, and then you see that giving it a name and specifying particular defining properties might be useful. The greatest example I have of this is compactness. I have the feeling that the rise of the concept of compactness began when someone realized that the finite subcover property is equivalent to the Hein-Borel theorem. Then, others noticed that this could be extended to arbitrary spaces. (As an aside, it seems to me that you really understand a definition– or at least its implications, and are they two different things?– at a gut level when you can present a sensible probable history for it.) Even though most definitions are made in hindsight, and therefore tweaked to streamline the presentation of the material, there are those that have been mercilessly twisted to serve for maximum mileage. These I don’t particularly like, e.g. Rudin’s definition of differential forms in terms of changes of variables; mostly these just beg the question: why is this definition this specific? Ideally, the line drawn between being so specific that the motivating intuition is lost and being just specific enough that the definition leads efficiently to results should not be crossed.

Then there’s the challenge of trying to assimilate this new concept into your picture of the world; developing your mathematical weltanschuang. To a large degree, I think this is what mathematical maturity consists of: being able to take seemingly arbitrary definitions, identify the gist of what they’re saying, and integrate them into the rest of your knowledge. At least for me, a large percentage of the time I spend struggling through math books is trying to assimilate the definitions that are new to me in such a manner that I truly grok them. In my ideal world, theorems are just natural extensions of definitions, statements of the obvious. The world is far from ideal.

I love the Springer GTM series because of the way it presents definitions interleaven with remarks, theorems, and exercises. A typical book is a repitition along the following lines:

• Prelims– what’s the current question, and broad strokes overview of how this definition assists in answering it.
• Remarks– notes subtle points of the definition, reinforces it with a few canonical examples.
• Theorems– show the usefulness of the definition, and reinforce it even more. Add meaningful implications to it.
• Exercises– yet more reinforcement of the definition, and attach more implications to it.

At some point, the definition becomes replaced with implications, as it should be– who remembers the definition of linear independence word for word? Then you can truly work with the underlying concept, instead of struggling with a bunch of words. And you can reconstruct the definition at anytime be considering what you what it to mean; e.g. the definition of linear independence is easy to reconstruct from the idea that you want the set of points to all lie on different lines.

Possibly relevant posts:

• Topological spaces (6/21/2003)
• The Brave New World of Functional Analysis (3/8/2006)
• Ideals in Banach algebras (6/30/2008)

Just finished my real analysis final. As I feared, the hardest problems were on uniform convergence. In fact, I spent a full hour and a half on one such problem; here’s the culprit:

If is a continuous function on and the sequence is recursively defined by , show that uniformly on .

After I got it, as usual, it seemed pretty obvious, but while I was sweating over it, I tried some pretty weird things. The real question is whether it is objectively a hard problem.

Possibly relevant posts:

• very simple modeling of stocks (4/2/2007)
• Two Kernels (3/4/2005)
• Pointwise convergence implies weak convergence (9/26/2008)

One of the frustrating aspects of going to college is choosing which classes to take. It seems like I’m at the end of my college career and have not much flexibility left, and now people are starting to give me useful advice on which classes to take. For instance, in my digital electronics class, the professor mentioned that analog electronics is becoming a bigger field, and the vast majority of graduates have a background in digital electronics. Therefore it would behoove me as an engineer to take the courses at UH concerning analog electronics. The sad thing is, this is information the department should systematically disseminate much earlier— most people in that class have only two semesters left to go, which is not enough time to take three unplanned courses. Of course, this only marginally affects me, because I have no intention of working as an engineer unless I absolutely have to.

Instead, it is the math department that is pissing me off. They give us no advice on how to choose courses, much less belated advice! I took real analysis these past two semesters mostly because it interested me, and I had heard in an REU program that is an important course to have when applying to grad school. For the same reason, I took complex analysis this semester (and didn’t learn all I had been hoping, so will be taking the graduate sequence for the next two semesters). On the UH math club mailing list today, someone posted a list of the undergraduate courses that apparently form the core of the prereqs for graduate courses:

4331;4332: Introduction to Real Analysis
Cr. 3 per semester. (3-0). Prerequisite: MATH 3334 or consent of
instructor. Properties of continuous functions, partial
differentiation, line integrals, improper integrals, infinite series,
and Stieltjes integrals.

4377;4378: Advanced Linear Algebra
Cr. 3 per semester. (3-0). Prerequisites: MATH 2431 and a minimum of
three semester hours of 3000-level mathematics. Matrices,
eigen-values, and canonical forms.

4333: Advanced Abstract Algebra
Cr. 3. (3-0). Prerequisites: MATH 3330 and consent of instructor.
Direct products, Sylow theory, ideals, extensions of rings,
factorization of ring elements, modules, and Galois theory.

4337: Point Set Theory
Cr. 3. (3-0). Prerequisite: MATH 3333 or 3334 or consent of instructor.

I was fortunate enough to have wanted to take advanced abstract algebra; I just finished that this semester— its amusing to note that the only thing listed in the course syllabus that we did learn was factorization of ring elements, and that just barely. But, I did not think topology was a biggie, and had kind of closed my eyes to the question of advanced linear algebra— I’m interested only in the second semester After seeing this posting, I signed up for advanced linear algebra next semester, so I’ll be taking that over the next two semesters. Unfortunately, they aren’t offering topology at a time I can take it next semester. Hopefully they’ll offer it during my graduating semester.

Possibly relevant posts:

• Winter term courses (11/19/2006)
• Just about done with Math (6/13/2005)
• Summer reading list (5/15/2004)

Update: I realized that jsMath’s output doesn’t look too good in IE.

I’m writing a tiny plugin to load jsMath so I can use it in my posts. This should be a nice looking statement of the generalized integration by parts theorem:

vs.

Awesome! It finally works. I had to do a lot of digging in the WP code to implement this. First I found that the ‘wp_head’ hook isn’t sufficient, because jsMath must be loaded in the < body > , and wp_head loads in the < head > . Then, I looked and looked for an action hook that gets called at the top of the body whenever a page is rendered, but I couldn’t find one. Pretty strange that there doesn’t seem to be one. So, I spent a lot more time trying to track down how the plugin system works so I could add a hook myself. This may seem simple after the fact, but since I was looking for a more centralized architecture, it took me forever to figure out— I thought all the calls to do_action were registering hooks— but it turns out that the appropriate function will call do_action with the appropriate tag, to execute all the functions attached under that hook. Finally, I tracked down a sensible location for my additional hook: in the file template-functions-general.php as the last line in the function get_header, which returns the head of the file to be displayed, as well as the beginning of the body element.

The next thing to work on is making LaTeXRender use jsMath optionally— it’s a real pain to have to type in the start and end tags for a span or div everytime I want to use jsMath.

Possibly relevant posts:

• TeX support added to Commenting (1/23/2005)
• Code Highlighting Added (2/2/2005)
• Plugin idea: modifications to LatexRender (7/19/2007)