I'm taking the GREs tomorrow (today), so instead of sleeping I'm avoiding looking up the rules and instructions. To do well on tests, it's best to go in knowing the structure of both the individual questions and the test as a whole. I don't yet, because I've been procrastinating with such useful time-sinks as listening to all of these pieces (link from the most excellent TWF234, about math and music). Oh, and actually getting things done --- I've written thank-you cards, answered e-mails --- there's no better way to be truly productive than to avoid something you really, really have to do.
One of my major accomplishments was writing back to a mathematical physicists from whom I had asked advice about grad schools. My e-mail ended up doing a decent job of outlining both some of my angst and some of my intellectual excitement; I thought you might enjoy it, and I'd gladly here your advice as well:
I'm at what I imagine to be the hardest part of grad school applications (and academic life? I'm sure there are harder things that my fantasy of the "easy life after grad school" leaves out) before actually writing them: figuring out what I want to do. This seems to come in two parts: 1. what am I interested in (and how to formulate it, and how much to formulate it or leave interests undecided as yet)? 2. and what people and departments are right for me given my interests?
I know I want to go into a career in mathematics; I want to teach calculus, and, though I enjoy the amorphous lands between math and physics, I've been ultimately happiest in math departments. At the same time, I know that the mathematics I want to study should have obvious connections with the physics: I'm happiest when I can use language and ways of thinking from physical theories, and when it's clear how the mathematical objects I'm playing with are connected to various attempts at fundamental theories. I want an anthropologist to conclude that my epistemology involves a real world that I'm studying (as opposed to those mathematicians who study platonic, nonexistent ideals). I assume that such is "mathematical physics" --- I definitely enjoy the material in John Baez's This Week's Finds. I would not be interested in studying interesting applied math such as fluid mechanics (or cryptography).
More precisely? I've been devouring This Week's Finds recently, so have been enjoying Baez's fascination with n-categories. I could happily study those for a while. Mostly as a way for me to record and inspire my own thoughts, I've been working on defining linear algebra entirely in terms of Penrose's tangle notation for tensors. I generally feel like algebraic notations, in which ideas are strung in lines, is restrictive and doesn't take advantage of the page.
My favorite toy is the hyperreal numbers, invented more or less by Abraham Robinson. These beasties have the power to do all of calculus, and provide actual interpretations for divergent sums, concepts like "much smaller than", and other important tools that are generally treated with intuition rather than rigor in most of math and physics. I would love to work on various projects to interpret and rigorize the mathematical footing of modern physics with this type of under-used tool. I hold that Robinson's calculus is more powerful than Cauchy's --- it's a conservative extension, so it can't prove anything that Cauchy can't, but it provides much more elementary meanings to a lot of the intuition. Vector fields really are infinitesimal, etc. The problem is that Robinson didn't get very far in constructing a user interface for his operating system. He can do Leibniz calculus, but no better than Cauchy can, and he didn't go farther. Cauchy is Windows to Robinson's Unix; I want to write Macintosh, incorporating QFT and the like.
What interests me most, beyond the actual mathematics, is the methods and institutions of mathematics and physics, and bettering those. I'm fascinated by the ways people think about math and physics, and the language they use, in a normative way: I want to find ways of understanding objects that get at their meanings, and specifically by combining math and physics intuition. It seems that the physicists are much more willing to take a cavalier attitude towards rigor, instead inventing formalisms that _might_ work in order to answer hard, hands-on questions. Whereas mathematicians may be better able to think extremely abstractly and provide the rigor, thereby arriving at a deeper meaning for the physicists' doodles. I want to help the mathematicians think in terms of particles, local processes, and effective theories (not to mention in terms of two-dimensional diagrams rather than "linear" equations). So what I would actually like to do is act as a translator.
So I've gotten some of the way towards an answer to my first question. Of course my interests will change as I continue to learn more math and physics. But my second question? I need all the advice I can get.
I think that I'm a strong applicant. I've taken the undergrad Intro to String Theory, and I'll be taking QFT this year. In math, I've taken a fair amount of algebra and analysis; most of my math knowledge comes from reading (including lots of math and physics blogs) and attending (now as a counselor) Canada/USA Mathcamp, which tries to expose its students to a wide variety of graduate-level math. So I'm looking at the top: strong departments, with people doing what I'm interest in.
But which are those? And who are the people?
Are there mathematical physics journals I should be reading or glancing at, because they're interesting or because they will suggest people and places I should pursue?
If you have any other advise for an aspiring (and presumably as-yet naive) mathematician with physics envy, please do share. Thank you so much.