Today's News

Today's headlines in the New York Times:

• Lockheed Martin got another government contract.
  
• Bush said something he said last week too.
  
• Folks post stuff online.
  
• Someone in Chicago wants to be mayor.

No news is good news?

Young boys and a man

While looking out the window at a rainy Newark Airport and waiting for a very delayed flight, I found myself standing next to a young boy — perhaps five or six — eating a large roll of bread. I struck up a conversation, and we were soon joined by his older brother — six or seven. I let the conversation go wherever it wandered, and learned quite a lot: that their father is a pilot; that the Yankees are the best baseball team, pitching is the best position, and next year they won't use the tee until you get six strikes; that the bushes below the hotel in Hawaii with the big rooms (three balconies in the suite!) now house a favorite action figure; that the police climbing the stairs into the jet-way were probably entering the airplane, because if there were a bad guy in the terminal, the security would have caught him in the initial screening (in fact, they were there to escort a very drunk passenger, who had repeatedly opened an alarmed door, from the terminal to the hospital).

After a while, their farther joined us at the window. "Tell the man next to you" — me — "what the kind of plane with the bump on top is," he asked his younger son. "I'll give you a hint: it starts Seven...."
"Um, Seven Seven?"
"No, Seven Forty-Seven."
"Seven Forty-Seven."
"And if there are [a particular kind of wing flaps]" — here my memory of the technical terms, which he used, has gone — "then it's a 747-400."


What I found most memorable about this discussion was not the ease with which we changed topics — an ease I normally associate with the uniformly brilliant kids at Mathcamp; an ease often pathologized as ADHD and ruined with drugs such as speed ritalin — nor the freedom with which these kids would talk to a complete stranger. What stuck with me was one particular piece of language: "Tell the man next to you..."

Those who've known me for a while may remember previous discussions I've had (though I think not here) about the different words "boy", "man", "kid", etc., which I find fascinating. I've intentionally used some throughout this entry: Mathcamp students and five-year-olds I've both described as "kids," for instance, whereas my first companion was a "young boy." I generally insist that periodicals refer to high school, and certainly college, students as "men" and "women": my freshman roommate was on the men's swim team, and in my brother's CS class there are only six women, as opposed to "boys'" and "girls." Mathcampers, on the other hand, and even my housemates, I often think of as "boys and girls". Not "children," perhaps, but "kids."

What's hardest, though, is self-identity — I'm good at holding multiple contradictory beliefs about the external realty — I had never before defined myself as someone who could be a "man [standing] next to you." Perhaps, when discussing sexual and gender politics, I've identified myself as a "(suitably adjectived) man," but more often as a "male." Categories like "men who have sex with men" are so entirely foreign and don't seem to apply to me or any of my peers. People in my socioeconomic class don't become "adults" until closer to 26, but I'm definitely no longer a "young adult." I'm a "student" or a "guy," not a "man."

One reason for my sojourn to New York was to attend a ninetieth birthday party and family reunion, where I spent some time chatting with various second cousins whom I haven't seen in ten years. My father, an older brother, is younger than his cousins, so while I played cards and board games with my fourteen-year-old cousin, the majority of "my generation" were three to ten years older than me. One announced the wonderful news of her pregnancy, making the matriarch whose birthday we were celebrating extremely happy. I'm used to my peers consisting of younger siblings and students exactly my age; I'm used to understanding those classmates only a few years older than me as significantly closer to adult, since they tend to be grad students when I'm an undergrad, or undergrads when I'm in high school.

But I'll be graduating in four months, and dreaming of my own apartment, and, eventually, house and family. I watch my fresh-out-of-college friends with their jobs in Silicon Valley, and can't help but think how similar that life is to college — they have roommates, come to campus, go on dates. They're no more "adults" than I am.

I have no trouble being "mature", or "old", or even relatively "grown up". But I'm twenty-one years old, and have a hard time thinking of myself as an "adult". Identifying as a "man" is impossible, and it is my current self-descriptor.

Angst and graduate school

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.

A conventional question

I'm in the progress of writing up what I understand about tensors, defining them from scratch, using only intuition and Penrose's graphical notation. Eventually, perhaps I will write a version of my notes for Wikipedia, since their current article on the subject is laughably bad. I first read about them in this post by jao at physics musings; I had started reading Penrose's most recent book, The Road to Reality: A Complete Guide to the Laws of the Universe, which explores them in some depth. I am going to shamelessly reproduce jao's picture of such diagrams, so that you have some idea what I'm talking about:



Incidentally, I wonder what the history of such doodles really is. I hear talk of "einbeins", "zweibeins" and "dreibeins", lit. one-leg, two-leg, and three-leg, and if I knew more German, multi-legs ("mehrbeins"?), which sound like these tensorial pictures. Based on skimming the discussion here, it looks like einbeins are related, but not fully formulated. I wonder why someone would refer to an objects legs, though, unless it had legs.

Anyway, the notational question I wanted to ask was this:

We generally write "vectors" (as opposed to covectors) with raised indices, and covectors with lowered indices. This has physical significance: the Poincare group acts differently depending on whether the index is raised or lowered: on lowered indices, symmetries act by the adjoint, and so it's really a "dual" action, in the sense that it happens in the backwards order. So, although in some sense vectors and covectors are interchangeable, interpretations of diagrams are not.

Since vectors' indices are raised, Penrose proposes that a vector ought to have one "arm" (an edge coming out of the top), whereas a covector ought to have one leg. This makes sense, and closely matches how he thinks of contractions: contracting indices corresponds to drawing curves from the tops of the vectors to the bottoms of the covectors.

On the other hand, as soon as you start playing around with Penrose's diagrams — well, as soon as Josh H. started playing with them, when I introduced them to him over IM — you notice the connection between these diagrams and various ideas from quantum topology. In particular, diagrams like this look an awful lot like tangles.

This is actually no surprise. A n,m-tensor (one with n arms and m legs, so e.g. a vector is a 1,0-tensor), by definition, is a map from V tensored with itself m times to V tensored with itself n times. (By convention, V tensored with itself 0 times is the ground field — no, not a generic one-dimensional vector space, because I do in fact need the special number "1". This is so that there is a natural isomorphism between "V^0 tensor W" and W.)

But this, then, is a problem, because this commits me to reading my morphisms as going up. But my friends who study TQFTs think of their cobordisms as going down (see, for example, the many This Week's Finds starting at Week 73, in which Baez gives a mini course on n-categories).

So clearly one of us is right, and the other is wrong. Either we should think of vectors as having a head and a leg (more than half the time I catch myself drawing them this way anyway), or we should think of cobordisms, tangles, and their cousins as transforming the bottom of the page into the top of the page.

I'm leaning towards the latter, but only because there's one more, very established, case in which this matters. Diagrams of Minkowski space, and more generally of, for example, light cones in curved space, the positive time dimension is always drawn going up the page. And if our conventions are to have any sensible physical meaning, morphisms must correspond to forward time evolution.

So only typesetters and screen-renderers (and English language readers), who insist and putting (0,0) in the upper left corner of the page, have it backwards. Then again, mathematicians have known that for years.

But, of course, we do live in a democracy. If you tell me that infinitely more people and publications think time flows down, then I'll happily switch my conventions.