Category Archives: Mathematics

A Joke.

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Disclaimer: Okay, I admit it: despite the title of this post, no one, not even me, will think that this story is funny. It made me smirk at the time, but I sneak into this class. I happen to know that none of you do. I’d see you. There are only eleven of us, and it’s a small room. But if you feel brave, or are work wasting time, or are my mom (Hi, mom!), I invite you to read on. But know that what follows is in no way funny, so don’t take it out on me when you don’t laugh.

In mathematics, as in physics, it is sometimes useful to treat the electric and magenetic fields separately. Historically, the electric field gets the symbolic designation E, while the magnetic field gets B. [I’m not especially sure why. Magnet in German is der Magnet. And Maxwell, the guy who settled the theory way back when, was British. I’m not sure who fixed the notation, but it’s screwy. And to make matters worse, sometimes physicists use an alternate quantity, the magnetic strength, H, instead. But now I’m starting to confuse myself. I hated electrodynamics. The important player in this story is E, anyway.]

I know, I know. Why would ever want to split up the E and B fields when we could combine them into a single, more manageable tensor? Well, it turns out to be useful when constructing the Ernst potential when studying the geometry of spinning black holes, and that’s exactly what we were doing in class on Wednesday.

Yau had, as he always does, scrawled several chalkboards worth of equations for our benefit and understanding of a rather subtle and technically difficult proof of the uniqueness of a charged, stationary, axially symmetric black hole — the so-called Reissiner-Nordstrom-Kerr black hole. These equations made use of the aforementioned fields E and B. By this time he had introduced another important player, the guage potential — a sort of secret symmetry [It ammounts to the relabelling of space. Even old New York was once New Amsterdam, when they changed the name, however, the geography was pretty uneffected. Nature doesn’t care what you call it. That’s the whole point of guage symmetry.] — which he denoted by capital lambda. While this notation is conventional, it’s not universally accepted. Some people also use A instead. Yau does. But the text he was lecturing from does not. As a result, he mixed As and uppercase lambdas freely. Eventually this bugged someone enough to ask about it.

But Yau had already overloaded E, too, using it simulateously for the electric field and the Ernst potential, which [indirectly] depends on the the electric field! [I told you this wasn’t going to be funny. Stop rolling your eyes.] A little braver now that someone else had expressed his confusion, another man spoke up.

“This E over there and this one over here aren’t the same E, are they?” he asked, pointing accordingly.

Yau realized his abuse of notation and set to redress the error of his ways.

“Oh. I’ll just erase this one, then,” he said. And then he did. The whole affair had a touch 1984 to it. Problem solved. Now there was only one, unambiguous E. Inconvenient history never happened. Not if it’s not recorded. The Founding Fathers knew this well. Orwell knew it. And apparently, Yau knows it, too.

Even if it’s not funny, you can understand, perhaps, why I smiled, though.

Odd Man Out.

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Tonight I spent some time with my neighbours who happen to be friends from high school, though they are a bit older than me. They are married and have two kids, both boys. The older one, Kyle, is six, which puts him at just the right age to start kindergarten, something he did, in fact start, last September.

I love asking kids questions. Hell, I like asking anyone questions. But kids are usually the best because things which we, the grown-ups, would consider old hat are, to them, brand new. After slowly walking towards Kyle in what he called “a chase” around the back yard — I like to walk slowly rather than run. Because it’s unexpected, it tends to freak them out a little more — we sat down inside at the kitchen table to prepare our hamburgers with Caesar dressing, grilled red bell peppers, and all the other fixings. I took this time to ask Kyle what he was doing at school. “Oh, just some math. But I’ve already seen math, so it’s not hard,” he replied as a matter of fact. I smiled. Everyone in the room smiled, but no one gave it away. They didn’t know what, but they knew it was coming.

“Yeah? Sounds like you’re ahead of the game, then,” I answered. It wasn’t time yet.

“Yeah, but not always. We learned about odd numbers and I didn’t know about them before,” Kyle offered. He’s a good, helpful kid. He’s constantly trying to help his younger brother, Luke, who’s just about to turn 14 months in a few weeks, do whatever kids that age do: throw the phone on the floor from on top of chairs and rip CDs out of their cases, I suppose.

Now it was time. Kyle had given me something to play with. I couldn’t resist, so I started out, “So what is an odd number?”

He thought about it and after a moment he responded, “It’s a number that doesn’t have a pair. If it’s an even number then there is always a partner, but in an odd there is one all alone.” Hey, it even made sense, at least to me. To see what sense he had made of it, I asked him for examples of odd numbers. He gave me one and three. And the next? Five.

“Okay, what is the biggest odd number you can think of?” I thought I had won, but you can’t ever underestimate little kids. If you do, they’ll prove you wrong. Kyle pondered my question.

At last he spoke, “There isn’t one.” Foiled, I smiled and regrouped.

“You’re right, but can you tell me which is the largest one you can name?” I’m sure that we can glean some fact about cognitive development or learning theory or maybe just that people can be tricked even if they themselves have supplied enough information not to be — and in my experience most people, not just children can be fooled even if you tell them “This is a lie:” — Kyle answered anyway.

He told me that “one-oh-one” was the biggest odd number he could name. It’s certainly odd, I agreed, but which odd number came next? He quickly gave one hundred three. Kyle would be the last one to finish his cheeseburger. His was cold before I downed two. By now I was working on a Sam Adams Boston lager, which I had saved for last.

We continued in this way until we made it up to nine thousand eleven. [I stopped after only one beer, though. Kyle told me that he is allowed to drink root beer, which is like beer except that it’d didn’t have alcohol.] Not satisfied with our latest contender, nine thousand thirteen spoiled its chances, we gave up.

Math and Sex

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Last night, after church choir, I headed into town to visit with Michelle on her last day of vacation this week. I planned for a crazy night and therefore brought the aforementioned ACEE evalutation results with me to read. But there, in the kitchen, Matie [MAE-TEE] — that is her legal, given name — and her friend Molly had already claimed the table for their lending letters. You see, they’re canvassing financial support for the sex shop they hope to open in Albuquerque. Now, before you guys pass judgement — and I know you already have — Matie holds an advanced degree in social development and non-profits or something germane. And she happens to be pretty on top of her stuff. In our now more frequent exchanges, we discovered something bizarre. The general public treat us and our fields in identical ways.

I explained that the world is mathphobic. If I were to go to a bar, say, and someone asked me, “Oh, which school did you go to?” Harvard gets you the first strike. You can approach the question “Yeah, really? [I had an aunt who went there in the 60s, she…] What did you study there?” in a couple different ways. If I don’t want them to talk to me I can say math, but if I feel a little more sociable, I can answer science. Science is vague enough that it might mean biology and therefore be less threatening. Everyone has a biology, few people carry around their math. But you can only dodge the question for so long. No matter when you pitch it, it’s always strike three: math.

Math makes people uncomfortable. And most people have no idea what math is and an even worse conception about what mathematicians actually do. It isn’t easy [for some reason] for people to hear that math is really just like anything else, that anyone can do it, and that they’ve probably never done it themselves. Adding and substracting isn’t math. [It’s almost computer science.] People forget that the content is secondary. It’s the relations that exist within the content that’s important. That’s why when I draft my socially responsible, angry letters a classicist and a sociologist can read them and there’s a pretty good chance that they would’ve written something similiar. [Though I’m sure we’d argue the grammar until the cows come home.] The same style of argumentation you use to write a paper on phonology is the same you use when discussing Engels dialectic approach is the same you use to investigate spin cobordism. The words look different. The language looks different. But the processes that govern them all, they’re the same. [Probably not, but close enough.]

But to overcome the discomfort I and my math present, people always share with me these impromptu anecdotes to justify or demonstrate or something, I’m not especially sure, maybe just to connect however awkward a connection it may be with math and therefore with me. “I was really good at math in high school, until calculus;” or “it’s not my thing. I can’t even add at the grocery store;” or, “I had this teacher and he was really good at explaining math. I really wish I stuck with it.”

It turns out that Matie gets the same response:

  • Sex [Math] makes people uncomfortable.
  • Most people are uneducated about sex [math] and get the issue confused, perhaps, to the detriment of themselves and those who do sex [math].
  • People offer unsolicitated, personal disclosures about sex [math] to those who profess to know anything about the subject.

We need to do something about the current state of affairs, even if it does make for some hilarious chit-chat.

“No, sir, I don’t want to know about what you do with your wife or your ‘really great’ high school math teacher.”

Two Real-life Jokes.

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Real-life Joke #1: Last night I went to a presentation given by the new (and I mean that this position is new) so-called Alcohol Czar of Harvard at Leverett. It’s not exactly clear what his job description is, but it must include talks on responsible drinking. So we invited him to speak in our Pizza, Pop, and Port series, which seemed apt enough. During the Q&A portion of his talk, one student questioned the validity of the “hair of the dog” hangover recovery strategy. Rather than answer straight away, the speaker asked, in return, “Who invented those drinks [mamosas and bloody Marys]?”

Without giving it a thought, Jenn raised her hand just above her head and slightly forward, and with one earnest swoosh yelled, “The British!” timing her comment with an abrupt stop which made for quite the dramatic response. Ryan, the speaker, was surprised but not undone. He tactfully posed a follow-up.

“Yes, sure. But more specifically?” I started to think. There’s got to be a trick to it. Who drinks these things? I do, when I can, and when it’s funny. Ian does, too. It was a Sunday afternoon. People go to church on Sundays. Ah, ha! I had the answer.

So it was my turn to scream a stupid response, this time after thinking it over. With just a tiny bit less histrionic gesticulation, I pointed my right index in front of me and proclaimed proudly, “Old women.”

Ryan was looking for us to say, “Alcoholics.” No cause is a lost cause like ours.

Real-life Joke #2: In Math 235: Minimal Surfaces, Professor Yau has been using the Kerr metric — a stationary, rotating black hole — to introduce various topics in general relativity. Today he wanted to discuss gravitational radiation and Bondi mass, even though the Kerr metric doesn’t radiate on account of its being stationary. [No stationary black holes radiate; that’s the point of them.] But he proceeded somehow even still. One of the magical things about the Kerr metric is that in the right coordinate system, its wave operator actually admits a solution by separation of variables. This is a suprising and blessed [though still tedious] fact. In some high schools, AP calculus students learn this method. To dream that it could work in the case of Kerr is unbelievable. To remind us how to perform the trick, Professor Yau wrote the following mnemonic on the board:

C0u(r)an(t).

He then laughed for about forty-five seconds. This is a long time for Yau during lecture, and an even longer time for anyone who saw the joke. [Richard Courant was the famous mathematician after whom the Courant Institutue of Mathematical Sciences at NYU is named. So, you see, it’s funny. Courant worked on partial differential equations and functional analysis and the calculus of variations, so this joke is not only funny, it’s appropriate — even more than it is funny.]

Out of a Job Even Before I Get Out of School

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In the fifth or sixth grade, I had to do a report. And much like my fifth grade science fair project, which I do remember: I passed in a paper on resistors with a small experiment that my dad all but typed up for me, I had no idea what I was writing. The topic was Einstein’s special relativity. The whole thing was lifted from the appropriate volume of Encyclopedia Britannica kept in my dad’s home office. But what was so special about special relativity? The encyclopedia article explained that the theory was the result of a simplifying assumption or something in Einstein’s general theory, and that this whole thing was really about gravity. I have to admit, I still find the entire enterprise of relativity and gravity mystifying. One of the more outrageous predictions of classical GR are those objects popularized by scary movies like Event Horizon and other popular science fiction called black holes.

Black holes are tricky to define mathematically. Physically, they’re a place where mass becomes “infinitely” dense. [The quotes are there because infinite anything is a physical no-no. If you were to squeeze the mass of the earth into a ball a few millimeters across, then the force of gravity would take over and compress it even further. The math predicts a formation of singularity — the thing at the center of a black hole.] The space around such points acts funny. Because of the strong gravity associated with these objects, if something, a rocket, a lampshade, or a photon of light, for example, gets too close, then it gets drawn in ever closer until it meets collides with the singularity. Then all bets are off, and nobody can say with any amount of certainty what happens. The boundary in space beyond which nothing can return is called the event horizon of the black hole. Because not even light can escape, the structure will look, well, black; hence the name.

Black holes have always made people feel a little uneasy. First off, they’re scary. When I was small, I hated the drain in the bath tub. It was only a matter of time, I thought, before it took me down with the bath water. Black holes evocate the same sort of fear. And according to the big bang, there are tons of tiny, primeval black holes floating around the universe. The thought of it petrifies me. Secondly, black holes cause a few problems. Most notable is the information paradox, something that Stephen Hawking both proposed and recently resolved. The old saying goes that black holes have no hair. To avert the paradox, it turns out that black holes must be fuzzy, that things can escape. The problem stems from a butting of general relativity against quantum mechanics. They both work in their regimes, so what gives?

George Chapline has an answer: there is no such thing as a black hole. Instead, he proposes something whose geometry looks outwardly very similar to a black hole. He calls this something a dark energy star. I met him last spring when he came to give one of the Friday colloquium talks. He motivated his quantum critical points — a concept which neither I nor the New Scientist article I link explains — with the following scenario. [Okay, I will a little: usually we think of temperature as the master of phase transitions. Cool down a gas, like water vapor, and you get a liquid, like water. Cool down further, freeze it, even, and you get a solid, say, ice. Now keep going, cool it down all the way to just above absolute zero. When things that cold, quantum mechanical effects are the dominating factor in phase transitions, not temperature. In this condition weird things can happen, like superconductivity.]

Consider a long cylinder filled with a superfluid. The pressure gradient will be small nearer to the top, at the bottom, it will be large. At the top of the tube attach a speaker which sends out a sound wave. As the wave travels through the liquid it will slow down as the gradient increases. At some height, the wave should stop. What happens, he asks, as the wave meets this surface? If you’re a classical general relativist, you might look at the math and think, “Ah, ha! That’s just like the event horizon of a black hole. So, nothing, the wave will just pass through.” Classical GR lets anything just fall into a black hole. Once you’re inside you can’t send emails or make outgoing phone calls, but outside of that, nothing happens. You wouldn’t feel a thing. The earth could’ve just passed through the event horizon of a super, ultra massive black hole right now and you wouldn’t even know it. But Chapline does some quantum mechanics and says that’s not what happens. Instead, we might expect the magical height at which the wave stops really to represent a quantum critical surface. And the phase transition effects are wild.

In his talk he explained that a sort of Georgi-Glashow process could occur, causing quarks to split into an electron and a positron. This could account for all the anti-matter we see at the centers of galaxies. Using the liquid superconductor analogy, he conjectured that the vortices like those that form when liquid helium might also explain relativistic jets we observe spitting out anti-matter, too. The exciting [or threatening] implication of Chapline’s idea is that there is no singularity, no black hole, just lots of dead stars.

On the other hand, Penrose and Hawking have their names attached to the famed Singularity Theorems. They say that given certain assumptions on the causal structure of a universe, assumptions that we think our universe satisfy, then there needs be a singularity some where in that space-time. What of that? I’m not sure, and I’m not sure I care. Luckily, mathematical general relativity is replete with really interesting questions that are completely divorced of whatever’s going on in this universe. As I like to say, “Physics is the study of this universe, mathematics is the study of all possible universes.”

The Plan

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Last night on the way to Cambridge, I had a small panic attack whereby I form a plan of attack on the rest of my immediate life lest it overtake me and I black-out in public. [It has never come to that, but I’m sure that’s what would happen.] In case you are wondering — if you’re not, you should stop reading now — I have organized my life into three hour days. Each day I must do the following:

Calculus of Variations. (1 hour) First I’ll work through Fomin’s translation of Gelfands lectures. They work out variational problems in flat space with a few examples from field theory, classical mechanics, and geometry. Mostly it’s a journey into analysis. Once I’m done with that, I’ll move on to Jost’s book and learn about minimal surfaces properly. If at some point I make it through Jost, I’ll hit up Morrey, but he’s a long ways off.

General Relativity. (1 hour) For this I’ve chosen an obscure but very good book by Barret O’Neill called Riemannian Geometry with Applications to General Relativity. The whole point of the book is to give a proof of the Penrose-Hawking singularity theorems: given some very weak causality assumptions, there needs be a black hole or a big bang. Along the way he fleshes out symmetric spaces, Lie algebras, and other good and fundemental stuff. He also takes on variational problems, mostly that of geodesics. Hawking and Penrose all but force him to. It’ll be good to visit the calculus of variations from two very different points of view.

Cognitive Theory. (1 hour) Now this is less straight-forward. It’s very difficult to make a good curriculum if you don’t know the material already. Even still, my [general] plan is this: Piaget’s Psychology of Intelligence to learn about assimilation and adaptation. Then to learn why he’s wrong, I’ll read the boldly titled Getting it wrong from the beginning: Our progressivist inheritance from Herbert Spencer, John Dewey, and Jean Piaget by Egan. It was written in 2003, so it must be righter than Piaget who wrote the Psychology back in 1950. Afterwards, it’s time for Vygotsky and situated learning in the Mind in Society and some other stuff about activity theory. There are a number of journal articles I’ll read, but I’m not sure which ones just yet. And there’s this book called Cognition in the Wild and I should reread stuff by Papert. Why can’t all disciplines write textbooks like math textbooks?

And math education! Gelfand wrote a number of books on algebra and trigonometry and arithmetic designed specifically elementary school students. I should check them out. I know next to nothing about elementary school math education. And then Paul Sally wrote those nice geometry books for little kid teachers, too. And if I’m not mistaken, Schmidt is big into math education, too. I wonder if he’d talk to me. And Judah Schwartz! and Andrea diSessa. There are a lot of people I should read.

Also, each day I must either do two hundred push-ups or go swimming. Last night I swam with Laura Chapman at Blodgett. It was the first time I’d been in the water since I had to cut off my jammers with a Swiss Army knife to avoid an extended, wet, and naked wriggle in the locker room. It was also the first time since I had thrown six-year old Robert on his head, landing myself on my knees. It took several days for them to bruise. That’s how you can tell it was really bad. Despite these physical and emotional trauma, I was able to pull along nicely. I cut my work-out in half, swimming only sets of 250 yards rather than the full 500. Laura tried to teach me the breast stroke, something I find rather unnatural. I kept defaulting to the dolphin kick, so eventually I just switched to butterfly. [Not for long, of course. I went 25 yards; she went 25 yards. Not to be outdone, I went another 25 yards. By this time we were sufficiently tired and left.]

Putting the Simple in Supergravity.

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Despite the title of this entry, I am not going to talk about supergravity outside of this: Cabot recently purchased a book I may’ve used for my thesis. However they processed my order over a week late. I’ve got it with me now, just for kicks — personal edification and so that I can impress you, the reader, and the people who see me with it at lunch, like Luke and Lixin. On my way back from the Science Center, I flipped through the table of contents, I came across a chapter called “Geometrical Gravitational Theories,” which is why I asked the library buy the book in the first place. This book,Geometry, Spinors, and Applications, makes heavy use of — wait for it — spinors. There are lots of books on geometry; tons on applications; a sizable number on both geometry and applications. But there are surprisingly few on geometry, application, and special mention of spinors. And under that chapter on gravity, there’s a subheading: Simple supergravity. It reminded me of a conversation I had with Eda a few weeks ago. She finds that mathematicians are superficially humble, but in an oblivious and therefore endearing way. The idea of supergravity ever being simple is sort like a slap in my face, but in an endearing kind of way.

I quit you now to take up a programming assignment Paul has given me. He has resurrected that automated inspection-announcement-general purpose-web-based-email-program-thing project for me. He doesn’t like me to admit to him that I can’t do things, like, we’ll say for example, program. I’ve got a copy of PHP3: Programming Browser-Based Applications to my right. Somehow I think the supergravity would be simpler.

Squashed!

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Kirkland wrecked me, and the Leverett team at large, tonight at IMs. David showed me how to alter my swing to get more power without using more. This came only minutes before my match, allowing little time to perfect my new technique. Even still, I’m not sure it would’ve helped much. My opponent had excellent court position and placement. I am encouraged, therefore, to take private lessons from the varsity coach once I graduate, find a job, and determine the feasibility based on time and money.

In the meantime, I took this quiz on the NOVA website about that silly E=mc^2 equation that everyone talks about. I was tricked by one of the questions.

Einstein appears to be the theme today. I met with Professor Strain to talk about possible final paper topics. After rejecting his initial offer (a proof of the previously thought impossible global solution to the vaccuum field equations by wave coordinates; they were thought — reasonably, it seems —, to diverge at inifinity.), we mulled over a casuality theorem due to Jacobi fields, or applications of PDE to mean curvature, finally arriving at the Hamiltonian formulation of GR. This is something that I really ought to understand. ADM mass, a quantity I now know and love, was originally proposed as a good definition of total mass because of its appearance by variational techniques. When explaining it, I like to use the nonlinearity of the field equations to eek out the self-interaction of gravitation. (This amounts to linearizing the theory and noting that these equations obey a linearized Bianchi identity, suggesting a conserved quantity.) I don’t know the calculus of variations, and, as a geometer, I should. And since I want to go into GR, this couldn’t make more sense.

Also, it seems, as my grandmother mentioned to me today, that Newton was an alchemist! I can’t but believe she’ll soon be concocting her own Philosopher’s Stone before long.

And mom, dad, and Janice, you can, if you want, purchase the complete DVD set of Sportsnight for me for Christmas.

Kaitlyn’s Twenty-third Birthday.

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I just came back from Tequila Rain on Landsdowne Street. There was a five dollar cover, but no matter where I go, I always end up spending precisely thirty dollars. Tonight was no exception. So things were okay. Whenever there are only bad beers on tap, I opt for a Maker’s Mark and giner ale. The Ryan’s were there to celebrate. And Myers got me a Bud Light. I bought Nick one, so things evened out. Ryan owed Nick. I payed Nick. I owed Ryan. All is right in the world.

Lindsey and Rita and Jacqui — I no longer call her Qui — where there. Qui and Nick recently took the LSATs together. While I enjoy knowing that Joe is sitting next to the man in the fedora and is not the hit man, I still can’t imagine going to law school. Nick couldn’t give me a good reason for applying, but he works at a law firm. Maybe he has a reasonable idea. It’s hard to say.

I’ve signed up for another Persian birthday party. This time it is Anahita’s birthday. Last time we celebrated Adelle’s [please excuse the spelling]. Before her birthday dinner and cake, Verena and I are slated to play squash. Halvar and David are invited once again. I can only hope they join us, and I can only hope I wake up in time.

Verena explained how Cambridge is a catch-all, at least for Harvard folk. At MIT we ran into Tse-tse, who is not a fly but a recent graduate from Eliot House now in an MArch of PhD program in course four, architecture for you lay-men. No one seems to go too far. At least most people don’t. Boston and New York are only fifteen dollars away, and email is free no matter where you are. Growing up here makes the world all that much smaller. And Nick and Kershner promised tonight to travel as far as Durham to see me. England is about as far. If only I could make some headway on my thesis. Complete Banach spaces with respect to horrible hypersurface Dirac operator-like norms, here I come.

By the way, I’ve included a picture of that horrible, famous man, Mandelbrot. He mentioned that some have called the Mandelbrot set the most complicated structure in mathematics. I wrote an elementary paper for my complex analysis class showing that it is, in some sense, no more complicated than a circle. [For those in the know, it is a routine fact that the Mandelbrot set is simply connnected; it is, as McMullen showed, also universal. It’s quadratic, after all. So are you really surprised?]

Benoit Mandelbrot.

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The Limit Set of a Schottky Group

Last night I dragged Venera and her, and now our, friend Anahita to the Museum of Science to the first in their Lectureology series. To kick things off, they brought in mathematician Benoit Mandelbrot to talk about roughness, cauliflower, and fractals. I’m not sure he did any of these things. He did manage to tell us just how great he knows himself to be more than a few times, however.

I was fairly shocked just how quickly he was to dismiss, mostly be flagrant omission, the other “fathers of fractals.” True, Mandelbrot did coin the phrase. He told us so three times in less than one hour. He was not, as he claimed last night to be, the first one to consider roughness. Julia, famed for his aptly named Julia sets, was Mandelbrot’s own teacher. In fact, the Mandelbrot set is a catalogue of Julia sets. Of course we could throw in Douaday, Hubbard, and Fatou while we’re at it. Then, there was another branch of complex dynamics going on which resulted in Bers slices, the Maskit slice, Fucshian groups and all that. I happen to think that the other school made cooler pictures. (Compare the limit set of a Schottky group [top] with the Julia set [bottom].) The point is Mandelbrot didn’t go it entirely alone.

Perhaps the only thing I took away from my tenth grade English teacher Mr. Tony Baxter is this: when writing a story, you can’t just tell the reader what’s going on. It is incumbent upon the author to show the reader what’s going on. Mandelbrot should’ve taken Tony’s class. He’s still a teller, not a shower. And then, not even a good one. He didn’t once tell us what the Mandelbrot set is, let alone a fractal. I do remember his throwing around the term Hausdorff dimension somewhere in the middle of his talk. But he didn’t spend nearly as long on that as he did some honorary doctorate he received from somewhere in Germany.

Verena and Anahita and I left for the food court at the Galleria in the middle of the question and answer session, just after he was explaining how much metereology he knows and named-dropped infinite dependence.

At least we got into the Museum for free.

An example of a Julia set