Chapter 2. Newtonian Growth and DecayThe growth-decay formulæ were developed in the trivial fashion by Isaac Newton's famous brother Phigg. His idea was to provide an equation that would describe a quantity that would dwindle and dwindle, but never quite reach zero. Historically, he was merely trying to work out his mortgage. Another versatile equation emerged, one which would define a function that would continue to grow, but never reach unity. This equation can be applied to charging capacitors, overdamped springs, and the human race in general.
-- fortune (Yes, I've included this quote just as an excuse to use the letter æ.)
Welcome to my home page ;-) It is not at all clear, a priori, why I should do an absurd thing like put up information about myself. Aside from the general information I have stated above, there's nothing particularly interesting to note about me =). Nevertheless I shall make a lame attempt, and I'll try to spice it up with things like pictures in the future. I'm a third-year graduate student of math; I graduated from UCLA in math and computer science. My interests are primarily geometric, analytic, and a little topological. So I'm working in geometric analysis, complex geometry, and stuff related to Ricci Flow.
Spring 2007: Math 10A (Calculus)
Fall 2006: No teaching duties (Yay! [Not to say, of course, I don't enjoy the company of students which I often refer to as my math homies])
Spring 2006: Math 20D, Differential Equations
Winter 2006: Math 20E (again)
Fall 2005: Math 10A (Calculus)
Spring 2005: Math 20E (again)
Winter 2005: Math 20E, Vector Calculus: Vector Fields, their derivatives, and the integral theorems
Fall 2004: Math 20C, General Multivariable Calculus
Spring 2006
Math 220C, Complex Analysis
Math 231C, PDEs
Math 221A, Several Complex Variables.
Winter 2006
Math 220B, Complex Analysis: Fun With Residues, Harmonic Functions & Lots of Conformal Mapping
Math 231B, PDEs: Fun With Sobolev Spaces and Distributions
Math 257B, Topics in Diff. Geom.: General Relativity: Fun With The Universe
Fall 2005
Math 220A, Complex Analysis: Fun With Complex-Analytic Functions
Math 231A, Partial Differential Equations: Fun With Waves, Diffusions, Transports, and Laplace
Math 257A, Topics in Differential Geometry (this quarter's topic: Metric Geometry)
Summer 2005
Graduate Summer School on the Ricci Flow at the MSRI in Berkeley
(I have made some notes for Bennett Chow's lectures, a rough version should be available here soon).
Spring 2005
Math 240C, Real Analysis: Fourier Analysis, Distribution Theory, Applications to PDEs
Math 250C, Differential Geometry: Basics of Lie Groups, Theorems about Curvature
Math 290C, Algebraic Topology: Poincaré Duality, Products
Not much stuff up here yet, hehehe.
Official Publications
C. Ratsch, C. Anderson, R.E. Caflisch, L. Feigenbaum, D. Shaevitz, M. Sheffler, and C. Tiee, "Multiple Domain Dynamics Simulated with Coupled Level Sets," Applied Mathematics Letters, 16, 1165-1170.
Forthcoming will be notes on the Ricci Flow, a series of lecture notes given by Ben Chow at the MSRI, appearing in the proceedings of the Clay Math Institute (CMI) (I may be able to post older versions of these notes online here, but I'll have to check with the CMI).
Perhaps the above publications list is a bit meager because I have too much fun with math... Exploring is all part of the fun. Here's some miscellaneous stuff I've picked up on various mathematical voyages.
Course Notes
A geometric explanation of why vector fields don't have curl, as explained to students of vector calculus.
Some notes on covering spaces, prepared for the Topology Qual review. If you ever thought that covering spaces could be confusing... I might be able to help.
Some notes on tensor analysis. Update: Finally, I pretty thoroughly understand orientation issues (NO NO NO, not that kind of orientation issue), and what pseudotensors are (Never heard of them? Even more the reason to check these out!). Still a work in progress, please send your thoughts, criticisms, etc.
I gave a talk on the Gauß-Bonnet Theorem in the Food For Thought seminar in Spring '06. The notes are here (it's a little rough around the edges, though).
Here are some expanded notes I made from a seminar at MSRI on some simple solutions to Ricci flow.
Inner tubes make another comeback in today's amusement. Visualizing 4D space, or 3D space that curves, can be a little, shall we say, mind-bending. One of the venerable ways to visualize the unit 3-sphere (x2 + y2 + z2 + w2 = 1) is by stereographic projection to plain ol' 3-space, and its decomposition into a bunch of nested tori (in our previous installment involving inner tubes, we further broke these tori down into a bunch of linked circles, but we won't need them this time around). The fancy-schmancy term is to say that the 3-sphere is foliated by Hopf tori. The Hopf tori are cartesian products of circles of radii a and b, such that a2 + b2 = 1, so by definition, any point on such a torus actually lives on the 3-sphere. When stereographically projected to 3-space, the Hopf tori look like inner tubes.
In the movie above, we start with an inner tube representing a Hopf torus both of whose circle-factors have radius 1/sqrt(2). The red and blue circles represent each one of these circle-factors. Now, the amazing thing is that there is a "continuous rotation" (rigid motion) of the 3-sphere (which can always be realized as a rigid motion of the ambient 4-space containing this 3-sphere) which will move the torus within in a way that the beginning and ending configurations look the same, except that the red and blue circles have been swapped. If we restrict ourselves to 3-space, such a rigid motion is impossible, but if we allow ourselves to let the torus pass through itself, then it can be done. However, visualizing the 3-sphere version in stereographic projection, with the funky 4-space rotation, we effectively allow ourselves to distort distances (actually the 4-space distance is not distorted; the distortion we see is an artifact of the stereographic projection), and add a "point at infinity." What happens is we inflate our inner tube, so a part of it gets puffed up to infinity, and wraps back around, turning the torus inside-out. In fact, after wrapping back around, we're "inflating" the outside of the torus. Or equivalently, getting back to donuts with frosting, the dough gets bigger and bigger, and when wrapping back around, almost all of space (plus a point at infinity) is dough, and the frosting bounds an inner-tube-shaped pocket of air (sort of like how complex inversion in Bubi's nose makes everything outside a circle nothing but nose).
Anyway, the full turning inside-out (which also swaps the red and blue circles, as promised) occurs exactly halfway through the movie (the rotation continues to restore the torus to its original state in the second half). Notice how the stripes on the torus which started out horizontal now are vertical, and what used to be the "apple core" shape which surrounds the donut hole now has become a "donut segment." Plus it just looks totally awesome!
Obviously haven't updated in a while. As an old roommate one said, grad school isn't exactly Disneyland. Nevertheless I managed to find a reasonbly fun gem worth others' amusment.
I really fleshed out my knowledge about tautological bundles over Grassmannian manifolds. Now that sounds like completely abstract nonsense that has nothing to do with what possible readers could relate to. Besides, who would expect much excitement to come out of collections of linear subspaces and the like? But I discovered one gem of a theorem.Consider all possible straight lines in the plane (by straight line I mean those that are infinitely long). My latest math musings have brought me to consider making a catalogue of all of these lines. A catalogue of all straight lines? What does that mean?
First off, surely even the most math-phobic readers remember at one time or another, and not necessarily with fondness, that almost all lines in the plane are uniquely characterized by the following famous equation, called the slope-intercept equation:
y = mx + b.
The variable m if you recall is the slope of a line, and b is the y-intercept which means where the line crosses the y-axis. However few people realize that what they're doing, really, is indexing every non-vertical line in the plane with some point in another plane (called, say, the mb-plane rather than the old traditional xy-plane). That is, we have a correspondence between points in one abstract mb-plane to the lines in the xy-plane, where m is the slope, and b the y-intercept. It doesn't treat vertical lines because they have infinite slope, and you can't have a coordinate on any plane with an infinite value. That is, you have charted out the space of all possible nonvertical lines with points in a different plane.
Of course how would we catch the vertical lines? We chart it out using different coordinates. There's the possibility of using inverse slope and x-intercept, which merely means we write the equation using x as a function of y. In other words, all non-horizontal lines are given by:
x = ky + c.
It's essentially obtained by reflecting everything about the diagonal line x = y and finding the usual slope and intercept. In other words we can chart out all non-horizontal lines on this new kc-plane. So you could represent the collection of all lines in the plane by having two charts... an atlas or catalogue of all lines, with these two sheets, the mb-plane and the kc-plane.
However, note that most lines in the plane--ones that are neither vertical nor horizontal, can be represented in both forms. That is, they correspond to points on both sheets. So one thing we could do is try to glue together both sheets to form just a single catalogue. The method of gluing is that we glue together the points that represent the same line in the plane. There is a nice, exact formula for the gluing, which is very simply determined: (m,b) and (k,c) represent the same line if and only if y = mx + b and x = ky + c are equations of the same line. All we have to do to convert from one to the other is solve for x in terms of y, that is, invert the functions. So we solve
y = mx + b
y-b = mx
y/m - b/m = x
x = (1/m) y - b/m
that is, if k = 1/m and c = -b/m, then (m,b) and (k,c) represent the same line in the two sheets. So to "glue" the sheets together, we glue every (m,b) in the mb-plane to (1/m,-m/b) in the kc-plane. Obviously the sheets need to be made of some very stretchable material, because it is going to be awfully hard to glue points together. Actually it's pretty hard to physically do this so, I guess, don't try this at home, but just try to imagine it (don't you just love thought experiments?). For example, points in the mb-plane (1,1), (2,2), (3,3), (4,4) get glued to the corresponding points (1,-1), (1/2,-1), (1/3,-1), and (1/4,-1). You glue them in a very weird way, but if you suppose for a moment, that you allow all sorts of moving, rotating, shrinking, stretching in this process (called topological deformations), but without tearing, creasing, or collapsing, you can preseve the "shape" of this space, and yet make it look like something more familiar. This would be our new catalogue of lines. In addition, the catalogue has an additional property: nearby points in the catalogue translate to very similar-looking lines.
One should wonder what kind of overall "shape" our nice spiffy catalogue has, after gluing together the two possible charts we've made for it. As it turns out, its shape is the Möbius strip! That's right, the classic one-sided surface (without, as it turns out, its circle-boundary). Let's pause a moment for a bit of sober reflection:
THAT IS TOTALLY AWESOME!!!
That is to say, if you give me a point on the Möbius strip, it specifies one and only one line in the plane. One would not, initially, be able see why non-orientability enters the picture. But a little interpretation is in order. First, if we take a particular line and rotate it through 180 degrees, we get the same line back. Everything in between gives every possible (finite) slope. It so happens that as far as slopes of lines is concerned, ∞ = -∞, and if you go "past" this single "projective inifinity" as they call it, you go to negative slopes. In other words, if you start on a journey on rotating a line through 180 degrees, from vertical back to vertical, you come back to the same line, except with orientation reversed (because what started out as pointing up now points down).
If you fix an origin and declare that it correspond to a certain special line in the plane, and then select a "core circle" for the Möbius strip, then as you travel around this circle, the distance traveled represents rotation angle for this special line. Traveling from the origin along the core circle and making one full loop should correspond to rotating the special line by 180 degrees. If you instead move up or down on the core circle, you instead end up sliding the line along a perpendicular direction, without changing its angle. So moving up and down the strip corresponds to parallel sliding of lines, and moving around the strip along a circle corresponds to rotating a line. (Watch the movie above.)
Technically, we need an "infinitely wide" Möbius strip for this, but we can always scrunch it down into a finite-width strip without its boundary circle. It's just that the closer you get to the edge, the quicker things go off to infinity. The animation is an example "path" through "line space," The blue dot travels around the white circle, and the line in the plane that corresponds to it is the blue line. The red line is a reference line perpendicular to the blue one, and always passes through the origin. The distance the blue dot from the core circle (in turquoise) indicates how far from the origin that the blue and red lines intersect. Because of the "scrunching down," though, the closer to the edge of the strip we get, the more dramatic the change in distance the blue line is from the origin.
Ok, ok, back to food analogies. You can think of yesterday's amusement as an expanding donut if you like. It's time to see the Incredible Shrinking Orange!
But that's not good enough! You've got to have the Incredible Shrinking Watermelon, too!
Yes, this actually has to do with math. In fact, they're 2D examples from my actual field of research, on the Ricci flow. What is happening here is that the curvature of the surface is being used to change the shape of the surface (which in turn generates different curvature, which in turn generates different shape, and so on... that's what's so fun about differential equations!). Unfortunately, a lot of interesting examples happen in higher dimensions so it's much harder to see. These are about the only interesting highly symmetric examples, i.e. those representable via analytic formulas that I could plug into Apple's Grapher [GREAT software, by the way... not to be confused with an older software program called Graphing Calculator. It has come with the system software on all Macs since at least 2003 and is widely overlooked by even my Mac/Math geek colleagues!! I created all the movies on this page using its animation feature! Though not as much a powerhouse as, say, Mathematica, the other math program I use every so often, you can cook up a lot of good stuff in Grapher pretty quickly, and with pretty results]. Nevertheless, Ricci flow is a powerful tool... it was instrumental in the proof of a recent million-dollar problem, the Poincaré conjecture.
Enough with donuts. Here's an inflating inner-tube (actually not a full inner-tube, but rather only a bunch of circles on the inner-tube called the Clifford circles.) This comes from trying to visualize the 3-sphere (not your usual basketball or orange peel, but one dimension up, representable as all points at a distance 1 from the origin in 4-space). The way it's done here is the 4-dimensional analogue to the stereographic projection. An ordinary 2-sphere can be (almost) entirely mapped in a one-to-one fashion with the flat plane (one point must be excluded; here it will be the north pole), by imagining our sphere as sitting on the xy-plane, and a light is shining at the north pole. The correspondence is established by tracing the path of a single any light ray from the north pole, so long as it has a little downward component in its direction of travel. It will intersect the sphere and the plane exactly once, these points are declared to correspond via the stereographic projection.
There is an analogous construction for the 3-sphere, namely to imagine it "sitting" on a 3-space "hyperplane" in 4 dimensions, and some light is shining from the "north pole" (the formulas generalize very nicely to show that it works). Anyway so all of 3-space can be put into 1-1 correspondence with all but one point of the 3-sphere. Inside the 3-sphere are interesting sets, for example, various cartesian products of circles. Namely, if we have a point of the 3-sphere (expressed in 4-space coordinates), say, (x,y,z,w), the first two coordinates lie on some circle of a certain radius, and the last two coordinates lie on another circle. The totality of points lying in the two circles determined by (x,y) and (z,w) form their cartesian product. But the cartesian product of two circles is topologically a torus (donut/bagel/inner-tube). Since they also lie on the 3-sphere, they must correspond to some shape in 3-space via stereographic projection... It turns out, they look like real inner-tubes. These tori fill up all of the 3-sphere in a well-defined way, which when carried over to 3-space, correspond to nested inner-tubes, getting larger and larger. I would not want to be standing in the center along the z-axis...
The circles have some major topological significance, too, forming the fibers of the famous Hopf fibration. It has some interesting applications in nature (having to do with exotic things like quantum spin).
After a good 6 months it's about time I got back in action. In the sprit of re-integrating myself into more concrete examples, after spending lots of time in the high-dimensional or even infinite-dimensional abstract clouds, I have gotten back to really studying 2- and 3-dimensional stuff more closely, namely surfaces and 3-manifolds. There's a fantastic theorem in 2 dimensions that says that all compact, closed, orientable surfaces are topologically classified by the number of holes they have. For example, a surface with no holes is a sphere, and one hole is the surface of a donut (the frosting!) or bagel, and two bagels glued together yield a surface with 2 holes, and so on. Topologically this is the only distinguishing feature. It's also instructive to look at these surfaces in terms of geometry, that is, with distances and curvature, rather than just topology which only describes the general shape (a sphere could be a blob, as far as topology is concerned, or the famous example of a donut being the same as a coffee mug).
However, there is more than one way to geometrize a donut. The usual way that yields (at least theoretically) familiar results with is to imagine cutting out a square of rubber, gluing the top and bottom edges together (getting a cylinder), and then gluing the ends of the cylinder together (which will require some kind of stretching of the rubber if you want to get the real donut shape and not cause it to collapse, as what would happen if you try this trick with a piece of paper). But instead, what you could do is just do with a piece of paper and "declare by fiat" that the top equals the bottom, and the left edge equals the right edge, without actually carrying out any gluing. This isn't just some bright idea from some lazy mathematician... if you then make this new abstract donut (surface) which we'll call the torus, and make it "inherit" the geometry from the sheet of paper, you have something genuinely different (geometrically, not topologically) different from the usual donut/bagel/innertube shape.
This geometry actually has been put to good use. If you're old enough to remember the 1980's... (ancient history... hey, it was back in the 20th century...) there was a video game called Pac-Man. And what would happen if you ran off the left edge of the screen, you miraculously reappeared on the right. Similarly going up, you miraculously reappear at the bottom. The thing is, if you think about it from Pac-Man's point of view, nothing really special happens when he goes off one side and comes back the other. The monsters still are perfectly capable of following him, he still eats the little dots in his path, etc. He can run around in this 2D world forever and never fall off an edge or run into an absolute wall (not the wall that contains him in the maze but a wall where everything must stop), and yet his world is still finite.
Now for the kicker.
Imagine instead of cutting out a square, you're floating inside a large virtual cube. By fiat, it is declared that if you walk through the front face, you come back in the back face, you go off the left and come back on the right and floating through the ceiling makes you come up through the floor. You are now living in what is called the 3-torus. Now imagine that light obeys the same rule, so one cube-distance away in front of you, you see the back of your head, and two cube-distances you also see another back of your head, and so on and so forth. In short, you see infinitely many copies of yourself (we'll ignore the speed of light for the moment... well, since you're floating, also gravity... apologies to Einstein), at every "lattice point." The point is, you won't notice anything when you walk through the front of the cube face and rematerialize in the back, because everything you see... all the light rays and so forth, all obey the same rule. Unless you really magically disappear and come back into the real world, you'll be trapped inside the cube and yet, again, you never run into a wall. You can go forever in any direction but you pass your starting point. For a 21st century example where this kind of thing has been applied... if you went to see Matrix Revolutions, when Neo is trapped in that subway station... runs off the left of the screen and comes back through the right, realizing he's not going to escape the obvious way... presumably he didn't need to do the running, because he'd see the back of his head repeated forever down the tunnel.
But now for the real kicker.
It is entirely possible that our entire universe is a cube like this. We don't know. What we do know is that if the universe is like a giant cube, it has to be, well, giant. The great distances involved and the slow speed of light makes this kind of determination much harder to detect than our fun example. Other possible shapes could be things like dodecahedrons with faces identified. Anyway... the inspiration of this amusement was stirred up from reading a very fun math book, The Shape of Space by Jeffrey Weeks, which tours many of these concrete, modern geometric examples, a full dedicated study of which have somehow been lost in even math education (too much focus on super-abstract stuff does have its costs).
I've been learning how to make fancy-schmancy math diagrams in Illustrator. Or so I claim. So far the only nontrivial thing I've made without using some big old math program to generate it, is a critter named Bubi:
Not completely divorcing myself from mathematics, though, I imported Bubi into a fancy-schmancy math program, anyway, to do some conformal mapping. For example, let's apply a conformal map that sends the square to the disk. This transformation involves more fancy-schmancy stuff, such as Jacobi elliptic function.
Didn't look all that different, eh? Let's get dramatic. Let's look at complex inversion, a much simpler and innocuous-sounding mapping, f (z)=1/z (with Bubi's nose at the origin):
Beyond the border of the pic, it's nothing but nose---the interior and exterior of the nose got swapped. In particular, notice that the whiskers point inward.
In my 20E (Vector Calc) section we are talking about parameterizations of surfaces... so I have a:
Pop Quiz! Match the following varieties of pasta to their respective parameterizations:
Note that generally s and t are the parameters, and other letters such as k, α, β, γ are constants.
In the course of TAing vector calc, obviously I get to talk a lot about vector fields. But it is nice to study vector fields in my own classes. It's kind of obvious we have to deal with vector fields in differential geometry, but what may surprise some people is that they are also very useful in complex analysis. In diff. geom we get so inured to the idea of vector fields in n dimensions that we think we're all sophisticated and anything in 2D is just boring! That kind of attitude is completely wrong, and I have somewhat learned it the hard way (in that, for example, I've put off studying complex analysis thinking all this 2D stuff would be easy-peasy). In complex analysis we learn to shun maps that operate on the real and imaginary coordinates explicitly, because analytic functions don't do those kinds of things. Nevertheless if you do look at them that way (as self-maps of the plane), you can see their special properties stand out geometrically. For example, they are conformal maps.
But there is another way: view them as vector fields (consider the image point as a vector sticking out of the domain point). Actually, ironically, is more informative to look at the vector field of the complex conjugate. This is called the Pólya vector field (if you write dz = dx + i dy and your complex function f = u + iv, and expand out the integral of f (z) dz over some curve nto its real and imaginary parts, you get you'll see why this vector field arises---the real part is the circulation about the curve, and the imaginary part is the flux!). Several great things emerge when thinking of complex functions "the vector calc way" (for example, the Pólya vector field of an analytic function is both sourceless and irrotational). And of course, several amusements.
This is a movie of two merging quadrupoles (the sum of the Pólya vector fields of two translates of the function 1/z3 and the distance between the singularities shrinking to zero). The totally merged field is called an octupole, but nevertheless from the picture you can see that the final result has only 6 "petals." The quadrupole comes from the merging of two dipoles, which in turn is the coalescence of two monopoles or point singularities (e.g. 1/z). Each of those guys has as many "petals" as their name would seem to indicate... (more pictures forthcoming... when I have all the QuickTime figured out).
(There is a fantastic book on this matter, by the way, Tristan Needham's Visual Complex Analysis).