Courses to teach

Some day, I hope, I'll get the chance to be a mentor at Mathcamp. In the meantime, I've started accumulating classes I'd like to teach. All are, not entirely unsurprisingly given my interests, motivated by physics.

1. Order-of-Magnitude Physics
   
   I'm taking a course this quarter for first-year physics grad students called "Back of the Envelop Physics". It comprises a ten-week review of undergraduate physics and an attempt to hone physical intuition and order-of-magnitude calculation skills. We're roughly working out of Sanjoy Mahajan, et. al.,'s online "notes" — really a textbook — titled Order-of-Magnitude Physics: Understanding the World with Dimensional Analysis, Educated Guesswork, and White Lies. These notes are based on the course taught yearly at Caltech.
   
   Versions of this course have been taught in past years at Mathcamp, sometimes by Sanjoy himself. In a year in which Sanjoy is unable to visit, I'd like to take a crack at teaching mathcampers some general physics.
   
   I could easily imagine this as a two-day course or one lasting all camp; probably one or two weeks is preferable. The course would probably be roughly two stars. I'd stick closely to a teaching style I've heard termed "Berkeley Standard" (although I might have misheard — someone actually from Berkeley, please correct me), which is very successful especially for problem-solving classes. And order-of-magnitude physics is indeed a problem-solving area. To wit: an hour class would consist mostly of time for the students to work (in small groups at the chalkboard) on problems presented either orally or on a handout, and time would be made for students to present their solutions. For new problem-solving methods, we might work a problem "lecture-style" first, to introduce the techniques. (For instance, I'd expect the students to be able to estimate how many trees there are in the U.S., but not to know a priori how to apply dimensional analysis to compute the size of a falling raindrop, or even to compute to drag-force on the raindrop.)
   
   After a few days of high-school level estimation and unit conversion, we would, however, need to go into some actual physics. So I would not be averse to spending half an hour occasionally giving lectures explaining the physics behind a phenomenon — surface tension, for instance, or sound or stars — so long as I can still spend part of class-time each day providing problems for students to solve. A good idea in order to get the right balance of lecture and work time would be to watch a few of Dorin's classes.
   
   
2. Differential Forms and Hamiltonian Mechanics
   
   This four- or five-week four-star course with homework is based largely on a course Yasha Eliashberg taught Winter and Spring of 2004 at Stanford, helpfully titled "Topics in Analysis and Differential Equations with Applications". The goal would be to arrive at the punch-line which I would give in the opening-assembly advertisement: say you have a box with a slider in the middle dividing the box, so that one one side of the box is one flavor of gas and on the other side of the box is another flavor, and say that you raise the slider at time t=0; the gasses will diffuse together under the second law of thermodynamics, but provided that the universe is static, eternal, and Newtonian, there will be some (bounded, and without any probabilistic argument at all) integral number of years from now in which the two gasses are completely sorted.
   
   The proof of this rather surprising assertion is straightforward. Say there are N total gas molecules; then the current configuration of the gasses in the box is some point in 6N-dimensional phase-space (3N for momentum, 3N for position). And the time-evolution of the system is given by some (in general very complicated) Hamiltonian, and its interactions with the canonical symplectic form \omega=dp\wedge dq. But the box is compact, and provided there are no outside forces on the box, total energy is conserved and finite, so the path of the system through phase-space is confined to some compact region. However, the vector-field (aka differential equation) determined by H and \omega (exactly \dot{p} = \d H / \d q; \dot{q} = - \d H / \d p) preserves the volume element \Omega = \omega^{3N}. Thus by pigeon-hole, the infinitely many images at time t = n years of a ball of size \epsilon around the initial state (\epsilon chosen so that all near-by states are also "sorted" states) cannot be all disjoint; running time backwards gives us a time many years from now in which the gasses are within epsilon of the current state. (There are, of course, some subtleties in the last step of this argument; you actually need a much more detailed description of volume-preserving flows in compact space, in which you, say, take the path through your starting position and consider its closure, and then argue that the flow restricted to that closure is still volume-preserving. But that's not the interesting physics.)
   
   I'd hope to present this proof at the end-of-camp "Thirty Proofs in Thirty Minutes", but to get there I'd need to develop quite a lot of mathematics, and it would, admittedly, be a challenge to fit it all in in four or five weeks. I would assume that students have seen high-school one-variable AP calculus, and are reasonably comfortable taking on faith that words like "vector", "volume", etc., all make sense in high dimensions. I'd probably begin by "reminding" students about tangent spaces and vector fields, and I'd do so rather intuitively. (I remember one class in which the teacher began with "review" and actually defined what that word meant: these are things that I realize you don't understand, but will pretend that you do anyway. I will probably begin my class that way too.) I'd talk about coordinate maps, fields, and partial derivatives, probably, and then I'd take some time to discuss dual spaces. Because what I want to be able to describe are cotangent bundles on (smooth) manifolds. I need concepts like the canonically-defined "d" operator, and I need to have differential n-forms at my disposal. My hope, of course, is to never evaluate an integral, or even define an integral of a differential form. I care that differential equations (which for my purposes are exactly the same as vector fields) have solutions, but I most likely will not prove this. No, this would be a lightning course based largely on pictures of line fields and anti-symetric bilinear functionals.
   
   Will we solve any actual physics problems? I'm not sure yet, since, as you can see, I have a lot of work before I'll understand the curriculum for the class. I expect that we'll work a few examples; say small oscillations of the coupled pendulum. And perhaps, if we're ambitious, we might derive Kepler's laws. It depends on how much time it takes to just develop the formalism.
   
   (Incidentally, I've only really worked this formalism in the Newtonian regime. How much carries over into the non-relativistic Quantum universe? I'd expect that quite a lot does, since there we also use words like "Hamiltonian", but now position and momentum are no longer simply coordinates. Indeed, all the previous scalar fields on phase space (and coordinates are (local) scalar fields that happen to intersect transversally and uniquely) are replaced by (Hermetian) operators which now no longer need commute. So even if many of the formulas hold, the argument would need to be completely reworked. And I can basically visualize stuff in 6N-dimensional phase space; time-perameterized paths in complex Banach space don't yet make sense to me.)
   
   
3. Linear Algebra and Quantum Mechanics
   
   Of my three courses, this is the one I'm least sure of, and the one that would be hardest to teach, especially in four weeks to high schoolers. It would also be four-star.
   
   The goal of the course would be to develop both linear algebra and quantum mechanics, based on a claim I made last summer: "When we mortals look at Quantum Mechanics, we see differential equations. When God looks at Quantum Mechanics, He sees linear algebra." And it's true: the differential equations in QM are all linear, and since we're working over complex space, all diagonalizable, etc. And the mathematics used in QM is all just linear algebra. I'd go so far as to claim that you don't understand linear algebra until you've used it in QM, in the same sense that Newtonian physics is essential for actually understanding high-school calculus, and Maxwell's electromagnetism for understanding a first course in multi-v.
   
   The course would probably start with Schrodinger's pet cat, in an effort to understand what physicists might mean by "the cat is half-alive in the box". We'd be uninterested in the philosophical underpinnings of the thought experiment, or even in what it means to "look at the cat". Then we'd probably ask what "spin" is, and I'd borrow discussion from the beginning of Feynman's third Lectures on Physics. This would lead nicely into comparison with geometric vectors: I can plot various spin-states of an electron as points in (complex) two-space. Pictures, intuition, and unjustified (and simplified) assertions taken on faith can lead us to a description of the classic "two-slit" experiment, based on the discussion in Feynman's QED.
   
   Then we'd move into the more interesting world of particles in free space and in a box. I'd like to compare the infinite-dimensional (Banach) space of states of a free particle (perhaps in a box) with the finite-dimensional (normed) \R^n (and also the finite-dimensional complex space of spins). Most concepts appear in both regimes: bases, change-of-bases, linear operators, "matrix" representation, non-commutativity and Lie brackets, inner products, orthonormal bases, eigenvalues and eigenvectors, diagonalization, and more. A ten-week course that meets every day, or a five-week ambitious course that assumes a basic introduction to differential equations, would solve the harmonic oscillator, and hope to solve the equations of light propagation in a vacuum. I would most likely avoid any discussion of angular momentum and of relativistic effects like the fine structure constant, because those would take the course too far astray from the central goal of teaching both QM and LinAlg. Besides, in a group that includes students who've never seen LinAlg, such discussions will not be particularly understandable.
   
   This course (in fact, all these courses) needs some rethinking before it can be presented to Mathcampers. I would give problem sets daily, and expect students to do some homework, because I'd hope to cover more material than is actually doable in twenty or so lecture hours. And there's no way in hell that I'd offer both this course and the one on differential forms in the same summer.