Brian Street's general exam My committee: Aizenman, Nelson, Ellenberg My topics: Stochastic processes and functional analysis Time 2 hours. We began with real analysis: A: Suppose you have a sequence of functions on the unit disk, converging pointwise, when do their integrals converge? (I give the standard answers) A: When do their derivatives converge? I had no good answer for this, when he pressed me about conditions on the second derivative, I said, by applying fundamental theorem of calc pointwise convergence of them, plus some domination would do it. He also wanted some result about the derivatives converging of convex functions. This confused me a little, and so we talked about convex functions for a minute or two. In retrospect the proof is easy, though I didn't think of it at the time. N: What can you say about the discontinuities of a function? (They're an F sigma) N: Why? I sketch a proof N: Can a function be discontinuous at just the irrationals? (no) N: Why? (Baire category [that's really all I had to say]) A: define the Lebesgue integral (I give the definition) A: Give an example of something that's Lebesgue Integrable but not Riemann integrable. (I give the characteristic function of the rationals in [0,1]) A: Why isn't it Riemann integrable? (All the lower sums are 0, and the upper sums >=1) N: Do the Riemann integrable functions form a vector space? (Yes) N: Why? (A function is Riemann integrable iff its set of discontinuities is Lebesgue measure 0. So... [Nelson stopped me there]) Now they moved onto complex, but really the rest of the exam was a blur of all the topics. A: Suppose you have a sequence of holomorphic functions on the closed unit disk, converging pointwise on the boundary. What can you say about the convergence of the derivatives in the disk? (I say I need some sort of domination or something) A: How would you prove it if you had that? (I start to write it down, once I write down the Cauchy integral formula things move on) A: Can you conformally map the unit circle to a square? (of course it's possible by RMT) A: How would you do it? (I don't know off the top of my head) A: How about the UHP (I write down the conformal map) A: How would you tell if two domains are conformally equiv? (I don't know a good general formula, I say something about annuli, and multiply connected regions, but mess that up a little. I mention the Dirichlet problem) A: Talk about the Dirichlet problem. A: Is there a functions on the annulus solving the Dirichlet problem with (and then gives some boundary conditions)? (I answer it using Brownian motion. Aizenman and Nelson laugh.) Aizenman wants to see the Dirichlet problem as the minimization of a functional in terms of Euler-Lagrange equations. I hadn't looked at physics in a few years and so didn't remember all this right away. They inform me that it's the minimizer of the gradient innerproduct, and that seems pretty obvious. Nelson asks why the gradient inner product is conformally invariant, I stumble on this a little and we move on. N: Suppose you have a function Holomorphic in a vertical strip, bounded by 1 on the boundary, what can you say about it? (Isn't there some sort of growth condition?) N: you tell me. (So I explain phragman lindeloff) A: Now how would you prove this using Brownian motion, like you talked about before? (I mention something about if it's bounded, I can do it) Aizenman wants something better. They hold my hand through an argument done by bounding the probability that complex Brownian motion will hit the edges of a rectangle of width L (L is large) and height 1 starting on the interior. I mess this up horribly. But we get through it, and get a phragman lindeloff for harmonic functions, they have to grow slower than some exponential that we came up with. Ellenberg's up for algebra now. We talk about the group of automorphisms of the disk, and SL(2,R) (and their relation). E: What's the stabilizer of a point in the disk under the group of automorphisms of a disk? (S^1) E: What group theoretic construct relates the stabilizer of two points? (conjugate) E: Prove it. We talk about conjugacy classes of SL(2,R). E: Consider SL(2,R) acting naturally on R^2, what is the stabilizer of a point? (I compute the stabilizer of (1,0)) E: Do you know what sort of subgroup this is? (Is it a Borel subgroup?) E: Yes it is. E: What is a tensor product? (I ask what category we're in. He says whatever, so I do it for vector spaces. I mention the universal property, and he asks what the elements look like, so I do that too.) E: Now we'll take the tensor product of two abelian groups, ie Z modules. Z/pZ and Z/qZ, p and q distinct primes. What is it? (I make a guess, and am wrong.) E: Well let's take an element from it and look at it. (So I write down a tensor b) E: Now multiply it by p, what is it? (0) E: Why? E: Now multiply it by q. (It has to be zero as well. I think for a minute and say "OH!", and say it has to be the zero element, and thus the group is trivial, and do a quick proof.) We talk about discriminants of polynomials. I define the discriminant. E: Why do you have that square there? (I had defined the discriminant in terms of the roots of the polynomial) E: How would you compute the discriminant for a quadratic? (Well I know that, it's b^2-4ac) E: Well that's right, does it always have to the a polynomial in the coefficients? (I don't manage to answer this one. I didn't review discriminants.) E: What does this have to do with symmetric polynomials? (I'm a little off on this one. Eventually he coaxes out of me what he wanted me to say. We talk about the symmetric and alternating group acting on the discriminant.) On to functional analysis. A: What topologies on the bounded operators on a Banach space are there? (Now I've forgotten, did he say Banach or Hilbert? I ask, and mention that I know more on the Hilbert space. I mention strong operator, weak operator, uniform, weak, and ultraweak (and mention that I only know ultraweak for Hilbert spaces)). Aizenman gives me an operator on l^2(Z), namely: D (a_n) = a_{n+1}-2a_n+a_{n-1}, the discrete laplacian. Adds to it a multiplication operator V. Now, he takes the projection onto the coordinates [-n,n], call it P_n. He asks in what topology does P_n(D+V)P_n converge to D+V. Nelson asks if V is bounded. I said I wanted it to be, so it was. I start proving away, and intermediately, see its clear for weak operator, and then keep going and show it for the strong operator. Aizenman is happy. Now they start asking about the laplacian. N: Take the laplacian on the real line, with domain smooth functions with support away from 0. Is it essentially self adjoint? (I talk about its deficiency indices, calculate them, mess up a little, eventually correct myself) We talk about the same problem in other dimensions. Specifically 3. N: Does the Laplacian with domain smooth functions with support away from 0 (call it D_0) have the full laplacian as its closure? (Boy do I mess this up. Eventually he coxes me into showing that functions in R^3 who are in L^2 and whose laplacian is in L^2 are continuous (Sobolev imbedding). Since they're continuous, everything in the domain of the closure of D_0, is 0 at 0 and we're done. This took awhile to coax out of me) A: What can you say about boundary conditions of the laplacian in terms of Brownian motion? (I do the semigroup generated by the Dirichlet laplacian in terms of Brownian motion) A: Let's go back to the above. And say how what you just said, and the case of R^3\0 relate. (Brownian motion never hits 0) A: Let's to back to the case R\0, use Brownian motion to talk about different self adjoint extensions of the laplacian, thereby giving an elementary proof that there is more than one. (I tell how to get Neumann and Dirichlet boundary conditions by using Brownian motion [reflect it and kill it respectively]) A: talk about the central limit theorem (I mention the iid formulation, and then go on to say Lindberg's condition, I can't remember what it is, they seem at least okay with that) N: I could never remember it either, until I saw it nonstandardly...read that section is Radically Elementary Probability Theory, and you'll never forget it again. A: What is Ascoli's theorem? (I state it) N: State something from stochastic calculus (I write down Ito's formula) N: What can you say about the generator of a Markovian semigroup given I'll let you have as nice a domain as you want. (I talk about a theorem from Dynkin that says when it's a diffusion, this is not what he wants. They write down some specific transition function. This doesn't help. Eventually he leads me to say that it's not a diffusion. I forget where all this lead.) N: Talk about quadratic variation of Brownian motion. What does this mean pathwise? (I state that it converges in quadratic mean. He wants an almost everywhere result, so I say if you do it on the dyadic rationals.) N: That's right, but why do you have to choose a specific sequence for convergence pointwise a.e.? (I don't know. Pretty soon I write down the modulus of continuity for Brownian motion, and they were happy and sent me out of the room and later came out and told me that I passed.) For how rocky things were at times, I was strangely comfortable. They were all very nice. Another comment: I read somewhere on this page to bring a bottle of water to the exam, so I did, and I am very happy that I did. My throat was killing me after about an hour. Book recommendations: Stochastic Processes: "Stochastic differential equations, an introduction with applications" by O/ksendal is an easy read. The problems are easy, too. Working through this book gave me a nice understanding of stochastic calculus and stochastic differential equations. If you don't know anything about those topics, but need to, I highly recommend this book. Durrett's book. I think it was called "Stochastic Calculus: A Practical Introduction", and it is essentially the same as his book "Brownian Motion and Martingales in Analysis". The former is newer and easier to read, but the latter does a little more in the way of solutions to PDEs in terms of Brownian motion. These books do some things in that vein that O/ksendal doesn't cover. Also, it has some nice proofs of properties of Brownian motion. Definitely worth a look, especially for the section on the Schrodinger equation. "An introduction to the theory of random processes" by Krylov has nice and easy proofs of such things as Donkser's invariance principle, and Komolgorov's criterion for continuity. If one does stochastic processes as a topic, one should know these proofs, and this book is an easier place to learn them quickly than Karatzas and Shreve. "Brownian Motion and Stochastic Calculus" by Karatzas and Shreve I find this book hard to read, but it has results in it that I didn't see done elsewhere, and so I am glad I looked at it. For instance, it is where I studied the modulus of continuity for Brownian motion, which came up on my exam. Plus it also goes into local time quite a bit, which is worth looking at. "Martingales and Stochastic Integrals" by P.E.Kopp This book does lots of nice things for martingales. The stochastic calculus part is better found elsewhere, though. "Introduction to Stochastic Processes" by Cinlar. This book has a nice easy overview of the Poisson process, if you just want to quickly learn it's basic properties, but not spend too much time with it (which is how I felt about the Poisson process). "Functional integration in quantum physics" by Barry Simon was recommended to me by Aizenman. It's worth looking at. It does a lot of Feynman-Kac type things from the Trotter-product formula perspective, and such proof methods are useful to know. Also, it talks a little bit about the semigroups generated by the laplacian with different boundary conditions in terms of Brownian motion. It doesn't really talk about this as much as I wanted, but there are a few tangential theorems on it. I ended up searching online a bit to learn about such things, which was good since it came up on my exam. Though, I was never able to find a general discussion of such things. "Markov Processes" by E.B. Dynkin. I read the first 6 chapters of vol. 1 and a bit of vol 2, and it helped me a great deal. He does a very formal introduction to Markov processes, which I enjoyed (though I know some people who like it much less than I do). Functional Analysis: I spent most of my functional analysis time with Rudin's "Functional Analysis" which served me well. But if I had to do it over again, I would spend most of my time with Reed and Simon, especially since my examiners had a physics bent. Either way, one should look at both, I think. Reed and Simon has some parts of the spectral theorem that don't appear in Rudin, and so should be looked at (like pure point, abs cont, and singular spectrum, and multiplicity and such things). Also, self adjoint extensions via quadratic forms is in Reed and Simon, but not in Rudin. Rudin does some nice Banach algebra stuff that isn't in Reed and Simon. Also, Reed and Simon makes some comments on the topologies that appear on bounded operators on a Hilbert space, which came up on my exam. Also, I read "A Short Course in Spectral Theory" which was a lot of fun, and quite useful. It's an easy read, and contains some good functional analysis, not even just things related to spectral theory. Dunford and Schwartz, vol II was quite useful to me. It says some nice things about the spectral theorem in the beginning, and I am very glad I read the chapters on self-adjoint extensions before my exam.