Ansh Saxena’s Generals transcript. Special topics: PDEs, Mathematical Physics Examiners: Bjoern Bringmann (chair), Michael Aizenman, Wei Ho Date and time: May 14, 2025, 1:00 pm - 4:00 pm. —— First some general remarks and advice: I decided on both my special topics quite late: at the start of the Spring semester, and hadn’t studied either before that. Strictly speaking, this isn’t the best thing to do. It helps to have at least one of your topics be something you are already comfortable with. I do think you should choose at least one topic that’s new to you so you get to explore around a bit. I learned quite a bit of math, and about myself this way. Here’s what I’d do differently if I could go back: I’d try to learn from my batchmates the math that they like. Learning math from friends is quick and enjoyable. See acknowledgements below. My topics were different from what anyone else was doing in my year — though I did PDEs, the matter that I was tested on was quite different from the usual syllabus. I think it would have been nice if I could have learned this material alongside a fellow student preparing the same topics, but I liked the content I covered and was happy with it at the end of the day. I routinely met Bringmann and Aizenman to discuss what I was learning. My committee was exceedingly nice, helped me through questions I needed help with, and made me feel at ease. Somehow I fared much better at harder problems than the trivial ones, but I guess generals make you think funny in that way. —— [B] - Bringmann [A] - Aizenman [H] - Ho I ran into [A] on the third floor a few minutes before my exam. He asked me how I was feeling, and I said I was nervous. He was very nice and tried to cheer me up. It had been raining outside, and seeing this he asked me if I’d rather be out there than in the room where I had my exam. [B]’s office was not very large, so the exam was in classroom 401. Everyone was there by 12:55 pm. I told them that their victim was ready, and so we began 5 minutes earlier than scheduled. Analysis: 1. [B]: What’s the hardy-littlewood maximal function? Can you show it doesn’t map L^1 to L^1? (I said any generic function should work, and indeed the indicator of the unit interval works.) What does it map L^1 to? Prove it. How do you prove Vitali covering lemma? (I said “greedy algorithm” and [B] was immediately satisfied.) What else do you know about the maximal function? (L^p -> L^p for p>1) 2. [B]: Set 0<\alpha<1. Look at integral f(y)/|x-y|^\alpha dy over the real number line, where f is non-negative and integrable. Show that this as a function of x is finite almost everywhere. (My first thought was just to integrate in x, but also half-guessed that [B] probably wanted me to compare with the maximal function — which he explicitly asked me to do right after. When y is distance 1 from x and beyond, we can bound the integrand by f. Otherwise, we can partition the interval around x dyadically, write the integral as a sum over the dyadics, and compare each using the maximal function. This finishes the argument.) After, [A] asked me to prove this by integrating in x! [B] said he was hoping I didn’t do it by integrating in x because he wanted to see me use the maximal function. (It’s trivial to do this by integrating in x, but the above technique of comparing with the maximal function goes further: it allows you to prove the Hardy- Littlewood-Sobolev inequality.) 3. [B]: Define the Fourier transform for integrable functions. What do you know about the Fourier transform? (I talked about it as a map on Lebesque spaces, and how it swaps regularity and decay.) [B] asked me in what sense the last statement was true, and to prove it. 4. Next [B] defined the Wigner transform for me, and asked me to show it’s an isometry on L^2. [A] asked me the connection with QM — I didn’t know, but guessed it’s taking the wave with definite momentum and tapering it so it’s in L^2, and they were happy with that. 5. [A]: Can a function and its Fourier transform both have compact support? We can say something slightly stronger, right? (Uncertainty principle.) I think [B] had more analysis questions prepared for me (he was carrying pages of prepared questions), but he chose to wind analysis up. [A] was asked if he wanted to ask anything in analysis, but he said he was satisfied. Your examiners will likely quickly move through stuff if they can tell you are comfortable with what they are asking. The section gave me much confidence and helped me feel at ease — it’s advisable to start with something that you are confident in. Complex Analysis: 1. [B]: Write down the Cauchy integral formula. Show that an entire function growing like a polynomial is in fact a polynomial. 2. [B]: Prove Schwarz lemma. I told them this can be used to classify automorphisms of the disk. 3. [B] wanted to make sure I knew how to do contour integrals. He asked me to compute the integral of cos(x)/(1+x^2) on the real number line. I said I could do it without complex analysis and [B] told me that he’d allow it but then would follow up with a harder integral that I wouldn’t be able to do without complex analysis. I decided not to test him. He was fine with broad ideas and didn’t actually make me compute the integral. 4. [A]: If you have a function on the real line, and I ask you to find an analytic function with that as boundary condition, how do you find it? What conditions do you need on the boundary function? What’s the Poisson kernel? Connections to Brownian motion? This last question was informally asked — I wasn’t expected to know it. Surprisingly, I wasn’t asked the usual in complex analysis: no proof of the Riemann mapping theorem or Picard’s theorems. Up next was algebra. This was definitely the section I was least excited about. I drank some water, and so it began. Algebra: 1. [H]: You talked about automorphisms of the unit disk. Can you identify the group? 2. [H]: How many elements are there in SL(2,F_p)? Whatever I did here was using hints from [B] or [H]. This took me a long while with much prodding. (First did it for GL(2,F_p), then was hinted at deducing the answer from that.) 3. [H]: Give a ring that isn’t a PID. 4. [H]: Prove that a PID is a UFD. Give a ring that isn’t a UFD. On the first problem I had the correct idea and wrote down the right thing, but was funnily having trouble concluding. [H] believed I could finish it and moved on. On the latter problem I had to be led to an answer because I didn’t remember off the top of my head — I hadn’t revised any UFD stuff! 5. [H]: How do you prove that 2 is not a prime in some ring (continuation of previous problem)? I said perhaps some notion of “size” might help. When asked to put more precisely what I meant by that, I said it should be 0 for 0 and satisfy the triangle inequality but didn’t know what else. She said she could see I was trying to reason about this like an analyst. I decided to write out the proof keeping track of the properties of “size” I needed to conclude my argument, and inferred the correct definition. 6. The above led to some discussion, including [H] saying something about number fields… Seeing that the norm is invariant under conjugation, she asked me if I can come up with another function that would be invariant. I didn’t know, so she told me it’s the “trace” and talked some more about it. 7. [H]: Give me a field extension and give its galois group. When is a field extension Galois? Give the subgroup corresponding to each subfield of a galois extension of your choice. I drew the lattice and showed what the group corresponding to each sub-field was. 8. [H] wanted to conclude algebra, but [B] asked her to ask me some linear algebra. My eyes widened in horror as I worried I might get some arcane question about modules, but thankfully that didn’t happen. I was still awkward as hell here for some reason, however. [H] asked me about rotations, about how to take projections, about diagonalizable matrices, and here’s the funny part — they asked me to give a non- diagonalizable matrix. I gave one. They next asked for one that was full rank, and I was stuck for far longer than I should have been(!!!). Somehow my mind managed to never conjure the words “Jordan canonical form” here. There was little diversity in the Algebra sections of Generals transcripts I’d read, and so I managed to miss questions on things like SL(2,F_p) and UFDs. I’d prepared Galois theory and Field theory much better, but wasn’t asked many questions there. We took a 5 minute break. Before my exam I’d thought it would feel like a long time between the start of the exam and the break after algebra, but the time actually just flew by! I was happy with how the exam was going modulo funny hiccups in algebra. Once we were all back, [B] asked me if I’d like to resume my exam. I joked that I didn’t think I had a choice in the matter. He replied that I could also just give up, but that he would suggest against it. I took his advice, and the exam resumed. PDEs (non-linear dispersive and parabolic) 1. [B]: What’s the critical regularity for the Schrodinger equation with a power-type non-linearity? 2. [B]: Write down the four-dimensional cubic defocusing Schrodinger equation. Next prove local well-posedness. Give the Strichartz estimate first. I gave the estimate, drew the Strichartz game board and showed how to close the iteration argument (he didn’t want more details about how to deal with the fact that this is critical). 3. [B]: Do you know any other spaces adapted to dispersive PDEs? (Bourgain spaces). Define them. Can you show well-posedness of the non-linear equation above in this? Yes or no. (I guessed yes without thinking and was told the answer is no. If I’d thought a little more I’d have remembered that you need to localize your solution in time to put it in the Bourgain space. But it was okay.) 4. [B]: Now suppose that you in fact have global well-posedness with the same bounds as above holding globally. Prove scattering for the Schrodinger equation. (You only need some well-posedness and decay to show scattering, both of which were given to me. I wrote out what it means to scatter and the proof followed smoothly.) 5. [B] said we will do some parabolic stuff now. He said he can ask either a maximum principle problem or a local well-posedness problem — I get to choose. I chose the former for some diversity in problems. He gave me the heat equation with cubic non- linearity on a nice bounded domain. Suppose the boundary condition is 0, and suppose you have two solutions u and v to the equation. Suppose also that u \leq v initially. Prove that u\leq v for all times. I concluded with a weak maximal principle argument (again, he didn’t want all the details). 6. [B] was happy, and asked [A] if he wanted to ask me anything. [A] asked me about connections between the Brownian motion and the parabolic question that [B] had asked, but I said I was aware of the connection but didn’t know about it. After generals he told me the connection, and it’s very nice. Using the Feynman-Kac formula you can see immediately that the result should be true. It’s a very satisfying way to do it. Mathematical Physics (QM) 1. [A]: Consider the Schrodinger equation on the real line. Suppose the potential is a compactly supported well. Prove that you will have at least one bound state. (He’d shown me this problem a week earlier, so I knew the idea already! I just filled in the details: you write down the lowest eigenvalue using the variational principle, form an approximate identity out of a compactly supported smooth function, and find what you are looking for by making the support large.) What happens in higher dimensions? (The above analysis is sensitive to the dimension, with 2D being critical – this was apparent from the above analysis. I said this shouldn’t be true for dimensions 3 and higher, and he said that was indeed the case, but didn’t ask me for a proof.) 2. [A]: Draw the spectrum for the potential above. Suppose the well is shallow enough to have only a single bound state. Say you start with Gaussian initial data. What happens asymptotically in time? (Solution converges to projection of the Gaussian onto the bound state — initially I told them the systematic way to do this, but didn’t zero in on that they just wanted to hear these words. I eventually did.) 3. [A]: Why is there an “i” in the commutator of position and momentum operators? Prove the Heisenberg uncertainty principle. The mathematical physics portion was short. [A] didn’t ask me more questions. Perhaps this was because it’d already been 3 hours and he didn’t want to keep me for longer. He’d told me a week prior that he would ask me questions about time dependent hamiltonians, but that didn’t happen — by the way, Reed and Simon volume 2 is a good place to learn this material. Don’t worry too much about the technical details, just look at the example(s) there. —— The exam concluded exactly at 4:00 pm, which gives it a total run-time of (precisely!) 3 hours. They asked me to go out for a few minutes so they could discuss my performance. I’d been told by friends that those minutes feel like the longest minutes of your life, but thankfully I was too gleeful to feel so. They let me in after a few minutes, told me I passed and congratulated me. I shook hands with them, and followed Bringmann to his office. He told me he was happy with how I did, but told me to brush up some linear algebra seeing I couldn’t immediately give a non- diagonalizable matrix, and would need to TA the course at some point! Kunal and Gautam helped me with mock exams which were helpful and built confidence. I studied analysis with Ben Scott. Many thanks to them. I especially want to thank three people: Stuti, who kept me sane throughout the generals prep, and my fellow first year comrades Alexey and Ines, who patiently taught me all the algebra I needed to know. I was surprised how quickly I learned with their help. Study for the generals with your friends!