Student: Ming-Yuan Chang Committee: Camillo De Lellis(Chair), Peter Constantin, Jonathan Hanselman Special Topics: PDE, Singular Integrals Place: Fine 205 (Professor Constantin’s office) Date and Time: 5/12/2025 13:00-14:40 I arrived 10 minutes early, and no one had arrived yet. Someone was taking a preliminary exam(PACM) in another room. Peter arrived first and started to chat about the exam with the front desk, and then he said that this guy(ME) would suffer more. I laughed a bit and knew that I’m dead. Peter asked me if I have a preferred order, I said maybe real, complex, algebra, PDE, singular integrals. Real [CDL] State Riesz representation theorem I asked which one should I state, Hilbert space one or continuous function space, then I was asked to state both. [PC] What is a Radon measure? Tried to state it but not so sure about some details, which led to a discussion about measurable sets. [CDL] Can you give me an example of a non-measurable set? I constructed a Vitali set, and they moved on before I dive into details. [CDL] Back to Riesz’s theorem. If you change R^n to torus, you can actually obtain the continuous one from the Hilbert space one. I was trying to obtain an L^2 bound directly (apparently impossible lol), then Camillo suggested that I consider an approximation. I used a mollifier and Young's to do it. [PC] State your favorite convergence theorem I stated the dominated convergence theorem. [PC] Do you know Fatou? Can you use it to prove DCT? I stated Fatou without realizing the functions need to be nonnegative. Peter caught me and made me find a counterexample. Complex(with some PDE interrupting) [CDL] State Liouville theorem. If it is bounded by a power of |z|, prove that it is a polynomial. I said I have seen this before, and he laughed. I used the Cauchy integral formula for higher derivatives. This made Camillo come up with a PDE question and cue Peter to ask one. [PC] If Laplace^2u+u=0, say u is in L^2 with compact support, what can you say about u? I used Fourier transform to show that it is zero, but this is not the method they expected. They said they both would do it another way but I solved it anyway. The office was filled with laughter and we moved on. I wanted to know their solution but we actually came back to this later. [PC] Do you know about Runge's theorem on approximating holomorphic functions by rational functions? I remembered seeing this before but had no idea what the statement should be. Peter hinted me quite a lot to lead me to say the word “simply connected” [PC] An equivalent condition of simply connectedness is that you can approximate holomorphic functions by polynomials uniformly on compact sets. Do you know other equivalent conditions? I stated Cauchy theorem and the topological definition of simply connected. He also stressed that he didn't want me to prove that they are equivalent. [PC] How many components can the complement of a simply connected domain have (in the Riemann sphere)? He then told me that there is a bounded component if not simply-connected and then you can construct something like 1/z Algebra [JH] How do you classify the linear automorphism of Z_2 cubed over Z_2? How many are there? I did the usual counting columns by columns, (2^3-1)(2^3-2)(2^3-2^2) [JH] How about classifying up to similarity? I used rational canonical form. [JH] There is a more general setting related to the theory you are using. Do you know what it is? I stated the structure theorem of a finitely generated module over PID, and discussed the relation with Jordan form and rational form. [JH] If you have a subgroup of a free group, what can you say? It’s free, but I don't really remember how to prove it. He suggested that I consider covering spaces of loops intersecting at one point. [JH] Do you know how to use this to show the subgroup free? Calculate the index? I pondered a while but had no idea. We moved on because he said it is a little bit unfair to ask this in algebra. [JH] How would you classify groups of order 15? Sylow, semidirect product. [JH] What is a Galois extension? What is the main theorem? What is the Galois group of x^3-2? Stated. Write down the automorphisms explicitly to get S_3. PDE [PC] Let A be a constant symmetric n by n matrix, consider the vector-valued equation u_t+Au_x=0, how would you solve this? Diagonalize to reduce to scalar, then use the method of characteristics. [PC] What happens to the L infinity norm as time goes? L^1 norm? I said both are preserved but he asked me to think about the vector equation, not scalar. I finally realized that the L^1 norm can increase! Interesting. [PC] If you have a sequence of harmonic functions converging uniformly on compact sets to u, is u harmonic? Yes by mean value property. I was going to write it down but he moved on as soon as he heard the keyword mean value. [PC] Do you know subharmonic functions? How do you prove that the maximum is attained at the boundary? I tried the second derivative test at first, but he wanted another proof, so I used mean value inequality. [CDL] About the Liouville theorem for harmonic functions, if you can bound it by a power of |x|, prove that it's a polynomial. I was trying to use the mean value property to get the same proof before, but proceeded it wrongly. They suggested that I use Poisson kernel representation. They also told me that I don't need to remember the exact formula, the ideas I sketched suffice. [CDL] Return to the question before, this time you have laplace^2 u+fu=0, f>0, what can you say? I want to multiply the equation by u and integrate, but hesitate about the validity of integration by part. So he gave me a hint related to Singular integral, the other special topic. [CDL] If u is in L^p and f is in L^infinity, what can you get? Of course I want to get the 4th derivative of u is also in L^p, but I didn't say it out and was trying to define laplace^(-2) by Fourier transform, and I didn’t proceed well. Then they just asked me the result I wanted without proof, and now I can do integration by part. Singular integral [CDL] Let F be a compact set in R^n. What is the Calderon-Zygmund lemma to the complement of F? Do you know an application? Stated. I said it can be used to derive weak (1,1) type estimates of Calderon-Zygmund operators, but they didn't ask me to prove it or even explain what it is. [CDL] Let me give you another application. Let u be C^1 on F in the sense that you have a linear approximation with a continuous vector field X on F as gradient. How do you extend u to a C^1 function on the whole R^n? I wrote down a definition which is really far from the answer (didn’t even use the decomposition lol), but Camillo said it’s a good try and guide me to the ideal answer. Then he only wanted me to show that this function has a bounded derivative. Got stuck at one term but again he guided me to the correct solution. These are all the questions. It is surprising to me that there are not many questions on singular integrals. I was told to wait outside the office. They closed the door and within one minute they opened the door again and told me I passed! Thoughts and suggestions to future test takers: If I need to do the exam again, I will probably speak up what I want to do or the result I want to get before I ponder into the details of proving it. I feel like sometimes they only want to hear you saying some key words and they know you know it. The three professors on my committee are all very kind. They said something like “yes” and “good“ when I presented the key technique they wanted me to show, which is very encouraging. They also gave me a lot of hints when I got stuck. I was very nervous before the exam but I felt good as soon as I’m thinking about their questions. It’s also fun to learn some new facts!