My Generals, by Yi Wang May 15 2007, 2:00pm, Fine 901 Commitee: Alice Chang (chair), Andre Neves, Gunning Duration: 2 hour 40 min Algebra G: describe abelian group of order 12 G: can you tell me the way to find all group of order 12 G: fundamental theorem of Galois theory G: when a field extension over Q has S_3 as Galois group? I gave the criteria for a cubic polynomial has Galois group A_3 and S_3, then say x^3-2 Let’s come back to something more elementary. G: what is Jordan canonical form, if not on C, can you also have it, then which form you can have? Me: Rational canonical form Complex G: Consider a function, holomorphic on a punctured disc, describe different type of singularities, the criteria of each type G: Can you give me an example of essential singularity Me: exp^(1/z) G: why? Me: I wrote down the Laurent series, but he want me use another way, then I showed that take different sequence converging to zero, the function has different sequence limit, thus the limit does not exist. G: On an annulus, if a function’s Laurent series has infinite terms in principal part, can you say it has essential singularity? Me: yes? (Saw Gunning shake his head) No. Me: it took me a while, then he gave me the hint to consider rational function P(z)/Q(z), finally I gave the answer 1/(z-1) on 2<|z|<3 G: You see, a good function may not appear to be good.(really cute) G: is holomorphic map always conformal? Me: no, only if f (z)!=0 G: if f (z)=0, locally how it looks like? I drew the picture G: should a conformal map always be a holomorphic function? Me: no, it could be the conjugate of a holomorphic function, but it reverses the orientation (He kept saying “good, good” during the time, really interesting and nice. ) G: Can you tell me the most famous theorem in complex analysis? Me: Riemann Mapping G: tell me the statement Me: Just wait a minute. G: It is too famous to remember. (Everyone laughed, when I was less nervous, I gave the statement) G: why you put the condition the domain couldn’t be the whole plane? Me: because otherwise, by Liouville’s theorem, it is a constant Gunning was satisfied, and Alice began to continue in complex part, I don’t know why, each of them like complex pretty much. C: construct the conformal map from a fan(a section of annulus) to a unit disc C: what is winding number, why the formula you give me defines an integer? C: what is Rouche’s theorem, what do you use to prove it? Me: the winding number I mentioned C: what is Schwartz theorem, prove it Me: maximum principle for f(z)/z C: can you give me another version of Schwartz theorem, (I wondered what she was asking, but finally figured out she was asking the following) Me: if your function is from B(z_0,1) to B(z_1, 1), then you have another version, C: do you know a function whose fourier transform is itself? Prove it Me: I proved it by Residue formula N: could zeros of a holomorphic function accumulate? Me: no, because of Taylor expansion I don’t know why they have so many problems on complex analysis, at this time Neves asked another, which is the only one in this part I failed to give answer. N: consider H^2 with negative curvature, whether there exists a non-constant bounded harmonic function (Laplace operator with H^2 metric) on it? Me: no?(he shook his head) yes Real analysis: C: Lebesgue differentiation theorem, C: could it be converge everywhere rather than almost everywhere? Me: if it is continuous function, yes, if L1_loc function, no C: briefly prove it Me: by using maximum function is weak type (1,1), and we can use continuous function to approximate L_1 function N: prove the Liouville theorem,(I asked whether it is the one in complex analysis, he said he wanted it to be on R^n) Me: then I need to use gradient estimate, I proved gradient estimate for harmonic function PDE C: Let’s talk about some PDE now. What is Sobolev embedding? Give me an example that a W^{1,2}(\Omega) function that is not L^{\infty}, where \Omega is a bounded set on R^2. C: Is the embedding compact? Give a non-compact example with p=2. Me: v(x)=(1/{1+|x|^2})^{n-2/2}, v_{\epsilon}=\epsilon^{(2-n)/2}v((x-x_0)/ \epsilon)Why this sequence is non-compact? C: What property does this function have? Me: I just know it is the radial solution of a PDE C: I mean the geometric meaning. Me: I don’t know. C: ok. Give an example in elliptic PDE of which a weak solution is not a strong solution. Why? C: ok. What do you think is the most important theorem in L_p theory? C: f\in L_1, Laplacian u=f, is u\in W^{2,1}? Me:no. C:give me an example. I don’t know. N: When the equation Lu=f, with u \in W^{1,2} and f\in L^2 has unique solution? Me:c(x)<=0. the maximum principle ensure the uniqueness, by Fredholm, uniqueness imply existence N: If we have general c(x)(bdd), why the equation’s kernel has finite dimension? Me: because the inverse of operator L_{sigma} is compact, on Banach space, compact operator’s eigenspace has finite dimension. Differential Geometry I did quite bad in this part. N: Synge theorem My version of this theorem is different from his. What he meant is even dimensional manifold, orientable, compact, sectional curvature positive, then it is simply connected. Then I gave the proof. N: give an example when it is not orientable then it is not s.c. Me:P^2 N:in particular, for 4 dimension manifold with above condition, give me two examples. Me:S^4. After a lot of try, he told me another example is CP^2, he then asked me the geodesic on CP^2. But he changed the topic when I could not answer. N:for compact surface, if g=2, can you give a metric s.t. curvature is positive every where? Chang gave the hint of Gauss-Bonnet. Then it is clear it is impossible. N: for compact surface again, what can you say if Gauss curvature doesn’t change sign? I don’t know. Finally he told me it is impossible for a manifold having negative curvature everywhere because the highest point has positive curvature. N: what is second fundamental form, what if it is positive definite? what is mean curvature, what if it is positive? Me:when second fundamental form is positive, all eigenvalues of the operator is positive, Mean curvature>0, the trace of the operator is positive. N: Draw the picture of a torus when its the mean curvature is >0. C: Bochner formula Me: I don’t know. C: calculate \laplcian |grad f|^2 in R_n, and curvature term will appear in manifold case. At this point, I think I really disappoint them with my geometry part. But finally they decided to let me pass. Throughout the process, they all appear to be very nice and helpful. When I cannot figure out anything, they kept providing hints, until I can say something myself. I do think it is a learning process as well as an important exam.