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.