Sophie's generals

Date: May 13, 2002
Committee: Yang (Chair), Fefferman, Gunning
Special Topics: Differential Geometry, PDEs
Duration: 1 hr 5 mins

I chose to start with PDEs.

(F) What is the Laplace eq? Talk about some properties.
  - Mean Value Property, Max Principle and analyticity.
    He asked me to prove them.
(F) What is the Laplacian on the manifold?
(F) Prove the Max Principle on the manifold.
  - I said the Laplacian was an elliptic operator
    so I proved the Max Principle for an elliptic operator.
(Y) How about the Strong Max Principle? 
  - I used Hopf's lemma to prove it, and then proved Hopf's lemma.

Then we went into Algebra.

(G) What is the Jordon Canonical Form?
  - I stated it. Then he asked me to prove it.
(G) What are abelian gps of order 16?
(G) How about gps of order 15?
  - He hinted that I should talk about Sylow thms.
    I stated them and said Sylow gps were normal in this case.
(G) What are the Galois gps of irr cubic polynomials?

We changed to Complex Analysis.

(G) What are types of singularities?
  - Removable singularity. I gave the criterion.
    Essential singularity. He asked me the property of it.
(G) Can you give an example of an essential singularity?
  - I couldn't answer then. Fefferman said it wasn't complicated.
    Finally, Gunning told me the example exp(1/z).
    I felt like an idiot.
(G) Prove it has an essential singularity. 
(G) What is the Laurent series? Give the coefficients for exp(1/z).
(G) What is the relation btw conformal mappings and analytic functions?
(G) If u'(z)=0, what is the geometric meaning?
  - I mentioned Jacobian but it wasn't what he wanted.
    Finally, I said the angle rotated.
(F) How about an analytic function for a punctured disc?
  - I wrote down a formula which was actually for harmonic functions.
    Yang pointed this out. However, I didn't answer this question though.
(G) What is the relation btw harmonic functions and analytic functions?
  - I mentioned Poisson's Formula for a disc.
(F) If we consider a punctured disc,say ln|z| for example, can you find
    the imaginary part?
  - I wrote down ln|z| + i (arg z). They all laughed. Though I didn't
    know the purpose of this question... :)
(F) What does arg mean?
 
Finally, Differential Geometry.

(Y) What is the 2nd fundamental form?
  - I defined the Gauss map and gave the interpretation of the 2nd 
    fundamental form.
(Y) What is the relation btw the 2nd fundamental form and the curvature?
  - I gave the definitions of the Gaussian curvature and the mean curvature. 
(Y) What's the relation btw the curvature and compact manifolds?
  - I said the total curvature was equal to 2*(pi)*x. (x is Euler char.)
(Y) Consider a compact convex surface. How to prove this formula directly?
  - I said the total curvature was equal to the area of the image of the 
    Gauss map. In this case, Gauss map was one-one and onto.
(Y) How to prove it is one-one?
  - I used a plane which touched the surface. He asked me to be precise. 
(Y) Prove the Gauss-Bonnet thm for a polygon on the plane.
(Y) then, for a simple region on the plane.
(Y) then, on the manifold.

Comment: There were no real analysis questions. Probably, they thought 
         PDEs could cover real analysis. :) I'm glad the algebra questions 
         were very easy! We spent most of the time doing Complex actually.
         The most important thing is that my committee was really nice.