My Advisor is Professor Gang Tian and so he was Chair of my committee.
My committee: Gang Tian, Rahul Pandharipande, Anna Wienhard
Special Topics : Riemannian Geometry, Complex Geometry
Date: 4/30/2008
We were there for 2 hours and half (I guess)
[P] Started by asking this question :
for any matrix A over complex number, does there exist a Matrix B such that A = B^2 ?
Then he asked to compute the number of invertible matrices over Zp ?
I think they were all the questions on algebra!
then [T] asked me
What do you know about harmonic functions ?
What can you say about the bounded harmonic functions on complex plane?
Then he asked me : Consider a good domain ( a disk ) and suppose that you are given
a good function f on it.
Does there exist a function U such that solves the equation Laplace U = f such
that U vanishes on the boundary?
also he asked me how do you prove the Dirichlet Theorem for constructing harmonic function
having specific boundary value?
during these discussions I moved to elliptic operator and what do we know generally for
solving the equation L u = f when L is an elliptic operator and f is some given function.
he asked about the domain and the range of this elliptic operator
then we moved to Complex Geometry :
[P] asked me to compute the Hodge numbers of Quinitic in CP4?
he asked me to construct an example of some Kahler Manifold which can not be embedded
in Projective Space.
also we discussed the theorem that all the curves are embedded in Projective space.
[W] asked me what do you know about fundamental group of Kahler manifolds ?
we then moved to Riemannian Geometry :
[W ] asked me about the definition of Covariant derivative and curvature?
She also asked me what do we you know when Ricci curvature is bounded from bellow?
What do you know when sectional curvature is non positive ?
What do we know when Scalar curvature is Positive?
What do you know special about 3-Manifolds?
[T] asked me : what are the isometries of H3?
also he asked me to classify constant sectional manifolds ?
We were there for 2 hours and half (I guess)
And finally I passed the exam and I am moving to start a real Job!
Thanks God