Zhongtao Wu's Generals

Date: May 22, 2006
Time: 1:00pm - 3:00pm
Committee: Szabo (chair), Grushevsky, Hang
Special Topics: Albebraic Topology, Algebraic Geometry

Algebraic Topology: (Szabo)

What is an Eilenberg-Maclane space?
What space is K(Z,1)? K(Z,2)?
What is CW complex?
What is the attaching map in the case of CP^n? cohomology group of CP^n?
What is cellular homology? its complex? its boundary map? (I had some trouble
here)
\pi_1 and \pi_2 of the space (S1)v(S2)?
\pi_3(S2)? \pi_4(S3)?
cohomology ring of S_2 * S_4 and CP3.  \pi_3 of them.

Algebra: (Grushevsky)

Classify groups of order 4. What is the multiplicative group of F_9?  (I said
it's abelian of order 8, so he let me state and prove the classification theorem
of finite abelian group. Then we returned to the original problem, and i proved
it's C_8)
Representation of S_3. its restriction to S_2. Induced representation.

Real analysis: (Hang)

State the fundamental theorem of calculus. (I stated the one for calculus) State
Radon-Nikodym theorem. What's the dual space of L1? (I confidently claimed it's
not L^\infty...)
What's the dual space of C(0,1)? (Riesz representaion)
What's BV function?  relation with monotone function.
Cantor set. The standard counterexample related to it

Complex analysis:

Classifying singularities. Picard's theorem.  Weierstrass factorization for entire
function
Find all positive-valued harmonic functions on the complex plane. (constant)
Find holomorphic map from Riemann sphere to torus? torus to riemann sphere.

Algebraic geometry: (Grushevsky)

meromorphic function on torus. Weierstrass p-function. Riemann-Roch. Hurwitz's
theorem.  Explicit equation of p and p'
Divisors of a curve. effective divisor, ample and very ample divisor.  Abel's
theorem
Definition of arithematic genus and geometric genus.  Are they equal for singular
curves? (i don't know)
Surface classification.  An example of K3, example of general type.

That's it.  Very nice committee as you can probably see...

zhongtao wu