Aaron Yip's General

Committee members : Klainerman, Christodoulo, Washnitzer
Time : 2.5 hr

Real Analysis 
- bounded variations, Helly Th (?!)
- why Leb integration is introduced ?
  main theorems of Lebegues integration theory, differention theory
  ( some example to show that this theory is better than Reiman theory )
- when is a Leb integrable function not Reiman integrable
- give an example of a dense open set in [0,1] such that its charac
  function is not Reiman integrable
- Fourier Transform, Plancheral theorem

Complex Analysis
- Riemann Mapping theorem, outline of proof ( very brief )
- Uniformization theorem
- Connection of harmonic function and analytic functions
- does there exist a harmonic function in the upper half plane such that
  boudary value : [0,1] = 1, others 0. if so, example
- can every surface be conformaly mapped to the complex plane(locally)

Algebra
- structure thoerem of orthogonal matrix, in the real case.
- every skew symmetric matrix has purely imaginary eigenvalues
- prove the fact that some combinations of skew symmetric matrix give
  a orthogonal matrix

Diff Geom
- Is Gaussian curvature conformal invariant ? Is Gauss map conformal ? 
- In R3, discuss the different compoent of the acceleration vector of
  a curve, prove some formula
- Gauss Bonnet formula, give "geometric interpretaion" of each term
  in the stated formula, like in terms of parallel transport, area of
  image under gauss map.
- "geometric meaning" gaussian curvature, in terms of length of curves
  state relevant formulas
- mean curvature and area, change of area under mean curvature flow
- what can you say about a compact surface in R3 with positve curvature
- Frobenius Theorem, in terms of vector fields and differential forms

Functional analysis
- Fourier Transform, in the case of Lp, Reisz convexity theorem
- Functional calculus
- Spectral theorem
- What are spectral measures
- Let E be spectral measure
  What can you say lim(n->inf) E(-inf, Rn), Rn decrease to R ?
- Laplacian, properties

Be careful what you say. The next question highly depends on your 
previous answer. Also, most of the questions are extremely vague.