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.