Arash Rastegar


Sarnak, Washnitzer,okikiulou

REAL AND COMPLEX ANALYSIS
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*Fn :D---C a seq. of holo. with L1 norm bounded by one.
Show that it has a subsequence converging uniformly on compacts
on disk of radius a half.
State and proof Montel's theorem.(I stated in the general case)
Prove it for the maps from D to C-{0,1}.
They stoped me in the middle and asked what is the main idea
(contraction of maps between hyperbolic surfaces)
Give a hyperbolic metric on say C-{0,1}.(Kobayashi metric is hyperbolic
but I 've forgotten how to construct it......... I don't know.)
prove any Riemann surface diffeo to sphere is biholo. with Riemann sphere.
(I started to do alg. geom. but they stopped me.
you are asking me to prove uniformization?)
Proof uniformization(I don't know. It's very difficult)
State and sketch the proof of Riemann mapping theorem.
Then Washnitzer said Uniformization theorem for bounded domains 
is as easy as this!
If F is a positive L1 function on R,define Fn(x):=F(x+n).
Show that there is a subseq. converging zero a.e.
(What?! trivial. If integral is bounded ,function tends to zero at infinity
atomatically)
Sarnak said give a continuous counterexample for what you said!
Hint:check if you can prove that for functions on interval.
(I showed that this suffices.and they stopped me to think more on it)
Give a harmonic function on the upper halfplane ,with limit equal one on [a,b]
and zero on the rest.Write in form of an integral(I was pushed to the answer)
Prove that Linf.[0,1] is dual of L1[0,1].(I missed this one!)
Prove that a convex (I said oh! Ok!,Sarnak siad:wait. what is the question?
I said she wants to ask if we have a convex domain in the Hilbert space
there is a unique pt. with minimum norm.She confirmed.Sarnak said you mean a 
bounded and closed convex set?....yes. and I showed uniqueness but missed the
existence!!!!!)

ALGEBRA
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Show that the group of nonzero elements of a finite field is cyclic.
State the two versins of fundumental theorem of finite! abelian groups.
Show that eigenvlues of a skewsymmetric matrix is not real(they are pure
imaginary)
find conditions for the existence of solution for exp(A)=B , B given matrix.
(I was pushed to the answer in most of alg. questions)

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They had only half an hour left to ask me about alg. geo. and No. theory!
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NUMBER THEORY
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Define class # and give examples for fields with class # nonequal to 1.
Prove class # is finite.( I wanted to do a nice zero-dimentional 
Arakelov theory proof,but they regected. I did the standard proof.)
Give an example of a #fid where unique factorization fails.(Q((-5)^1/2) )
What is the ring of integers of this.
State and prove Hermit theorem.(fortunately I was stopped in the middle)
State and prove Kroneker-Weber.
State and prove Minkowski's theorem.
State and prove Dirichlete unit theorm.


ALGEBRAIC GEOMETRY
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Proof x^5+y^5=1 is a nonsingular curve.
Give explicitly a basis for the set of differential forms of the first Kind.
(I used Poincare residue formula and Washnitzer shouted! and the exam finished)