General Exam

Carolina Araujo

1/26/01  2:00pm

Topics: Algebraic Geometry and Model Theory

Committee: Janos Kollar (Chair), Simon Kochen, Edward Nelson

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Real Analysis (Nelson)

- Let h(x) = \int_R f(x-y)g(y) dy. What can you say about it if both f
and g are in L^2?
(I proved it existed for all x, was bounded, continuous, and vanished
at infinity)

- Define different notions of convergence for functions on [0,1]. 
Which modes of convergence imply others? When does convergence in L^p 
norm imply convergence in L^q norm?

- Suppose you have a sequence of continuous functions on [0,1], 
pointwise convergent to f. Does it have to converge uniformly? (No) 
What if the functions decrease to zero? (Yes) Prove it.

- Consider the problem: (i) y' = f(y) (ii) y(x_0) = y_0. What are
sufficient conditions for the existence of a solution?
(First I showed that f Lipschitz implies existence and uniqueness of a
solution, then I gave a vague idea of how to prove that continuity of f 
is enough to guarantee existence)


Complex Analysis

- (N) What can you say about a holomorphic function whose real part is
bounded?

- (N) What is your favorite proof of Liouville's Theorem?

- (N) Suppose you have a holomorphic function on { 0 < x < 1}, 
continuous and bounded in absolute value by 1 on the boundary, and 
bounded in the interior of the strip. What can you say? (It's bounded 
by 1 on the strip) Prove it.

- (Kol) What is special about simply connected regions in the plane?

- (Kol) Prove the Riemann Mapping Theorem.


Algebra

- (Kol) Compute the Galois group of p(x) = x^7 - 3.

- (Koch) Classify finite fields, their subfields, and field extensions.
Which are the automorphisms of a finite field?


Model Theory (Kochen - he had helped me make the program for this topic)

- What exactly did you study?
(The basic stuff, plus Ultra-products, Morley Theorem, Kochen's paper 
on Model Theory of Local Fields - as an application, plus some general 
logic, like Godel's Incompleteness Theorems and Tarski's book on 
undecidability. He didn't ask me almost anything about these things...)

- Define Ultra-Products.
(I defined them, and stated the Ultra-Product Theorem)

- Is there any property of Ultra-Products that goes beyond the given
models? (omega-saturation) Do you know how to prove it? (I said yes, 
and proceeded to prove it, but he stopped me and said he just wanted to 
know if I knew it)

- Define model completeness. Use the model completeness of
Algebraically Closed Fields to prove Hilbert's Nullstellensatz.

- Define ordered fields, real fields and real closed fields. Use the 
model completeness of Real Closed Fields to solve Hilbert's 17th 
Problem.


Algebraic Geometry (Kollar - I had talked to him the whole semester
before the exam, and he had given me lots of questions to work on. I 
believe that's why he asked me so little here)

- Compute the genus of the curve x^3 = y^6+1
(I did it by blowing up the curve until I got a nonsingular one, and
computed its arithmetic genus. I had to prove the formulas I used: the
genus-degree formula and how the arithmetic genus changes when you blow 
up the curve. He also asked me what happened in characteristic 3, and 
I took a long time to realize the curve was nonreduced. That was really
embarassing...)


The exam took about 1h 50min.