Zhiyuan Chen
Friday, Apr. 29, 2022. 13:00--15:30. 
Committee: Chenyang Xu (chair), Lue Pan, Ana Menezes. 
Special Topics: Algebraic Geometry, Commutative Algebra. 

[Real Analysis]
Menezes: Can you define the L^p space? How do you prove that it is complete (sketch)? 
Menezes: Are there inclusions between them (for different p)? 
	(I answered that if X has finite measure, then L^q(X) is contained in L^p(X) when q > p.) 
Menezes: Can you give a counter-example when X has infinite measure? 
Menezes: Can you give a function on an interval [a, b] that is L^1 but not L^2? 

Menezes: Can you define the Fourier transform (on R^1) of an L^1 function? 
Menezes: What can you say about the Fourier tranform? 
	(I answered that if f \in L^1, then its Fourier transform \hat{f} is continuous and vanishes at infinity. I proved the continuity, but was unable to show that \hat{f}(t) \to 0 as |t| \to \infty.) 


[Complex Analysis]
Menezes: Can you state the Riemann Mapping Theorem? 
Menezes: Why do you need the region is not C. 
Menezes: Do you know Riemann's orginal idea to prove this theorem? Why is his idea not rigorous? 
Menezes: What is the proof you know (sketch)? 
	(I sketched the proof in Rudin's book: consider all the injective holomorphic maps from a simply connected region U to the unit disk D, and take the one that has the maximal derivative at a point.) 
Menezes: Do you need the Schwarz Lemma and Rouché's theorem in your proof? Can you state them? 
Menezes: Which one of them can you prove? 
	(I tried to prove the Schwarz Lemma, but got confused with some details; Menezes said the idea was correct and moved to the next question.) 

Menezes: Can you prove the Fundamental Theorem of Algebra using complex analysis?
	(I said if f is a polynomial of degree d with no zeros, then consider g = 1/f and estimate g using Cauchy's formula. I first said |f| \leq c|z|^d, but this didn't work, then Pan hinted that |f| \geq c|z|^d would work.) 
Menezes: Do you know the genus of a holomorphic function? 
	(I don't know. Menezes said that's fine.) 


[Algebra]
Pan: Let's do some finite field. What can you say about a degree n extension of F_p (p is a prime)? Is it a Galois extension? 
	(I said there is a unique one in a fixed algebraic closure of F_p, which consists of the roots of x^{p^n}-x. Then I said this implies that it is Galois.)
Pan: Is this sufficient? 
	(No. It is also separable, because x^{p^n}-x is a separable polynomial.)
Pan: Why? 
	(I said its derivative is -1, so it has no multiple roots.)
Pan: What is the Galois group of F_q/F_p (q = p^n)? 
	(Z/nZ.)
Pan: What is a generator?
	(The Frobenius Fr(x)=x^p.) 
Pan: Now regard F_q as a vector space over F_p, and the Frobenius as a linear transformation. What is its characteristic polynomial and minimal polynomial? 
	(I said the characteristic polynomial is f(t) = t^n-1 since Fr^n=Id on F_q and the dimension is n. I was unable to show that it is the minimal polynomial.) 
	(I finally worked it out under many hints: If g is a polynomial of degree d < n such that g(Fr) = 0, then g(Fr)x = 0 for all x \in F_q gives a polynomial of degree p^d < q which vanishes on F_q, which is a contradiction.) 


[Commutative Algebra]
Xu: What properties do you know about the polynomial ring R = k[x_1,...,x_n] over a field k? 
	(I said Noetherian, UFD,...)
Xu: How do you prove it is Noetherian? 
	(I said Hilbert's Basis Theorem)
Xu: Prove that. 
Xu: Prove that if p is a prime ideal of R = k[x_1,...,x_n], then height(p) + dim(R/p) = n. 
	(First I wanted to use Noether's normalization theorem, but was unable to deal with height(p).)
	(Then I tried to show that in any Cohen--Macaulay ring this equality holds. I needed that if A is a Cohen--Macaulay local ring and p \in Ass(A) then dim(A/p) \geq depth(A). I said this follows from some vanishing of Ext groups but I didn't remember that.)

Xu: Consider the ring R of continuous functions on the interval [0, 1]. Is it Noetherian? 
	(No. But I didn't know how to prove.)
Xu: What are the maximal ideals? 
	({f : f(t) = 0} for t \in [0, 1])
	(I first proved that any finitely many functions in a maximal ideal has a common zero. Then use compactness.)
Xu: For example the ideal I of all functions vanishing at 0, is it finitely generated? 
	(I didn't know how to prove, and then got a hint: if (f_1,...,f_n) generate I, then \sqrt{|f_1|+...+|f_n|} gives a contradiction.)

Pan: A discrete version of this problem. Let R be the product of infinite copies of F_p (say indexed by Z, the set of integers). What is the Krull dimension of R? 
	(I said it is 0, but I don't know how to prove.)
Pan: Let's prove it. What relations does F_p satisfy (Fermat's Little Theorem)? 
	(I realized how to proceed. x^p-x=0 for all x \in R, so for every homomorphism from R to a field, the image is F_p. Hence all prime ideals are maximal.) 
Pan: What is the residue field? 
Pan: If m is a prime ideal of R, what is the localization R_m? 
	(R is reduced, dim(R_m) = 0, so R_m is a field.)
	(So R_m is exactly the residue field R/m, which is F_p.)
Pan: Can you describe maximal ideals of R? 
	(obvious ones, and others given by non-principal ultrafilters.)
Pan: How do you prove that non-principal ultrafilters exist? 
	(Axiom of choice.)
Pan: Do you know other way to prove there are more maximal ideals than the obvious ones? 
	(Spec(R) is quasi-compact, so it cannot be a discrete set of infinitely many points.) 


[Algebraic Geometry]
Xu: What do you know about minimal model program on rational surfaces. 
	(Contract all the (-1)-curves, then get minimal rational surfaces. I said they are P^2 and F_n for n \geq 0, F_n is the projective bundle P(O \oplus O(-n)) over P^1.)
Xu: Why are they different? 
	(F_n contains a rational curve C with self intersection number (C.C) = -n, which is a section of the projective bundle.)
	(Then I realized that F_1 is the blow-up of P^2 at a point, so it is not minimal; others are minimal.)
Xu: How do you show they are connected by blow-up and blow-down. 
	(Blow-up a point on C on F_n, and contract the strict transfrom of the fiber F through p, then get F_{n+1}.)
Xu: What is the class of the canonical divisor on F_n? 
	(I said Pic(F_n) is generated by the class of C and a fiber F, so write K = aC + bF and then solve a and b using the adjunction formula.)
Xu: In general, consider a ruled surface f: X = P(E) \to B on a curve B of higher genus, E is a rank 2 vector bundle, what is the class of the canonical divisor. 
	(K_X = f*K_B + f*(det(E)) + O(-2).)

Xu: What is the class group of a smooth affine conic curve (say, in characteristic 0). 
	(I first said that over an algebraically closed field it is 0. In general there is a quadratic field extension such that the base change is an open subset of P^1. Using this I showed that it is either 0 or Z/2Z.)
	(But I was unable to classify in which case it is Z/2Z.)