Jiahui Gao
October 15, 2021

The professors in the committee are Shouwu Zhang(chair), Yunqing Tang and Boyu Zhang. My special topics are algebraic geometry and representation theory. My questions in the exam are as follows:

Algebra:

Tang: Why is finite field cyclic, the decomposition theorem of finitely generated abelian group, why is Z/2Z\oplus Z/4Z not cyclic. Why polynomials of degree n over a field have at most n roots.

The representation of diheral group D_8


Tang: (Galois theory) the split field of x^4-5.

Complex Analysis

Boyu: 

Given a bounded operator T on Banach space B, if B is finite dimensional then T is compact. Give examples of compact and a non-compact operator. 

Prove the Liouville's theorem. 

and Schwartz theorem.

Write down a biholomorphic map from the open unit disk to the open upper half-plane.

Write down all the biholomorphic self-maps of the complex plane and the open upper half-plane.


Real/Functional Analysis:

Write down the definition of compact operators.  Show that if a bounded linear operator has a finite-dimensional image then it is compact.  Construct a compact operator whose image is not finite-dimensional.

Show that C^\infty(S^1) is dense in L^2(S^1).

Find a function on R that is continuous at irrational numbers and discontinuous at rational numbers.

Representation theory

Zhang: 

State and prove Peter-Wyles theorem. (G is a compact group, G->U(H) is a unitary representation where H is a Hilbert space. I said the crucial step is to prove the sum of finite representations of G is dense in H)

Why are the irreducible representations of compact groups finite dimensional?(use the spectural theorem) Prove the spectural theorem.

The definition of Lie algebra

The representation theory of Lie algebra: sl2(C), sl3(C), what is the highest weight

\rho_n is representation of sl2(C) of highest wight n, what is the invariant subspace of \rho_k tensor \rho_l tensor \rho_m.

Tang: the representation of SU(2), SO(3), SO(4)

The induced representation, If H\subset G is a subgroup, what is the induced representation on G, the representations of \mathbb{C}[G]

Algebraic geometry

Shouwu: Tell me the definition of a scheme, give examples of affine scheme, projective scheme and a scheme that is not affine nor projective. 

Why is X=A^2-{0} not projective nor affine? what does the cohomology of affine/projective scheme look like? 

(I computed the cohomology of X)

So we have a map f from X to \mathbb{P}^2, there is a G_m action on X, 
f is an affine map? the property of higher direct image 

? The push forward of the structure sheaf of X has a decomposition? (direct sum of O(n))

So acturally we can compute the cohomology of P^2 by computing the cohomology of X.

Shouwu: Definition of curves(smooth) and genus. Classify the curves of genus 0, 1, 2, 3 (do not have time for 4, 5)

Tang: (when I give the expression of the elliptic curve) Why is it degree 3? (I use genus formula which is g=(d-1)(d-2)/2) Prove the genus formula?

 Give the map from elliptic curve to P^2 (given by which line bundle),  How to find that line bundle (use Riemann-Roch)


(For genus 2, hypperelliptic curves) 

Zhang: what is the definition of hyperelliptic curves


genus 3: for hyperelliptic curve y^2=f(x). I used the Hurwitz formula to compute the degree of f(x) . 


Boyu: So is there a meromorphic function from a curve of lower genus to higher genus?


Shouwu: compute the genus of the curve that Tang mentioned in her lecture which is y^5=x(x-1)(x-\lambda).