Maggie Miller's generals
May 2 2016, 1:10pm
Committee: Gabai (chair), Yang, Bhargava
Duration: 1.5 hours
Special Topics: Algebraic Topology and Differential Geometry
When the exam started, Bhargava said something like, "It's weird to be asking Maggie hard questions when usually she asks really hard questions," referring to a puzzle I told him last semester. So at the end of the exam I told the puzzle to Gabai and Yang.
Algebraic Topology (Gabai)
What are the homology and cohomology groups of the Klein bottle (Z coefficients)? What is the ring structure on the cohomology? Now do it all with Z/2Z coefficients.
Must an automorphism of a 3-manifold which is a rational homology sphere have fixed points? What if the map is orientation reversing? What is the Lefschetz number?
What can you say about the boundary of a contractible 4-manifold? Prove it.
What is the Mayer-Vietoris sequence? What are the maps? Prove that the sequence is exact. What is the sequence for a pair?
Must the normal bundle of a 2-sphere embedded into the 4-sphere be trivial? (I said, "no." then he looked at me for a while and I said, "yes." Then we talked about homology and normal bundles.) How do you usually classify bundles?
If you have a map from CP^2 to the torus, must it be homotopic to a constant map? (I started doing something crazy with Pontryagin-Thom before he stopped me to think about universal covers.)
What is the Pontryagin-Thom construction?
Differential Geometry (Yang)
Talk about the fundamental forms of a surface embedded into R^3. What are principal curvatures? What is the Gauss curvature? What formulas relate the Gauss curvature to the first fundamental form? Why is this important?
What is mean curvature? What can you say about a surface with constant mean curvature? What if it's compact? What is an umbillic point?
What is the Gauss-Bonnet theorem (in generality)? How do you prove it? Is there a generalization for non-compact surfaces? What about higher dimensions?
Algebra (Bhargava)
Classify groups of order 15.
Can a finite integral domain have order 15? What about 10? What can you say about the order of a finite integral domain? Prove it. (I proved that finite integral domains were fields and that finite fields have prime power order. I started with the fundamental theorem of finitely generated abelian groups. Bhargava was like, "Oh, I've never seen someone do it this way but I think it will work." So I froze up and he said something about units and I said, "It's a vector space!!!!!!!!!!" and we moved on.)
Show x^4-2 is irreducible. What is it's splitting field? What is the Galois group? What are all possible Galois groups of irreducible polynomials of degree 4?
What are all irreducible representations of D_4 (symmetries of the square)? (He helped me by telling me to do it for the Klein four first).
What is fundamental theorem of finitely generated abelian groups? Prove it. (I just proved the structure theorem for modules over a PID using Smith normal form). Explain how you get Smith normal form.
Complex Analysis
Y: State the Schwarz lemma. (I thought he said Schur's lemma and was very confused.)
B: Talk about Cauchy's theorem. Now the integral formula. Why are poles isolated?
B: Write down a conformal map from the sector {0