Shruthi Sridhar
General exam: May 7, 2018
Committee: (Chair) David Gabai (D), Zoltan Szabo (Z), Javier Gomez-Serrano (J)
Special topics: Algebraic Topology, differential topology.
Time: around 2.5 hours
Complex Analysis: (Textbook: Stein and Shakarchi)
(J) Prove argument principle
(J) Rouches theorem problem: number of roots with Re(z)>0 for t-z-e^(-z) with t in (1, infty)
(J) What can you say about an entire function satisfying (ref)^2 <= (Imf)^2?
(J) if f is holomorphic on a domain that contains the closed unit disk such that |f(z)|=1 on the boundary of the disk, show that f is constant.
Practice general:
(J) If |f(z^2)| <= |f(z)|^2 for all z for an entire function f, show it is a polynomial.
(J) suppose f is holomorphic in the rectangle R with vertices 0, 2, i, 2+i, and that f(z + 2m + ni) = f(z) for any integers m and n, show it is entire. If f was meromorphic, show f has atleast 2 poles in R
Real Analysis: (Textbook: Stein and Shakarchi real analysis + chapter on Lp spaces)
(J) Define fourier transform of L1. Prove continuous.
(J) If f*g =f for all f in L1, show g =1 a.e (Use fourier transform).
(J) Show that the fourier transform of an L^1 function dies out as we go to infinity.
(J) Suppose that f is a continuous function on [0,1] and that the integral of f(x)•x^n is zero for all n in N. Then prove that f is zero. (Approximate by polynomials).
Practice general:
(J) Show that bounded functions on a closed interval form a non separable complete normed vector space. What if we replaced bounded by continuous?
(J) I forgot the second problem. The solution involved some standard analysis argument by defining E_n = { x | |x|n? (Other than pi3(S2) I didnt know any other examole).
(Z) Some idea for the above using Hopf invariant. I hadn't studied Hopf invariant and said so. Zoltan provided a different track for this question:
(Z) Define Hopf fibration. What about if we considered quaternions?
Looked at pi_7 of LES of homotopy groups for the fibration S3 ->S7->S4 to show existence a non trivial element of group that \pi_n(S_m) tensor Q is non trivial. Apparently pi_7(S3) works.
(Z) Prove pi_4(S3) = Z/2?
(Z) What are K(G,n). Why is their homotopy type unique? How would you construct a K(Z,3)?
(Z) What is pi_3(S2 wedge S2) (messed up big time here. Only could show a subgroup Z^2))
(D) What's a gluck twist? (I took a while here to find a non trivial diffeomorphism of S2×S1. Included a discussion of pi_1 of SO3)
(D) What's its pi_1 (State van kampen, use it to show trivial pi_1), H_1, H_2 (Meyer Vietoris and a brief discussion of Alexander duality here). Now what do you know about the manifold?
(D) What is whiteheads theorem?
(D) What is the action of pi_1 on the set of pre images of single point in the covering space? (Got this wrong at first. Thought it acted by simple translation by lift of the element of pi_1).
(D) Work out the example of figure 8 considering its universal cover. I figured out my mistake and showed the action is by a kind of conjugation.
Differential topology:
Milnor-Stasheff:
(Z) What do you know about Stiefel whitney classes? (axioms, existence and uniqueness sketch).
(Z) What is the SW class of RP^n? Can you talk about some application of this? (Embedding into R^(n+k))
(D) Can RP4 have a 2 plane sub bundle?
(Z) What are chern classes, can they be defined axiomatically like SW classes too?
(Z) State Hirzebruch signature theorem for dimension 4.
H-cobordism and morse theory:
(D) Show that any component of S6\S5 is a smooth 6 ball.
(D) Why can you have critical points arranged by index?
(Z) Give a sketch of the proof of h-cobordism theorem.
(Z) More detailed discussion of eliminating index 1 critical points.
(D and Z) What happens to flow lines when you have a handle slide? (Struggled a bit with this one although I had seen the idea with a the height function of a torus when placed on its side)
(Z) What can you say about homology under the operation of handle slides?
They asked me to step out of the office for a few minutes to discuss and congratulated me on passing.
Remarks:
This committee is very friendly and helped me out through parts I got stuck on without any visible judgement.
Javier was willing to give me a practice general exam for real and complex a few days in advance and that helped a lot with nerves. Also Maggie and Irving gave me a practice exam for special topics (Thanks!). Highly recommend a practice exam if you are nervous about oral tests like me.
The constructive feedback I received for topology was to practice being more comfortable with hands-on topological methods of argument (thinking of maps more explicitly vs trying to kill all the problems with algebraic topology).