Béla Rácz's Generals
April 30, 2010
Special topics:
Algebraic Topology, Differential Topology
Committee:
Zoltán Szabó (chair)
William Browder
Robert Gunning
Complex Analysis
State Cauchy's integral theorem.
Geometric definition of conformal maps; relation with complex differentiability (characterize conformal maps; I mentioned that
orientation-changing maps can also be conformal, which was unnecessary). I made a short and also unnecessary comment on conformal
maps in higher dimensions.
State Riemann's fundamental theorem on conformal maps; why can the domain not be the whole C.
Behaviour on the boundary; I stated Carathéodory's theorem.
Talk about the Schwarz reflection principle (they gave me the hypothesis for Schwarz and asked "then what"; I commented that the
domain's boundary must not only contain a line segment, but also it mustn't be "jagged" on it, i.e. half disc neighborhood for
every bdy point.)
What if function is holomorphic on an annulus (Laurent series). Standard talk about isolated singularities.
Then I was asked if I knew anything about the classical theory of functions' Riemann surfaces with function elements, or conformal
mapping groups on the unit disc, modular forms, etc. I cautiously said I didn't know much, but I may be able to answer a concrete
question. They didn't press the point.
Real Analysis
I said I used big Rudin to review, which invoked a brief comparison of the merits of two constructions of the Lebesgue measure
(Riesz representation theorem vs. the standard machinery from outer measure to positive measure by taking "well-dividing" sets.)
Definition of measure, outer measure on the way.
L^1 class, Lebesgue dominated convergence theorem (I wanted to sketch a proof but was shot down before I could start). Does L^1
convergence imply a.e. convergence? Gave standard counterexample and discussed relations between convergence in L^p, a.e., in
measure.
What is a distribution? I defined C^r metric on the smooth, compactly supported functions on R^n and said distributions are linear
operators that are bounded for all C^r. I commented that they are nice for doing PDE, but before I could elaborate, the analysis
section was declared over.
Algebra
Structure theorem of finitely generated Abelian groups. (I did the fundamental theorem of finitely generated modules over PID,
proved via reducing to the Smith normal form, then let R = Z.)
Is the free component uniquely given? (Oops, no -- Z x Z_2 is already a counterexample.)
Is the torsion component uniquely given? (Yes.)
State the Sylow theorems. Can you prove them? (I got confused in the second part -- after a while Szabó told me not to worry about
it and go on.) In the end we discussed an exercise he told me earlier about: prove that groups of order 160 are not simple [if
2-Sylow is not normal, G acts nontrivially on the 5 2-Sylows]; what can you tell about their structure [semidirect product one way
or another]. He had refused my attempt to blow it away with Burnside (yes, even if I can prove Burnside).
Talk about simple groups. I mentioned their significance in inductive proofs; composition chains, Jordan-Hölder; and started to
explain about the classification of finite simple groups. I wanted to give a survey of the infinite classes, but was stopped at
A_n and asked to sketch a proof that it's simple. I gave the one with the 3-cycles. At this point I was told that since algebraic
topology also involves algebra, we don't need to do any more of it.
Differential Topology
I said I studied the Milnor: h-cobordism Theorem and the Milnor-Stasheff: Characteristic classes.
This resulted in a discussion of the h-cobordism theorem, right from the point of defining Morse functions and why they exist; I
referenced Thom's jet transversality theorem. I sketched the Morse homology and cohomology, existence of self-indexing Morse
functions, elimination of critical points of index 0 and n.
Then I gave a somewhat jumpy and disorganized roadmap of the proof of the h-cobordism theorem, including elimination of critical
points of index 1 and n-1, the algebraic lemmas (basis with cancelling pairs, transitionbetween integer bases via elementary
transforms and their geometric equivalents), first cancellation theorem, second cancellation theorem, Whitney trick). At the
Whitney trick I was shot down.
Application of the h-cobordism theorem? I derived the topological Poincaré conjecture for n>=6, and talked briefly about what
happens in other dimensions, also mentioned exotic spheres.
Characteristic classes: compute the Stiefel-Whitney classes of T^2 # RP^2 (torus connected sum projective plane).
I computed w_1
by discussing which curves were orientation changing and w_2 by using that the top Stiefel-Whitney class is the mod 2 Euler class,
after a hint. Szabó forced me to do it also using the Wu classes and the Wu formula; as I didn't remember it, he stated it for
me.
Embarrassingly, I was surprised to find out that w_1^2 = w_2 for all surfaces (this comes from Wu's formula, but also follows
easily from the fact that every surface can be immersed in R^3 -- I did this in my senior thesis!)
Compute cohomology ring of CP^1 x CP^2 (Künneth). Compute all possible characteristic classes on its tangent bundle (by
naturality; I was allowed to use c(CP^n) = (1+x)^{n+1} without proof). Discussion of how the product formula for the Pontryagin
class may not hold if there is 2-torsion. Comment: for complex bundles, w_{2k} is the mod 2 reduction of c_k (I was able to
prove it by naturality, the splitting lemma & checking for the universal complex line bundle).
Algebraic Topology
Talk about homotopy groups of spheres. I said pi_i(S^n) = 0 for i2, but only elaborated on Steenrod squares. Gave the axioms
(naturality, stability, product formula). Mentioned that every cohomological operation on x \in H^n(X, Z_2) is
f_x^* (\alpha), where \alpha \in H^*( K(Z_2, n), Z_2) is a fixed class and f_x is the homotopy class corresponding to x in
H^n(X, Z_2) = [X, K(Z_2, n)]. Very roughly I described the construction of Sq^n using the cohomology (Leray-Serre) spectral
sequence of the fibration * -> K(Z_2, k) with fibre K(Z_2, k-1).
Final comments
As fate would have it, this exam took place amongst quite chaotic circumstances, the details of which are best left to
oblivion.
This might be a reason why I didn't get very hard questions. The total net time of the exam might be around 1.5 hours.
Before most topics, I was asked which books had I been learning from, and the questions were given accordingly.
There are probably a lot of small questions that I forgot.
Advice: there is no reason to fear the Generals whatsoever, at least with this committee. As you can see, they only asked
questions they could well expect me to know. Also, everyone was really nice and the moment it began I instantly felt much much
better.
If you want to be well-prepared, try to know the basics well rather than cram a lot in your head, even though it never hurts to
have some perspective. It's OK if you forget some proofs, as long as you have the big picture under control, and can do some
calculations/examples/counterexamples.
Very important point: discuss which books/topics to learn well in advance with the corresponding committee member(s). Remember
that some professors may have a pretty unusual idea even about the three basic topics -- read past reviews to see what your
committee members like to ask, and also talk to them!