January 18, 2002
Committee: P. Ozsvath (chair), E. Stein, P. Yang
Bojan Gornik's Generals
Topics: Algebraic Topology, Differential Geometry
Duration: 1h30
Algebra
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Galois group of Q(sqrt(3),sqrt(2)).
Hilbert Nullstellensatz.
Give examples of maximal ideals in K=RxRxRx... (countably many Rs).
I gave I_1=(0,*,*,*,...) or I_2=(*,0,*,*,...) and so on.
Are there other maximal ideals in K ? Yes.
Name one. Take the maximal ideal that includes the ideal which contains
sequences with
almost all (=all but finitely many) 0's.
Real Analysis
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Talk about the fundamental thm of calculus and its proof.
The Cantor function.
L2 spaces, why are they complete ?
Complex Analysis
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The Riemann mapping thm. What are the ingridients in the proof ?
How does RMT relate to uniformization ?
Name some Phragmen-Lindelof thms.
Topology
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Give two manifolds (of the same dimension) X,Y that are not homeomorphic
but are homotopically equivalent ?
I was led to the answer: X=sphere minus three points and Y=torus minus one
point (they are both homotopically equivalent to the wedge of two
circles).
Why are X,Y not homeomorphic ? Again, after some hinting, it turned out
that their cup product structures on (relative) cohomologies aren't equal.
Give an example of two compact manifolds X,Y ! I gave an answer in 4
dimensions, but was told there's an example in 3 dimensions (lens spaces).
Now, why can't Z+Z+Z+Z be a fundamental group of a 3-manifold ?
There were some hints, but we didn't really get to the answer.
Geometry
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What does it mean that Gauss curvature is intrinsic ?
Give an example of two isometric embeddings of a surface in R3 with
different 2nd fund.forms.
Prove that X,Y from the topology section aren't diffeomorphic "in the
spirit of diff.geom.".