My Generals Exam
May 5,2005,1:00pm--2:15pm
Committee: MacPherson(Chair);Gunning;Pandharipande
Special Topics: Algebraic Geometry;Algebraic Topology
Complex Analysis:(Gunning)
If you have a function on the unit disk which is holomorphic outside the origin, what can you say about it?
---I said 3 types of singularities.
What's the behavior near an essential singularity?
---Big Picard.
Give an example of an essential singularity and prove it.
---exp(1/z).
Suppose a function is holomorphic on an annulus, what does its Laurent series look like?
---It can be written as a sum of two functions. One is holomorphic near infinity and the other holomorphic near 0.
Algebra:
(P)Over the finite field F_p, are there irreducible polynomials of each degree?
---Yes and I computed the number of them.
(M)If you have a Z/5 action on a complex vector space, what does this action look like?
---The vector space splits into 1-dim weight spaces.
(M)What about an S_3 action?
---I described representations of a general dihedral group.
(P)Classify groups of order 8.
---abelian,dihedral and quaternion.
Real Analysis: (Gunning)
We have various notions of convergence for a sequence of
functions, say pointwise, L^1, uniform, etc. What are the relations
between them?
---I state the relations and give examples to show those are all the implication relations.
L^1 convergence implies...
---a.e. convergence of a subsequence. And I gave the proof.
Algebraic Geometry: (Pandharipande)
Consider a cubic surface in P^3, what are its Hodge numbers?
---The only interesting number is h^(1,1)=7.
What's its intersection form?
---diag(1,-1,-1,-1,-1,-1,-1).
Give an example of a genus 4 curve.
---I write down a hyperelliptic one.
Are there any non-hyperelliptic ones?
---Yes. Compute the dimension of moduli spaces.
Give a concrete example.
---I thought about complete intersection of two surfaces in P^3,
but somehow convinced myself that it wouldn't work. Actually there are,
given by the complete intersection of a cubic and a quadric.
Algebraic Topology: (Mostly MacPherson)
What's the relation between \pi_1 and H_1?
---Abelianization.
Are there any space with trivial H_1 but nontrivial \pi_1?
---Pick any space with \pi_1 a non-abelian simple group.
How do you construct a space with \pi_1 isomorphic to a given group?
---Pick one loop for each generator and attach one 2 cell for each relation.
If the Chern class of a complex line bundle is trival, is the bundle trivial?
---Yes. And I gave a proof using Cech cohomology.
What's the first Stiefel-Whiteney class of the tangent bundle to the Klein bottle.
---I computed the mod 2 cohomology of Klein bottle and said this
w_1 is non-zero. But one cann't distinguish nonzero mod 2 classes.
(P)What's the cohomology ring of the Grassmannian?
---I wrote down in terms of Chern classes of the canonical bundle.
(P)Give a projective embedding of Grass(2 planes in C^4).
---Using Plucker coordinates.
(P)What does it look like?
---It's a quadric hypersurface in P^5.