Examens pour candidats au PhD

Bonjour,

vous savez, certainement, que les candidats au PhD (aux Etats-Unis) doivent passer une sorte d'examen oral devant un jury. Les questions sont diverses et variées. J'ai trouvé, sur internet, quelques questions posées à un candidat en 2005 à l'Université Princeton.

Je me demande combien de nos doctorants sont capables de répondre à de telles questions?

Bonne lecture :



May 5, 2005, 1:00 pm -- 2:15 pm

Committee: MacPherson(Chair); Gunning; Pandharipande

Special Topics: Algebraic Geometry; Algebraic Topology

Complex Analysis:(Gunning)

(1) If you have a function on the unit disk which is holomorphic outside the origin, what can you say about it?
(2) What's the behavior near an essential singularity?
(3) Give an example of an essential singularity and prove it.
(4) Suppose a function is holomorphic on an annulus, what does its Laurent series look like?


Algebra:

(1) Over the finite field ${\mathbb F}_p$, are there irreducible polynomials of each degree?
(2) If you have a $Z/5$ action on a complex vector space, what does this action look like?
(3) What about an $S_3$ action?
(4) Classify groups of order 8.


Real Analysis: (Gunning)

(1) We have various notions of convergence for a sequence of functions, say pointwise, $L^1$, uniform, etc. What are the relations between them?
(2) $L^1$ convergence implies...


Algebraic Geometry: (Pandharipande)

(1) Consider a cubic surface in $P^3$, what are its Hodge numbers?
(2) What's its intersection form?
(3) Give an example of a genus 4 curve.


Algebraic Topology: (Mostly MacPherson)

(1) What's the relation between $\pi_1$ and $H_1$?
(2) Are there any space with trivial $H_1$ but nontrivial $\pi_1$?
(3) How do you construct a space with $\pi_1$ isomorphic to a given group?
(4) If the Chern class of a complex line bundle is trival, is the bundle trivial?
(5) What's the first Stiefel-Whiteney class of the tangent bundle to the Klein bottle.
(6) What's the cohomology ring of the Grassmannian?
--- I wrote down in terms of Chern classes of the canonical bundle.
(7) Give a projective embedding of Grass(2 planes in $C^4$).
--- Using Plucker coordinates.
(8) What does it look like?
--- It's a quadric hypersurface in $P^5$.