Jon Pakianathan's generals:
Committee: Hsiang, Mather, Machedon.
Real Analysis:
-State Fourier Inversion Formula.
-Quick sketch of proof of Plancherel thm.
-sequence of functions converge pointwise to a function do integrals
converge? Give simple counter example.
-What is limit as n goes to infinity of 1/n + 1/(n+1) + ... + 1/(2n) ?
-Does the fourier series of a function f in C(T=circle) converge to it
pointwise? (No) Prove there exist plenty such f using Principle
of uniform boundedness(Ban. Steinhaus) and estimates on dini kernel as
in Rudin.
-Define Hardy Littlewood Maximal function.
-State its properties i.e maps Lp to Lp for p>1 takes L1 to weak L1
(mentioned interpolation)
-Give counterexample why does not map L1 into L1 (bump in R, max function
looks like 1/x far from origin)
Functional Analysis:
-What are distributions?
-What are distributions on R with support at the origin?
-Show tdelta'(t)= -delta(t) where delta is the delta "function".
-State Sobolev Embedding Theorem and give idea why useful.
-Define weak and weakstar topologies.
Complex Analysis:
-Prove Annulus theorem.
-Prove Riemann Mapping theorem.
-Prove Montel's theorem.
-State Arzela-Ascoli thm.
-If you know functions (holomorphic) values on boundary of region how
would you give estimates on nth derivative?(Discuss Cauchy integral
formula)
-Why can't 1/z on unit circle extend to holomorphic function on disk?
-If you have real function on bdry of domain how would you "extend" it to
be real part of a holomorphic function on domain? (Do for U say, discuss
Poisson kernel and harmonic conjugates)
- If f holomorphic on C\{0} when is it a derivative of F holomorphic
in C\{0} (Give answer in terms of Laurent series)
Algebra:
-State Fundamental Thm. of fin. gen. Abelian Groups.
-State Fundamental Thm. of fin. gen. modules over a PID.
-Prove subgrp. of a fin. gen. abelian grp. is fin. gen. with <= generators.
-Prove torsion free, fin. gen. module over PID is free.
-Talk briefly about rational canonical form and Jordan canonical form?
For Jordan what condition do we need on the field?
-Prove the alternating group on n letters is generated by the 3 cycles for
n>=3.
-Talk about normal subgroups of symmetric group on n letters. State
simplicity of An for n>=5.
-Sp generated by a p-cycle and a transposition for p prime(prove).
-Discuss sufficient conditions on a polynomial of degree 5 to have
splitting field S5 over Q and prove your statements(irred, 3 real roots
2 nonreal roots)
-What can you say about irreducibility in Z[x] versus in Q[x]. Stated
primitive poly in Z[x] irred in one iff in other. Stated Gauss lemma
which is what they wanted to hear.
Algebraic Topology:
-Define a graph. What sort of group is its fundamental group?(free)
-Prove subgroup of a free group is free.(Use fundamental grp techniques)
-OK, say we have figure 8 so Pi1 is free group on 2 gen. So every subgrp.
is free is it necc finitely generated? (No. exhibit a cover with
nonfinitely generated Pi1)
-Compute homology of RPn, Cohomology rings of RPn,CPn.
-State Universal Coefficient Theorems.
-Use Seifert Van-Kampen Theorem to compute fundamental grp of surface
of genus 2.
- Discuss Hopf fibration S1->S3->S2. Derive it from U(k)->Vn,k->Gn,k
bundle. Why do we know it's a Serre fibration? (Property of being
Serre fibration is local and locally a bundle map is just a proj.)
Calculate Pi3(S2) from this.
-Describe hopf map h:S3->S2 explicitly i.e as attaching map of 4 cell
to CP1 to get CP2. Notice h-1(D+) and h-1(D-) are solid tori whose union
is S3. Draw them.
-State Poincare Duality, Lefshetz Duality and Alexander Duality.
-Last Question(For "Fun" from Hsiang): How would you start determining
relationships for A,B,N where N is a "hypersurface" in Sn which has
Sn-N=union of A and B.(For example N=bdryA=bdryB so can use Lefshetz.)
General Comments:
In generals the committee won't usually want to see the whole detailed
proof to something so as you prove something will usually say I'll
believe that etc. so proofs take much shorter than if you did them
fully. Also don't worry too much if you get stuck anytime your
committee will help you out!!