Sucharit Sarkar's Generals
Date and time: 15th May 2006, 1-2:30pm
Zoltan Szabo' (chair), John Conway, Fengbo Hang
Topics: Algebraic Topology, Differential Topology
JHC: Greetings. Shall we start?
ZSz: what do you want to start with? (Topology)
JHC: Topology as a special topic? I thought we did special topics in the end. (Anything then)
Complex Analysis
JHC: What is Morera's theorem? Prove it. (Showed anti-derivative of f, and hence f, are
analytic)
ZSz: What is Riemann mapping theorem? (Proper open sets in C are bianalytic to U) Are you
sure that is what you mean? (Connected and simply connected, oops) Prove it. Where do you
use simple connectedness? (in showing the family of injections is non-empty) For annulus it
is non-empty, so where else? (I never realised that it is used at some other place, saw that
I need it to construct another function later on)
ZSz: Picard theorems statements (just statements, no proofs)
FH: Show roots are continuous function of coefficients for polynomials (proved from Rouche)
JHC: Prove Rouche. (Showed index of a certain curve around origin is 0)
JHC [to ZSz] Shall we move on?
ZSz: One last question. Integrate sinx/x from 0 to infinity. (This got me stuck for some
time, for exp(-ix) was behaving badly on my contour, I finally had to do the whole thing to
get Pi/2) What about |sinx|/x? (goes to infinity)
Algebra
JHC: State Sylow theorems. (stated them) I think that's about that.
JHC: Can there be a group of order 12 with 2 non-isomorphic subgroups of same order? Go
through the possible orders of subgroups. (not for 2,3,4, possibly for 6) Is there an
example for subgroups of order 6? (A4, doesn't work) Start with S3, what can you do to it
to make a group of order 12? (adjoin an element, wrote a group presentation with 3
generators) Do you know semidirect product? (Yes, so can do a semidirect product of Z/2
and S3) What specific semidirect product would you take? (Let me see)
ZSz: I think what John is saying is that you can take the direct product. (aha)
JHC: Right, now I guess I have to ask the standard questions, so show group of order 40
is not simple. (5-Sylow subgroup)
JHC: Let's move on to more classical realms of Algebra, what is Jordon canonical form?
(over alg. closed fields, form with eigenvalues in diagonal, 1 or 0 in diagonal over that,
0 elsewhere) Is that all? (Yes) Write down for 2X2 (Ok, whenever there is 1, the 2
eigenvalues are equal) Prove existence of Jordon decomposition. What about over
non-algebraically closed fields, like say over Q (some arbitrary irreducible polynomials
appear) What about uniqueness (gave statement, was asked very specifically whether the
submodules themselves were unique). Prove.
Real Analysis
FH: How would you construct meaure? (Riesz Representation) What are the conditions on
the functional? What is the domain for the functional? (C(X)) That's only for compact X.
(ok, C_0(X)) I think compactly supported works better. (ok) What do you mean by saying
bounded? What norm?
FH: Define L1. Differences between L1 and Lp for p>1 (I didn't know what he wanted, it
was something about duals) Do you know the dual of Lp (Lq if 11,
have K(G,n)) What is so special? (uniqueness) Why are homotopy groups for n>1 abelian?
What is K(Z,2)? (CP^inf) How do you construct K(Z,3)? (Start with S3 and attach discs to
kill Pi4(S3) and so on) What is Pi4(S3)? (Z/2)
Differential Topology
ZSz: What is the classification of closed 2-manifolds? (for oriented connected sum of
tori, for unoriented connected sum of RP^2) And also S^2 for the oriented (right)
ZSz: This is slightly open-ended, but what do you know about 3-manifolds (stated prime
and irreducible are basically same except S^1xS^2, Pi2 non-trivial means some non-zero
element is representable by an embedded sphere) Do you know Thurston's geometrisation
conjecture? (vaguely) What is it? (some 8 geometries for simply connected 3-manifolds,
named a few of them)
JHC: These manifolds are being considered as just smooth manifolds or something else?
(Well Riemannian manifolds, I don't know what exactly is meant by saying they are
geometries)
ZSz: So what about non-simply connected manifolds? (It is made up pieces which look
like these 8 things) What does that mean? (take a graph, vertices are pieces like this,
and edges are connected sum) You have to consider tori too. (ok) Let's move on
ZSz: What is Poincare conjecture in dimension 3? Higher dimensions? Can homeomorphism
be replaced by diffeomorphism? (no) Example of such exotic sphere (I know they exist in
dimension 7, can't construct them)
ZSz: Can you prove Poincare conjecture for dimension>4? (Yes) Prove it (stated
h-cobordism theorem) Prove h-cobordism theorem (gave outline, had to prove the
elimination of index 1 points in detail, showing exactly where we need simply
connectedness) So now prove Poincare conjecture (proved for n>5, I said take S^n,
remove 2 balls, it is a h-cobordism hence product cobordism, so when we join back
he balls, it is homeomorphic to S^n)
JHC: You said you start with S^n and then got back S^n, probably you mean you start
with the given manifold. (oops, yes I meant that)
ZSz: [to others] Is there anything else anyone wants to ask? [no answer] In that
case, could you go outside while we discuss a little bit?
ZSz: [after a few seconds] We would like to congratulate you, you passed.