_____________________________________________
From: Barth, Nils 
Sent: Tuesday, 22 May, 2007 11:09
To: Sen, Anindya
Subject: RE: What every math PhD should know..

I've actually been writing some such thoughts myself.
John Baez has a page on what every physicist should know
(physics is a small topic).

Briefly:
Math is a sprawling subject, and most pros know only their small corner.
There is a central core though, of highly interrelated topics.
I'd classify that as "discrete and continuous solutions and symmetries",
(discrete groups and Lie groups)
which arose in the 19th century from trying to solve polynomials and
PDEs,
and these overlap in algebraic geometry, which is central in this sense;
(I speculate that "infinite-dimensional non-commutative geometry" would
be a nice "theory of everything" in this area).

(Multivariable) calc and linear algebra
are fundamental tools, both in applications and in math.

Some great theories:
-	Galois theory
-	Lie theory
-	Fourier theory

Some useful (algebraic) tools:
-	(co)homology / homological algebra
-	category theory
-	representation theory

Bourbaki distinguished Algebra, Analysis, and (Geometry/)Topology as
main fields.
Terry Tao distinguishes structured and unstructured math.
Some distinguish continuous and discrete math;
I would rather say that "discrete math": combinatorics, logic/set
theory, and theoretical computer science
(and some aspects of algebra) are less central areas.

For my area (geometry):

There are a number of results / theories in geometry that I think
everyone should know:
theory of manifolds in dimension ... is ...
0, 1 : pretty trivial
2, 3 : geometric
4 : messy
5+ : algebraic

low dimension: every (closed) surface admits a constant curvature metric
(sphere is positive, torus is flat, higher genus are all negative)
(these also correspond to Kodaira dimension in algebraic geometry)
every 3-manifold breaks up into pieces that are geometrizable (there are
8 geometries in dim 3)

The algebraic theory is called "surgery"; you can cut up a manifold into
handles, just as you cut up a CW-complex into cells.
Surgery works topologically in dim 4, but not smoothly, so you get lots
of exotic smooth structures in dim 4.

In general there's a difference between smooth, PL (polyhedral), and
topological definitions of manifolds and maps; indeed, S^7 admits 28
different smooth structures (and more in higher dimension: S^{4n-1} has
lots), and R^4 admits uncountably many (!).

Positive curvature imposes strong constraints on topology;
(these are called "sphere theorems")
negative curvature doesn't (as you see even for surfaces).

		_____________________________________________
		From: Sen, Anindya 
		Sent: Tuesday, 22 May, 2007 09:14
		To: Barth, Nils
		Subject: What every math PhD should know..

		
		Well, what should they know ?

		Have been trying to make a list of topics (and learn up
the new things on it). Reckoned you're pretty good at this sort of
thing. ;-)
		 Want to start if off ?

		I got a similar list from a physcis PhD once. Was a bit
surprised that the majority of them don't learn any GR or QFT.