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From: Sen, Anindya
Sent: Tuesday, 22 May, 2007 11:48
To: Barth, Nils
Subject: RE: What every math PhD should know..
Good stuff, good stuff.
I once wrote up a "something every math major ought to know".
I have this idea of math education as having these steps. Climbing up
these steps is a pain, but once you've done this, you have relatively
easy access to large swathes of math.
So, in undergrad,-- you need multivariable calc and linear algebra (with
rigorous proofs), point set topology ( with metric spaces, Banach
spaces etc.)
and an intro. to abstarct algebra( at least at the level of Artin).
Maybe an introductory course in complex variables as well.
I think the major climb at the next step is, indeed:
Some useful (algebraic) tools:
* (co)homology / homological algebra
* category theory
* representation theory
An enormous amount of modern pure math uses these and
you are seriously handicapped (like me) if you are not very
comfortable with these tools and ideas.
They should try to focus math PhDs on picking up these
tools, rather than randomly wandering around in math space, like at U of
C.
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From: Barth, Nils
Sent: Tuesday, May 22, 2007 11:09 AM
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