Re: differential/algebraic geometry
- From: hale@xxxxxxxxxx
- Date: 20 Jun 2005 20:43:20 -0700
Orion wrote:
> Is differential/algebraic geometry and multilinear algebra basically
> one and the same?
I'm currently reading about algebraic geometry, so I am just a
beginner in that area.
Dieudonne has written a brief article (but very technical) on
the history and development of algebraic geometry which is
worth reading. I don't have the reference at hand though.
It might give some answers to your question.
Of course, differential geometry and algebraic geometry are
not basically the same since the former deals with smooth
functions on R^n while the latter deals with polynomials
on k^n (wher R is reals and k is a field).
However, I get the feeling that they (or their underlying ideas)
are related or are similar. For example, both have a version of
the Riemann-Roch theorem that relates the genus g with algebraic
properties (in the algebraic geometry case).
Also, both consider the germs of functions at a point. Both
consider the tangent space at a point.
One major feature of differential geometry is that the
geometry is considered to be locally Euclidean at a point
and that charts are used to globally patch sections together.
This part I am not sure about: it also appears that algebraic
geometry is considered to be locally affine and that charts are
used to globally patch secitons together. If this is true, I
think this would provide reasons to consider them both to
be related or similar.
-- Bill Hale
.
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