Subject: [HM] Weil conjectures and algebraic topology
From: Volker Eisermann (eiserman@math.uni-bonn.de)
Date: Fri Mar 31 2000 - 05:01:13 EST
I would like to make a remark on Colin McLartys challanging answer
to my posting: "20th century mathematics and knot theory":
> Well, maybe more famous are the Weil conjectures, which Weil
> and Serre and Grothendieck understood as algebraic topology (perhaps
> Grothendieck had the greatest faith in this view, and Weil the least).
> Algebraic topology became cohomology, and all kinds of objects have
> cohomology--topological spaces (in the point set sense), algebraic
> varieties, groups, field extensions, everything.
>
That's very interesting and I would like to hear more about this.
I have never dreamed of considering the Weil conjectures as
algebraic topology. It would be great, if you can give me a reference for
Weil/Serre/Grothendieck having this point of view.
Let me just explain, how I wanted to use the term algebraic topology:
I did not want to make any claims about something like
"the true nature of algebraic topology", or about what algebraic topology
should be, but about algebraic topology as a part of reality:
There are textbooks on algebraic topology (like Dold, Spanier, Bredon),
there are courses at universities on algebraic topology, there are
conferences on algebraic topology, at general conferences (like ICMs)
there are sections on algebraic topology, there are specialized journals
(like "Topology"), there is the AMS-subject-classification ... So, I
want to take into account all this things, getting a much more traditional
point of view on algebraic topology.
From this traditional point of view the Weil-conjectures should just be
described as follows: Weil had the vision that one could apply a methode
from algebraic topology (the Lefschetz-trace-formula) to a problem, that
belongs to an intersection of algebraic geometry and algebraic number-
theory.
So I want to distinguish between something beeing the point of view of
some great individual mathematicians (even if they are as great as
Weil/Serre/Grothendieck) and a point of view generally adopted
(and used!) by the mathematical community.
I would appreciate it very much, if you could write more about this
things. Anything about Weil, Serre or Grothendieck is quite interesting.
And what they thought about a topic as important as the Weil conjectures
is especially interesting. Please write more about this.
Best regards,
Volker Eisermann
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