Непрерывное математическое образование

https://sbseminar.wordpress.com/2009/07/28/topology-that-algebra-cant-see/

«Let X be an algebraic variety over ℂ; that is to say, the zero locus of a bunch of polynomials with complex coefficients. We will consider this zero locus as a topological space using the ordinary topology on ℂ. One of the main themes of algebraic geometry in the last century has been learning how to study the topology of X in terms of the algebraic properties of the defining equations.

In this post, I will explain that there are intrinsic limits to this approach; things that cannot be computed algebraically. In particular, I want to explain how from a categorical point of view, we can’t even compute the homology H₁(X,ℤ). And, even if you don’t believe in categories, you’ll still have to concede that we can’t compute π₁(X). This is a very pretty example and it should be more widely known.

Absolutely none of the ideas in this post are original; I think most of them are due to Serre.»

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5.44K viewsedited  Mar 17 at 13:53

https://sbseminar.wordpress.com/2009/07/28/topology-that-algebra-cant-see/

«Let X be an algebraic variety over ℂ; that is to say, the zero locus of a bunch of polynomials with complex coefficients. We will consider this zero locus as a topological space using the ordinary topology on ℂ. One of the main themes of algebraic geometry in the last century has been learning how to study the topology of X in terms of the algebraic properties of the defining equations.

In this post, I will explain that there are intrinsic limits to this approach; things that cannot be computed algebraically. In particular, I want to explain how from a categorical point of view, we can’t even compute the homology H₁(X,ℤ). And, even if you don’t believe in categories, you’ll still have to concede that we can’t compute π₁(X). This is a very pretty example and it should be more widely known.

Absolutely none of the ideas in this post are original; I think most of them are due to Serre.»

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