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The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
4
votes
Characteristic polynomials of trees and E8
Using sage, and up to 16 vertices included, one finds only two trees satisfying $|det(A+2I)|=1$, namely $E_8$ and $E_{10}$.
EDIT: I also confirm that there are 18 such trees with 18 vertices. Here th …
1
vote
Computation of the Euler characteristic of a specific real variety
One possible way to compute Euler characteristic is to use its properties:
$\chi$ is additive on disjoint unions
$\chi$ is multiplicative on fibrations
$\chi$ of the point is $1$
So one has to eit …