Bill Cordwell writes: << One of my students asked me what my favorite theorem was, a question that I found interesting and surprisingly difficult to answer. So...what's your favorite theorem or proof (and why)
Great question. There's probably not one single favorite, but many I'm equally fond of. The question is slightly muddied for me by the fact that many theorems that were the cat's pajamas when I first learned them have become so familiar by now that their wonderfulness seems tarnished by time. But a few are * Real division algebras occur only in dimensions 1,2,4,8. * The only number fields Q(sqrt(-d)) whose ring of algebraic integers has unique factorization (for d > 0) are d = 1, 2, 3, 7, 11, 19, 43, 67, 163. * Bott periodicity, especially for O(oo), the limit of the isometry groups O(n) of R^n: (I.e., O(oo) = the union of all the O(n)'s with O(n) a subset of O(n+1).) Bott proved: The homotopy groups pi_k(O(oo)) depend only on k mod 8: k mod 8: 0 1 2 3 4 5 6 7 ____________________________________________________________ pi_k(O(oo)): Z_2 Z_2 0 Z 0 0 0 Z * Goedel incompleteness * Cantor: Infinitely many infinities * The 3-sphere S^3 admits infinitely-differentiable foliations by surfaces, but no real-analytic foliation. * The Chern-Gauss-Bonnet theorem: The (integral of the Gaussian curvature of a closed surface S) = 2pi * X(S) -- (where X(S) = the Euler characteristic of S) -- and its higher-dimensional analogues. * Classification of regular polytopes in all dimensions: dimension: 2 3 4 5 6 7 8 9 . . . ______________________________________________ # regular polytopes: oo 5 6 3 3 3 3 3 . . . ------------------------------------------------------------------------------ * Classification of surfaces under connected sum, as a commutative semigroup with 0: Two generators T, P with one relation: P + P + P = T + P. ------------------------------------------------------------------------------ * The kissing number in n dimensions (the maximum number of unit spheres that can all be simultaneously tangent to another one while all having disjoint interiors): Known cases to date: dimension: 1 2 3 4 8 24 _______________________________________________ kissing number: 2 6 12 24 240 196560 ------------------------------------------------------------------------------ --Dan