More often than not, seemingly disparate fields of math talk to one another and prove amazing theorems: theorems in topology or combinatorics proven by logicians, theorems in discrete math proven via Fourier analysis. I'm going to try my hardest to keep this page updated whenever I learn some unexpected connection where methods from field A are used to prove a result in a seemingly entirely different field B, either from a friend, professor, or in my own reading and research.
Did you know...
that the fundamental group of a path-connected, locally path-connected, compact metric space is either finitely generated or has cardinality continuum, but this was originally proven using techniques from descriptive set theory?
that for H an infinite-dimensional complex Hilbert space, all homotopy groups of GL(H) are trivial?
that if X is a compact Hausdorff space, then the conjecture "every algebra homomorphism from C(X) into a Banach algebra is continuous" is independent of ZFC?
that if G, H are non-bipartite graphs with a graph homomorphism H → G, then the decision problem "given a graph J and a promise that J → H or J ↛ G, output Yes if J → H and No if J ↛ G" is NP-hard when G has no 4-cycles, but the proof relies on a necessary detour through algebraic topology?
that if F is a set family on [n], then we can get a lower bound for the chromatic number of the Kneser graph of F by figuring out into what dimension of Euclidean space its missing faces complex Σ(F) = {σ ⊆ [n] : no τ ⊆ σ is contained in F} embeds? (5.7-5.8)
that every injective vector polynomial ℂ^n → ℂ^n is surjective, and the proof involves model-theoretic methods?
that the compactness theorem is a surprisingly powerful technique to have in your toolbox as a concrete mathematician?
that we can use the set-theoretic notion of an ultrafilter to prove Tychonoff's theorem, generalize Hall's matching theorem to locally finite graphs, and find finitely additive translation-invariant probability measures on groups?
that the Post correspondence problem reduces to the puzzle game Baba is You, and hence the latter is undecidable?
that some of the closest approaches we have so far to figuring out the Kakeya problem of geometric measure theory is through additive combinatorics?
that Tsirelson's original construction of his eponymous Banach space was motivated by Cohen's forcing arguments in set theory?
that we can prove the first-row periods of Laver tables are unbounded assuming a rank-into-rank exists (basically the strongest known large cardinal axiom not known to be inconsistent with ZFC), we cannot prove it in Presburger arithmetic, but otherwise we have no idea?