We give a broad-brush account of some milestones in the theory of 3-dimensional manifolds, particularly the Poincaré Conjecture and its proof. We also discuss Thurston’s Geometrisation Programme.
We will generalise the concept of angles in Euclidean space to any arbitrary metric space, via Alexandrov (upper) angles.
We will consider a variation of Hilbert’s hotel, within which guests may not be relocated too far from their current room.
I will present a short argument on an upper bound for \( r(s) \), the Ramsey Number associated with the natural number \(s\).
I present an inefficient yet novel way of recursively counting derangements of a set, and generalise this to counting permutations without short cycles.