this lecture will be about a remarkable law of nature discovered by george polya. consider a particle initially situated at a given point of the d-dimensional integer lattice. suppose that, at each tick of the clock, the particle jumps to a neighboring lattice site, with equal probability of jumping in any direction. polya's law states that the particle returns to its initial position with probability one in dimensions d = 1,2, but with probability strictly less than one in all higher dimensions. thus, a drunk person wandering a city grid will always return to their starting point, but if the drunkard can fly s/he might never come back.