On loops intersecting at most once
How many simple closed curves can you draw on the surface of genus g in such a way that no two are isotopic and no two intersect in more than k points? It is known how to draw a collection in which the number of curves grows as a polynomial in g of degree k+1, and conjecturally, this is the best possible. I will describe a proof of an upper bound that matches this function up to a factor of log(g). It is based on an elegant geometric argument due to Przytycki and employs some novel ideas blending covering spaces and probabilistic combinatorics.