We will talk about a recent result of Jeff Kahn, Bhargav Narayanan, and myself stating that the threshold for the random graph G(n,p) to contain the square of a Hamilton cycle is 1/sqrt n, resolving a conjecture of Kühn and Osthus from 2012. For context, we will first spend some time discussing a recent result of Keith Frankston and the three aforementioned authors on a conjecture of Talagrand (a fractional version of Kahn-Kalai "expectation-threshold conjecture").