In this talk, I will discuss the problem of determining the time to steady state or mixing time of stochastic evolutionary dynamics in finite populations. Such processes lie at the core of evolution and, in the recent past, have been used to model viral populations from the point of view to design drugs to counter them. Here, the time it takes for the population to reach a steady state is important both for the estimation of the steady-state structure of the population as well to determine the treatment strength and duration.
I will show that, for a broad class of such dynamics, the underlying Markov chain mixes fast resolving a central problem in this area. Technically, the result relies on a novel connection between Markov chains arising in such evolutionary dynamics and dynamical systems on the simplex. More generally, this result sheds light on how quickly life could have evolved.