The presentation will cover new results on modal clustering. We first provide a unifying view of this topic, which includes two important non-parametric approaches to clustering that emerged in the 1970s: clustering by level sets or cluster tree as proposed by Hartigan; and clustering by gradient lines or gradient flow as proposed by Fukunaga and Hostetler. We will draw a close connection between these two views by 1) showing that the gradient flow provides a way to move along the cluster tree; and 2)  by proposing two ways of obtaining a partition from the cluster tree—each one of them very natural in its own right—and showing that both of them reduce to the partition given by the gradient flow under standard assumptions on the sampling density. We will then establish some consistency results for various methods that have been proposed for modal clustering, including the famous Mean Shift algorithm proposed by Fukunaga and Hostetler in that same article. If time permits, we will conclude by a broader discussion of what is meant by clustering in Statistics, and suggest a set of axioms for hierarchical clustering that lead to Hartigan's definition. 

Joint work with Wanli Qiao (George Mason University) and Lizzy Coda (UC San Diego).