  We consider the estimation of multiple modes of a (multivariate) density. We
start by proposing an estimator of the $H_0$ persistence diagram. We then
derive from it a procedure to estimate the number of modes, their locations and
the associated local maxima. For large classes of piecewise-continuous
functions, we show that these estimators achieve nearly minimax rates. These
classes involve geometric control over the discontinuities set and differ from
commonly considered function classes in mode(s) inference. Although the global
regularity assumptions are stronger, we do not suppose regularity (or even
continuity) in any neighborhood of the modes.
