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.