In this talk, we consider the symmetric inclusion process on a general finite graph. Our main result establishes universal upper and lower bounds for the spectral gap of this interacting particle system in terms of the spectral gap of the random walk on the same graph. In the regime in which the gamma-like reversible measures of the particle system are log-concave, our bounds match, yielding av ersion for the symmetric inclusion process of the celebrated Aldous’ spectral gap conjecture – originally formulated for the interchange process and proved by Caputo, Liggett and Richthammer (JAMS 2010). Finally, by means of duality techniques, we draw analogous conclusions for an interacting diffusion-like unbounded conservative spin system known as Brownian energy process, which may be interpreted as a spatial version of the Wright-Fisher diffusion with mutation. Based on a joint work with Seonwoo Kim (KIAS, South Korea).