We consider the symmetric inclusion process on a general finite graph. In the log-concave regime, we establish universal upper and lower bounds for the spectral gap of this process in terms of the spectral gap of the single-particle random walk, thereby verifying the celebrated Aldous' conjecture, originally formulated for the interchange process. Next, in the general non-log-concave regime, we prove that the conjecture does not hold by investigating the so-called metastable regime when the diffusivity constant vanishes in the limit. This talk is based on joint works with Federico Sau.
