| Abstract: The study of monotonicity within renormalization group flow has attracted considerable attention over the years, and gradient flow is the strongest possible version of monotonicity one could demand. After a brief review of monotonicity theorems in general and gradient flow in particular, I will detail the advances in the study of gradient flow in multiscalar systems. Perturbatively, gradient flow can be reduced to a set of constraint equations on the coefficients appearing in the beta function, and using recent results for the generic multiscalar beta function I will show that these constraints are satisfied through all known loop orders. The monotonic quantity appearing in this solution is connected to a weak monotonicity conjecture by Fei, Giombi, Klebanov and Tarnopolsky in d=4-ϵ, suggesting that this conjecture can be strengthened. Interestingly, gradient flow produces a natural metric on the space of couplings, and I will investigate the geometric properties of this metric by writing down its Ricci scalar at next-to-leading order. Link to the Event Video |