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This informal high energy physics talk explores the concept of gradient flow and its implications for understanding the curvature of theory space. William Pannell from Kings College London begins with a comprehensive review of monotonicity theorems and gradient flow fundamentals before delving into recent advances in multiscalar systems. Learn how gradient flow can be reduced to constraint equations on beta function coefficients and discover how these constraints are satisfied across all known loop orders. Examine the connection between the monotonic quantity in this solution and the weak monotonicity conjecture by Fei, Giombi, Klebanov and Tarnopolsky in d=4-ϵ, suggesting potential for strengthening this conjecture. Explore the geometric properties of the natural metric produced by gradient flow on the space of couplings, including an analysis of its Ricci scalar at next-to-leading order.
