Formal mathematical definition of renormalization group flow I was watching some lectures by Huisken where he mentioned that one-loop renormalization group flow was in some analogous to mean curvature flow.  I have tried reading up the exact definition of what this flow actually is, but could not find anything suitable and was wondering if anyone could explain it to me.
I have tried reading texts on QFT, but I don't really want the physics behind it and I don't want vague descriptions. I just want to know what the precise mathematical definition of the flow is, similar to the one for Ricci flow or mean curvature flow.  Is there a manifold, what is the PDE involved etc.?  I am familiar with the idea of loops in the context of Feynman diagrams and integrals if that helps, so I know what a one-loop Feynman diagram is.
 A: Classical field theories (Lagrangian variational principles) sometimes come in families. The families may be finite dimensional, or also infinite dimensional. One could even take the family to consist of all possible Lagrangians, say with the fields (dependent variables) as well the source and target manifolds (domains of the independent and dependent variables, respectively) fixed. The notion of a symmetry of a single theory is straightforward: a local transformation of the fields that leaves the Lagrangian fixed (up to total derivative terms). The notion of a symmetry of a family of theories is similar: the action of the transformation of the fields must keep any Lagrangian from the family within the same family. When quantizing the family of classical field theories, lets just take it as given that one can lift field transformations from the classical to the quantum level (this involves technical subtleties and is not automatic, but that is not the point here). Ideally, one would like to preserve the symmetries of classical theories, but in general the quantum lift of the classical symmetry may not be a symmetry of the quantum theory. Strictly speaking, one should then say that the symmetry is anomalous. However, when considering quantization as a deformation problem (parameter $\hbar=0$ corresponds to classical, while $\hbar \ne 0$ to quantum), it is natural to also to allow quantum corrections ($\hbar$-parametrized deformations) of the lifted classical symmetries. So the term anomalous symmetry is reserved for those symmetries whose quantum lifts cannot even be quantum corrected. Even more confusingly, the quantization procedure is not unique (in the context of perturbative quantization, specific quantization procedures are referred to as renormalization schemes). Thus, if one can change the quantization procedure to make an anomalous symmetry into a non-anomalous one, then one says that the anomaly in the symmetry can be cancelled.
Now, getting more specific, suppose that the family under consideration has a scaling symmetry (basically, an action by the multiplicative positive reals $\mathbb{R}_+^\times$). Lets call the infinitesimal version of this action the classical scaling flow. Obviously, the scaling flow has an action on the parameters of our family of theories. The quickest definition of renormalization group flow is that it is the quantum corrected lift of the classical scaling flow (provided that the quantization procedure has been chosen to cancel potential anomalies in the scaling symmetry). One could consider specifically the action of the renormalization group flow on the parameters of our family of theories, and call that renormalization group flow as well. The latter meaning is the one encountered most often in the literature and the one appearing in the OP.
To make matters a bit more muddy, a given quantum lift of a classical symmetry strictly speaking depends on the quantization procedure. So the quantum lift changes when the quantization procedure changes (of course restricting ourselves to changes for which the symmetry remains non-anomalous). So some people say that the renormalization group flow (or its restriction to some parameters) is non-trivial only if there is no choice of quantization procedure for which no quantum corrections are needed.
Of course, the discussion I gave in the first paragraph is quite heuristic. When reading about mathematically rigorous treatment of the renormalization group flow, most of the details deal with making what I described mathematically precise. There are different approaches to doing that and the number of technical obstacles is not small, which is what makes such treatments difficult reading for outsiders to the field.
Finally coming back to Ricci flow, one could say that it coincides with the renormalization group flow of a 2-dimensional Euclidean non-linear sigma model (quantized in a reasonable perturbative way), when restricted to the target space-metric, as a parameter of the Lagrangian). One can find at least two mathematically precise approaches to making the above statement rigorous:

Nguyen, Timothy, Quantization of the nonlinear sigma model revisited, J. Math. Phys. 57, No. 8, 082301, 40 p. (2016). ZBL1351.81089. arXiv:1408.4466
Mauro Carfora, Claudio Dappiaggi, Nicolò Drago, Paolo Rinaldi, Ricci Flow from the Renormalization of Nonlinear Sigma Models in the Framework of Euclidean Algebraic Quantum Field Theory, Commun. Math. Phys. (2019). arXiv:1809.07652


