Singularities of renormalization group flows

We discuss singularity formation in certain renormalization group flows. Special cases are the Ricci Yang–Mills and B$B$-field flows. We point out some results suggesting that topological hypotheses can make RG flows much less singular than Ricci flow. In particular we show that for rotationally symmetric initial data on S²×S¹$S^2\times S^1$ one gets long time existence and convergence of RYM flow, in stark contrast to the case for Ricci flow [S. Angenent, D. Knopf, An example of neckpinching for Ricci flow on Sⁿ⁺¹$S^{n+1}$, Math. Res. Lett. 11 (4) (2004) 493–518]. Other results are given which allow one to rule out many singularity models under strictly topological hypotheses. A conjectural picture of singularity formation for RG flow on 3-manifolds is given.