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Abstract: Ricci flow has been extensively studied, and most results are either specific to dimension 3 or valid in any dimension. However, given the potential topological applications, a theory specific to the 4-dimensional situation is desirable. In this discussion, I will present tools and techniques that are unique to the dimension 4.

Together with A. Deruelle, we introduce a notion of stability for orbifold singularities. This notion helps to explain the formation of orbifold singularities along Ricci flow and lets us construct large classes of ancient Ricci flows. In collaboration with K. Naff, we utilize self-duality in dimension 4 to simplify the evolution equations of curvature. Among other things, this approach lets us to uncover intriguing links between Ricci flow and Yang-Mills flow.