In this talk I will survey current results on the long-time existence and behavior of Ricci flows in dimensions 2, 3 and higher. Moreover, I will point out analogies with construction techniques for Einstein metrics.
In dimension 3, the Ricci flow together with a certain surgery process has been used by Perelman, amongst many others, to establish the Poincaré and Geometrization Conjectures. Despite the depth of this result, a precise description of the long-time behavior of this flow has remained unknown. For example, it was only conjectured by Perelman that it suffices to carry out a finite number of surgeries and that the geometric decomposition of the manifold is exhibited by the flow as $t \to \infty$. Recently I was able to confirm Perelman's first conjecture and I partially answered his second one.

I will first give a brief overview of Ricci flows with surgery and explain the finite surgery theorem. Next, I will present long-time existence results in dimensions 4 and higher and describe possible further directions in this field.

Since the foundational results of Chung-Graham-Wilson on quasirandom graphs over 20 years ago, here has been a lot of effort by many researchers to extend the theory to hypergraphs. I will present some of this history, and then describe our recent results that provide such a generalization in some cases. One key new aspect in the theory is a systematic study of hypergraph eigenvalues first introduced by Friedman and Wigderson. This leads to the study of various extremal questions
on hypergraphs, for example, spectral Tur an problems and spectral packing
problems. This is joint work with Peter Keevash and John Lenz.

Although the Ricci flow with surgery has been used by Perelman to solve the Poincar�© and Geometrization Conjectures, some of its basic properties are still unknown. For example it has been an open question whether the surgeries eventually stop to occur (i.e. whether there are finitely many surgeries) and whether the full geometric decomposition of the underlying manifold is exhibited by the flow as $t \to \infty$.

In this talk I will show that the number of surgeries is indeed finite and that the curvature is globally bounded by $C t^{-1}$ for large $t$. This confirms a conjecture of Perelman. Using the new curvature bound, it is possible to give a more precise geometric picture of the long-time behavior of the flow.