Stability is the prototypical model theoretic dividing line. One interpretation is that a binary relation is stable if it is "close to unary": if the question $(x,y)\in E$ can be answered, at least most of the time, by knowing enough information about $x$, and separately enough information about $y$.

One natural question is asking how this can generalize to ternary (and higher-arity) relations. The connection to hypergraph regularity suggests an approach to identifying ternary stable-like properties, and also that there should be several versions, since a ternary relation could be almost unary, or almost binary, or a combination of these properties.

In this talk, I'll survey some of what we know about several of these "stable-like" ternary notions.

The Lindelöf hypothesis is known to be weaker than the Riemann hypothesis and one way to assess the difference in their strength is to consider what can be said about the zeroes of the zeta function under the assumption of the Lindelöf hypothesis. Viewing this question in the context of zero density estimates, we prove that $N(\sigma,T) \leq T^{\frac{4(5-6\sigma)}{3(3-2\sigma)} + o(1)}$. This improves the currently known estimate conditional on the Lindelöf hypothesis, $N(\sigma,T) \leq T^{2(1-\sigma)+o(1)}$ based on the mean value theorem, for $\sigma$ near $3/4$.