In a riemannian $n$-manifold $M$, consider non-contractiblethose that cannot be continuously deformed to a point closed curves; their shortest length is the systolefrom Greek συστολή, “contraction” of $M$. Consider also all embedded hypersurfaces that can not be written as the boundary of an $n$-dimensional domain in $M$; call the infimum of their $(n-1)$-dimensional volumes the cosystole of $M$. With Hannah Alpert and Larry Guth, we proved that for any $\varepsilon>0$, any closed riemannian $n$-manifold $M$ of bounded local geometrythe sectional curvatures are between $-1$ and $1$, and the injectivity radius is at least $1$; this can always be achieved by scaling up the metric obeys the following systolic estimate:
\[\text{systole}(M) \cdot \text{cosystole}(M) \le c_{n,\varepsilon} \text{volume}(M)^{1+\varepsilon},\]as long as the left-hand side is finite ($\iff$ the first cohomology $H^1(M; \mathbb{Z}/2)$ is non-trivial).