Appears in collection : Not Only Scalar Curvature Seminar

In this talk, I will discuss two results that illustrate some counterintuitive metric behavior, related to positivity of scalar curvature, for total spaces of circle bundles over large manifolds: (1) For every dimension $d\geq 4$, there exists closed enlargeable $d$-manifolds possessing nontrivial circle bundles over them, whose total spaces admit metrics of positive scalar curvature. This answers a question posed by Gromov. To construct these examples, we use Donaldson-Lefschetz symplectic divisors. (2) For every dimension $d\geq 4$, and constants $C$, $\varepsilon > 0$, there exists a closed Riemannian $d$-manifold $M$ that is a total space of a circle bundle over a large manifold, such that (a) the Urysohn $(d-2)$-width of $M$ is bounded below by $C$, and (b) the volumes of concentric unit balls inside the universal cover of balls of radius 2 in $M$, are bounded above by $\varepsilon$. These provide counterexamples to the macroscopic version of Gromov's Urysohn width conjecture, posed by Alpert-Balitskiy-Guth. To establish these examples, we use a novel estimate on the codimension two Urysohn width of circle bundles over $\mathbb Z/2\mathbb Z$-enlargeable $\mathrm{Pin}^{-}$-manifolds. The talk is based on joint works with Aditya Kumar.