We show the validity of the minimal model program (MMP) for threefolds in characteristic 5.

We consider the case of scattering by several obstacles in ${\mathbb{R}}^d$ for $d\ge 2$. In this setting, the absolutely continuous part of the Laplace operator Δ with Dirichlet boundary conditions and the free Laplace operator ${\mathrm{\Delta }}_0$ are unitarily equivalent. For suitable functions that decay sufficiently fast, we have that the difference $g(\mathrm{\Delta })-g({\mathrm{\Delta }}_0)$ is a trace-class operator and its trace is described by the Krein spectral shift function. In this article, we study the contribution to the trace (and hence the Krein spectral shift function) that arises from assembling several obstacles relative to a setting where the obstacles are completely separated. In the case of two obstacles, we consider the Laplace operators ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ obtained by imposing Dirichlet boundary conditions only on one of the objects. Our main result in this case states that then $g(\mathrm{\Delta })-g({\mathrm{\Delta }}_1)-g({\mathrm{\Delta }}_2)+g({\mathrm{\Delta }}_0)$ is a trace-class operator for a much larger class of functions (including functions of polynomial growth) and that this trace may still be computed by a modification of the Birman–Krein formula. In case $g(x)=x^{\frac{1}{2}}$, the relative trace has a physical meaning as the vacuum energy of the massless scalar field and is expressible as an integral involving boundary layer operators. Such integrals have been derived in the physics literature using nonrigorous path integral derivations and our formula provides both a rigorous justification as well as a generalization.

We prove that many spaces of positive scalar curvature (psc) metrics have the homotopy type of infinite loop spaces. Our result in particular applies to the path component of the round metric inside ${\mathcal{R}}^{+}(S^d)$ if $d\ge 6$.

To achieve that goal, we study the cobordism category of manifolds with positive scalar curvature. Under suitable connectivity conditions, we can identify the homotopy fiber of the forgetful map from the psc cobordism category to the ordinary cobordism category with a delooping of spaces of psc metrics. This uses a version of Quillen’s Theorem B and instances of the Gromov–Lawson surgery theorem.

We extend some of the surgery arguments by Galatius and the second-named author to the psc setting to pass between different connectivity conditions. Segal’s theory of Γ-spaces is then used to construct the claimed infinite loop space structures.

The cobordism category viewpoint also illuminates the action of diffeomorphism groups on spaces of psc metrics. We show that under mild hypotheses on the manifold, the action map from the diffeomorphism group to the homotopy automorphisms of the spaces of psc metrics factors through the Madsen–Tillmann spectrum. This implies a strong rigidity theorem for the action map when the manifold has trivial rational Pontryagin classes.

A delooped version of the Atiyah–Singer index theorem proved by the first-named author is, moreover, used to show that the secondary index invariant to real K-theory is an infinite loop map. These ideas also give a new proof of the main result of our previous work with Botvinnik.