Abstracts for Conference on "Interactions between higher algebra, manifolds and functor calculus"

The action of the absolute Galois group of the rationals on the little disks operad has several important implications, including for the study of the embedding tower of knots, as shown by Boavida de Brito–Horel.

The generalized Hilbert’s third problem asks about the invariants preserved under the scissors congruence operation: given a polytope P in $\mathbb{R}^n$, one can cut P into a finite number of smaller polytopes and reassemble these to form Q. Kreck, Neumann and Ossa introduced and studied an analogous notion of cut-and-paste relation for manifolds called the SK-equivalence ("schneiden und kleben" is German for "cut and paste").

I will explain a new point of view on the subject of homological stability inspired by aspects of chromatic stable homotopy theory. In this worldview, the usual stabilisation map plays the role of multiplication by $p$ on the $p$-local sphere spectrum $S_{(p)}$, and describing the stable homology (i.e., inverting this map) therefore plays the role of calculating the rational stable homotopy groups of spheres.

Given a smooth manifold with action of a finite group $G$ and an equivariant tangential structure $\Theta$, the spaces of $\Theta$-framed equivariant disk embeddings form a "$\Theta$-disk $G$-presheaf" on the underlying $G$-category of a $G$-symmetric monoidal category of disks in $\Theta$-framed orthogonal representations. This occurs as the $G$-symmetric envelope of a $G$-operad, called the "$\Theta$-framed little disks $G$-operad" $\mathbb{E}_\Theta$, i.e. $\Theta$-disk $G$-presheaves should be thought of as right $\mathbb{E}_\Theta$-modules.