We examine a class of semi-Riemannian manifolds that undergo smooth metric signature change—from Riemannian to Lorentzian—across a hy- persurface with a transverse radical. This class includes physically mo- tivated cosmological models such as the Hartle-Hawking “no-boundary” proposal, in which the universe transitions smoothly from a Euclidean to a Lorentzian phase. We show that these manifolds admit isometric embeddings into higher-dimensional pseudo-Euclidean spaces and, in par- ticular, prove the existence of global isometric embeddings of the canonical model into both Minkowski and Misner spaces. This framework provides a mathematical setting for studying smooth signature change and its role in higher-dimensional and cosmological models.

For G a finite group, Malle’s conjecture predicts the asymptotic growth of the number of G extensions of a fixed global field. In joint work with Ishan Levy, we compute the asymptotic growth of the number of Galois G extensions of F_q(t), for q sufficiently large and relatively prime to |G|.  In the first part of the talk we will introduce Malle’s conjecture and explain how to reduce the above result to a homological stability result for spaces of G bundles. In the second part of the talk we will explain some of the key ideas in the proof of this homological stability result, which uses tools from higher algebra.

TBA