In 1983 Hartle and Hawking put forth that signature type-change may be conceptually interesting, leading to the so-called no-boundary proposal for the initial conditions for the universe, which has no beginning because there is no singularity or boundary to the spacetime. But there is an origin of time. In mathematical terms, we are dealing with signature type-changing manifolds where a positive definite Riemannian region is smoothly joined to a Lorentzian region at the surface of transition where time begins.

We utilize a transformation prescription to transform an arbitrary Lorentzian manifold into a singular signature-type changing manifold. Then we prove the transformation theorem saying that locally the metric \tilde{g} associated with a signature-type changing manifold (M, \tilde{g}) is equivalent to the metric obtained from a Lorentzian metric g via the aforementioned transformation prescription. By augmenting the assumption by certain constraints, mutatis mutandis, the global version of the transformation theorem can be proven as well.

The transformation theorem provides a useful tool to quickly determine whether a singular signature type-changing manifold under consideration belongs to the class of transverse type changing semi-Riemannian manifolds.

Remez-type inequalities bound the suprema of low-degree polynomials over some domain K by their suprema over a subset S of K. Existing multi-dimensional Remez inequalities bear constants with strong dependence on dimension. In this talk we will prove a dimension-free Remez-type estimate when K is the polydisc D^n and S is from a certain class of discrete subsets. As a direct consequence we also obtain a Bohnenblust-Hille-type inequality for products of cyclic groups, which in turn has consequences for learning algorithms. Based on joint work with Lars Becker, Ohad Klein, Alexander Volberg, and Haonan Zhang.