Supplementary Information. Undecidability of the spectral gap

This is the supplementary information associated to the article: T.S. Cubitt, D. Perez-Garcia, M.M. Wolf, Undecidability of the spectral gap, Nature 528, 207-211 (2015), doi:10.1038/nature16059

Linear non-local diffusion problems in metric measure spaces

The aim of this paper is to provide a comprehensive study of some linear non-local diffusion problems in metric measure spaces. These include, for example, open subsets in ℝN, graphs, manifolds, multi-structures and some fractal sets. For this, we study regularity, compactness, positivity and the spectrum of the stationary non-local operator. We then study the solutions of linear evolution non-local diffusion problems, with emphasis on similarities and differences with the standard heat equation in smooth domains. In particular, we prove weak and strong maximum principles and describe the asymptotic behaviour using spectral methods.

On the size of the fibers of spectral maps induced by semialgebraic embeddings

Let S(M) be the ring of (continuous) semialgebraic functions on a semialgebraic set M and S*(M) its subring of bounded semialgebraic functions. In this work we compute the size of the fibers of the spectral maps Spec(j)1:Spec(S(N))→Spec(S(M)) and Spec(j)2:Spec(S*(N))→Spec(S*(M)) induced by the inclusion j:N M of a semialgebraic subset N of M. The ring S(M) can be understood as the localization of S*(M) at the multiplicative subset WM of those bounded semialgebraic functions on M with empty zero set. This provides a natural inclusion iM:Spec(S(M)) Spec(S*(M)) that reduces both problems above to an analysis of the fibers of the spectral map Spec(j)2:Spec(S*(N))→Spec(S*(M)). If we denote Z:=ClSpec(S*(M))(M N), it holds that the restriction map Spec(j)2|:Spec(S*(N)) Spec(j)2-1(Z)→Spec(S*(M)) Z is a homeomorphism. Our problem concentrates on the computation of the size of the fibers of Spec(j)2 at the points of Z. The size of the fibers of prime ideals "close" to the complement Y:=M N provides valuable information concerning how N is immersed inside M. If N is dense in M, the map Spec(j)2 is surjective and the generic fiber of a prime ideal p∈Z contains infinitely many elements. However, finite fibers may also appear and we provide a criterium to decide when the fiber Spec(j)2-1(p) is a finite set for p∈Z. If such is the case, our procedure allows us to compute the size s of Spec(j)2-1(p). If in addition N is locally compact and M is pure dimensional, s coincides with the number of minimal prime ideals contained in p. © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.