Activities This Week

L-functions of motives over Spec Z encompass both L-functions associated to smooth projective varieties over Q and zeta functions of schemes over Spec Z. In this talk I will state a conjecture about special values of such L-functions. This conjecture shares a kinship with Poincaré duality or Artin-Verdier duality. It is equivalent to the conjunction of the conjectures of Beilinson, Soulé and Tate and allows to take advantage of the fact that motives over Spec Z form a triangulated category.

Multiscale substitution tilings are a new family of tilings of Euclidean space that are generated by multiscale substitution rules. Unlike the standard setup of substitution tilings, which is a basic object of study within the aperiodic order community and includes examples such as the Penrose and the pinwheel tilings, multiple distinct scaling constants are allowed, and the defining process of inflation and subdivision is a continuous one. Under a certain irrationality assumption on the scaling constants, this construction gives rise to a new class of tilings, tiling spaces, and tiling dynamical systems, which are intrinsically different from those that arise in the standard setup. In the talk, I will describe these new objects and discuss various structural, geometrical, statistical, and dynamical results. Based on joint work with Yaar Solomon.