2:00 pm, Wednesday, 21st February


The number variance N_R of a point process in d-dimensional Euclidean space is defined as the variance of the number of points in a centered Euclidean ball of radius R. For the most random of all point processes, the Poisson point process, N_R is proportional to the volume of the ball, i.e. N_R grows like R^{d}. On the other hand, for the least random point processes, the periodic ones, N_R grows like R^{d-1} (on average). Following the seminal works of Stillinger and Torquato from the early 2000s, we say that a point process is hyperuniform (or exhibit long range dependence) if the number variance N_R is o(R^d). Besides periodic point processes, determinental point processes (coming from projections) and (certain) quasi-crystalline point processes are hyperuniform. In this talk I will discuss recent works with Tobias Hartnick (Karlsruhe) and Mattias Bylehn (Gothenburg) which explore different number-theoretic and geometric aspects of hyper-uniformity for quasi-crystalline point processes in Euclidean spaces, in Heisenberg geometries and in hyperbolic spaces, and highlight a few surprises on the way. All notions from point process theory will be defined.


Michael Bjorklund

Research area

Number Theory


Chalmers University


Wednesday 21 Feb 2024, 2.00 pm


Room 4082 (Anita B. Lawrence Center)