Activities This Week

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.

I will discuss my joint work with Fu-Tsun Wei from Tsing Hua University in Taiwan.

Let $K$ be the completion of $\mathbb{F}_q(T)$ at $1/T$ and $r\geq 2$ be an integer. In an ongoing project, we study modular units on the Drinfeld symmetric space $\Omega^r$ over $K$, harmonic cochains on the edges of the Bruhat-Tits building of $PGL_r(K)$, and the cuspidal divisor groups of certain Drinfeld modular varieties of dimension $r-1$. In particular, we obtained a higher dimensional analogue of a well-known result of Ogg for classical modular curves $X_0(p)$ of prime level.