I am interested in a few different topological invariants for tiling spaces and the interplay between them, namely cohomology, K-theory, (etale) fundamental group, and character variety.
Papers:
New results on tilings via cup products and Chern characters on tiling spaces, with Jonathan Rosenberg and Rodrigo Treviño, Algebr. Geom. Topol., to appear.
K-theory of two-dimensional substitution tiling spaces from AF-algebras, J. Geom. Phys. 194 (2023), 105016.
I wrote a script in Sage that computes, in dimension 2, the K-groups and induced maps I am interested in. The examples that have been computed include the chair (and a few of its variants), half-hex, Robinson triangle (Penrose), Tübingen triangle, Danzer's 7-fold (and a few variants from Gähler--Kwan--Maloney), Ammann A2, Ammann A5, and some of the metallic mean Wang tilings.
There's also a function that draws these tilings, provided that it is given a patch.
Talks:
The étale fundamental group of substitution tiling spaces, Huntonfest, Durham University, September 2025.
The étale fundamental group of substitution tiling spaces, Directions in Aperiodic Order, BIRS, July 2025.
Étale topology for substitution tiling spaces, Grimm Network, University of Glasgow, December 2024.
K-theory of (rank 2) substitution tiling spaces, Groups and Dynamics Seminar, The University of Texas at Austin, January 2023.
K-theory of two-dimensional substitution tiling spaces from AF-algebras, 36th Summer Topology Conference, University of Vienna, July 2022.