Pattern-equivariant Cohomology of Tiling Spaces With Rotations

Pattern-equivariant cohomology theory was developed
by Ian Putnam and Johannes Kellendonk in 2003, for
tilings whose tiles appear in fixed orientations. In
this dissertation, we generalize this theory in two
ways: first, we define this cohomology to apply to
tiling spaces, rather than individual tilings.
Second, we allow tilings with tiles appearing in
multiple orientations - possibly infinitely many.
Along the way, we prove an approximation theorem,
which has use beyond pattern-equivariant cohomology.
This theorem states that a function which is a
topological conjugacy can be approximated arbitrarily
closely by
a function which preserves the local structure of a
tiling space. The approximation theorem is limited to
translationally finite tilings, and we conjecture
that it is not true in the infinite case.

Autorentext

Betseygail Rand was born in Gainesville, Florida, on February3rd, 1978, to proud parents Kenneth and Colleen Rand. Shortlythereafter, she became an Assistant Professor at Texas LutheranUniversity. She is married to Jamaal Fraser, and has one lovelydaughter, so far. Her parents are still proud of her.

Klappentext

Pattern-equivariant cohomology theory was developedby Ian Putnam and Johannes Kellendonk in 2003, fortilings whose tiles appear in fixed orientations. Inthis dissertation, we generalize this theory in twoways: first, we define this cohomology to apply totiling spaces, rather than individual tilings.Second, we allow tilings with tiles appearing inmultiple orientations - possibly infinitely many.Along the way, we prove an approximation theorem,which has use beyond pattern-equivariant cohomology.This theorem states that a function which is atopological conjugacy can be approximated arbitrarilyclosely bya function which preserves the local structure of atiling space. The approximation theorem is limited totranslationally finite tilings, and we conjecturethat it is not true in the infinite case.