A domino is the union of two adjacent unit squares (or cubes).
Domino tilings in dimension 2 have been extensively studied
and there are several deep and remarkable theorems.
Almost without exception, similar problems in dimension 3 or higher
are much harder.
In this talk we consider the simplest local move
among domino tilings of a given compact region:
a flip consists of removing two dominoes
and placing them back in a different position.
In dimension 2, Thurston proved that any two tilings
of a simply connected region can be joined by a finite sequence of flips.
In higher dimension, the corresponding question is far subtler.
There exists an invariant under flips known as the twist.
The twist can be considered a variant of the Hopf invariant.
For domino tilings in dimension 3, the twist assumes integer values.
For domino tilings in dimension at least 4, the twist assumes values in Z/(2).
In dimension 3, a slab is a box of dimensions 2x2x1;
for slab tilings in dimension 3,
there exists a version of the twist assuming values in Z^3.
For many regions, there are explicit examples of tilings which admit no flip,
and these give us examples of pairs of tilings with the same twist
but in different connected components under flips.
A cylinder of dimension n is the cartesian product of
a contractible region of dimension n-1 (the base) and an interval.
Given a base D, we construct a CW complex, the domino complex,
and study its fundamental group, the domino group.
The base D is called regular if the domino group has a certain structure.
We prove that many bases are regular.
For instance, tileable rectangles of sides at least 3 are regular.
If D is regular it follows that it is almost always (but not always) true that,
if two tilings have the same twist then
they are in the same connected component.
For dimension 3, if D is regular then
the sizes of flip connected components follow a normal distribution.
For each value of the twist, there is a giant component.
For higher dimensions,
the numbers of tilings with each value of the twist are almost equal.
Thus, there are two twin giant components of almost equal size.
This includes joint work with C. Klivans, J. Freire, P. Milet and many others.
The talk will be via Zoom at:
https://utoronto.zoom.us/j/99576627828
Passcode: 448487
arXiv:2007.09500,
arXiv:2007.08474,
arXiv:1912.12102,
arXiv:1702.00798,
arXiv:1410.7693.