About two months ago I gave a talk at the AMS Fall Western Sectional Meeting. While I could have safely rehashed my QPL talk, I decided to push forward instead. That may or may not have been the best idea – the results were certainly less polished. On the other hand, I was able to describe controllability and observability of a control system in terms of string diagrams. This is something that was painfully missing from my QPL talk’s results in July. Seeing the nontrivial constants in the string diagrams the quantum folk were using provided the key insight, and I wanted to capitalize on it as soon as possible.
The punchline of the AMS talk is that the duality between controllability and observability noticed by Kalman in the late 50s and early 60s can be expressed in terms of a PROP, which is a kind of symmetric monoidal category. In particular, this PROP includes a subPROP of finite-dimensional vector spaces and linear relations, which is basically what Paweł Sobociński deals with here under the name of Interacting Hopf monoids. Okay, so the actual punchline is that the duality Kalman noticed six and a half decades ago between controllability and observability? it’s simply time-reversed bizarro duality.
Bizarro is Sobociński’s term (seen in episode 7 of his blog), but I’m kind of partial to it.
For my inaugural post here, I would merely like to explain the title chosen for this blog. Symmetry is all around us, and many important things can be related to observing symmetries. Two basic symmetries that often go hand-in-hand are reflection and rotation. My language below will have a bias towards two dimensions, but a lot of it does generalize to higher dimensions.
Rotations can be made from reflecting twice, but along different axes (that go through a common point). What happens if you keep adding more reflections through axes that go through that point? Three reflections will give you something that may be a reflection, but it also may not be either a reflection or a rotation. I don’t know of a standard word for a combined reflection and rotation, so I made a suitcase with two equal-sized compartments to put them in: roflection.
What happens if you reflect again? In two dimensions, a reflection with a roflection will always combine to make a rotation, since four reflections about a common point is equivalent to two rotations about that point, which is another rotation. In higher dimension, it is still considered a rotation, though not necessarily a simple rotation. For instance, in 4D, a general rotation leaves a point fixed, and will have two orthogonal planes fixed, in the sense that those planes are closed under the operation of applying that rotation any number of times.
There is a nice pun value to the term ‘roflection’ as well, thanks to the penchant towards abbreviating in texting and internet culture, which include a number of phrases that indicate amusement. ‘Rofl’ is one such abbreviation, and it is my hope that there will be occasion for jocular posts.