Last week I attended a conference at the Perimeter Institute for Theoretical Physics, titled “Hopf Algebras in Kitaev’s Quantum Double Models: Mathematical Connections from Gauge Theory to Topological Quantum Computing and Categorical Quantum Mechanics”. My flight from LAX (Los Angeles) to YYZ (Pearson-Toronto) was delayed due to a maintenance issue, leaving me sitting in the plane on the tarmac for about an hour and a half before takeoff, right close to midnight. My flight to Oxford for the QPL conference (Quantum Physics and Logic) two years ago had a similarly long delay… Hmmmn. Despite this conference being fairly small—40 people on the list of participants—there were several people present that I met at that QPL conference (Ross Duncan, Stefano Gogioso, and Pawel Sobocinski), and a couple more that I met while I was a grad student at Riverside (Tobias Fritz and Derek Wise). It turns out Ross and Stefano are sharing an apartment on the floor above my room. Continue reading
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.
On July 17 I gave a talk at Oxford in the Quantum Physics and Logic 2015 conference. It was recently released on the Oxford Quantum Group youtube channel. There were a bunch of other really cool talks that week. In my talk I refer to Pawel Sobocinski’s Graphical Linear Algebra tutorial on Monday and Tuesday, as well as Sean Tull’s talk on Categories of relations as models of quantum theory. There are benefits to speaking near the end of the conference.
I’m giving a talk tomorrow afternoon at the CSUSB Mathematics department. I will explain some of the stuff I’ve been working on, then shift gears to show how the general machinery can be applied to another mathematical landscape: knots. Sticking purely with knots is a bit restrictive, so I almost immediately skip ahead to the open version of knots, i.e. tangles. Luckily John Armstrong posted an article on arXiv in 2005, where he showed (among other very nice things) that tangle groups really are tangle invariants.
While I probably will not mention it tomorrow, another landscape that is being opened up (in the same sense that tangles open up knots) is Markov processes. Blake Pollard is doing exciting work on this front, and the same general machinery I’ll be talking about should have some interesting things to say about Markov processes as well.
A friend of mine is teaching engineering-type stuff at another university, and he relayed a question to me, which I think he said came from his students. He asked me if I could prove why a certain transformation works. After a false start in interpretation, wherein I proved that it works, which he was already convinced of and for which didn’t need any further corroboration, I think I understand the spirit of his question well enough to provide a (hopefully) satisfactory answer. And since I will be writing the answer up anyway, I may as well blog it up here.
The problem: You have two coordinate frames, each Cartesian, with one frame rotated and displaced relative to the other. Let be the matrix that describes the rotation, and let be the vector that gives the displacement of the origin. A point, , with coordinates given in the second frame (as a column vector) can be expressed in the first frame by the transformation . If you picked the wrong direction to rotate or translate, replace with or with . Regardless, this transformation is annoyingly affine.
The solution: Augment with as a new column, and row filled with zeros in all entries except the last, which will be 1. Let’s call this augmented matrix . Append a 1 to the bottom of , too. Let’s call that . Now will also have a 1 in the bottom entry, and can be read off by ignoring that extra 1. The question of whether this will work is left as an exercise to the reader; it is not difficult to convince oneself it will always work.
The puzzle: Why does adding a dimension like this convert our transformation from one that is affine to one that is linear?
An aside, as to the engineering significance of the problem, suppose you have several rods connected by rotating joints. If you want to know the position of the end of the last rod, relative to the base of the first rod, this kind of transformation, possibly composed several times, would be a way to determine that position. The potential for composing the transformation several times is very good reason why it is so nice that the affine transformation can be converted to a linear one.
The justification: Generalize the problem to the case where the direction of the displacement is fixed, but the magnitude is not. That is, , where . The reason for this generalization is that it makes the affine transformation we are interested in be a special case of this transformation, with , and just as importantly, this transformation is a linear combination of linear transformations! The value of is independent of the rotation, which means we have the side effect of increasing the dimension of the transformation by 1, as seen in the solution.
A calculation analogous to the exercise above shows for this generalization, the matrix representing it is , augmented by . This is not a square matrix, but square matrices are rather convenient, since the vector space in the codomain is the same as in the domain. We can make it square by adding a row. This row will be filled with zeros except the last column, which will be 1, in order to make the determinant 1. This is exactly . Now let’s take a look at why is what it is. In order to translate by , is appended to instead of 1. But as we have already seen, the transformation we are actually interested in is when . Therefore the linearization ought to have the form prescribed by the solution.