AMS talk 10/24/15

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.

Grading exams (a modest^H^H^H^Hck proposal)

Exams are some of the most time-consuming parts of being an instructor.  They have to be written, proctored, and finally graded.  Given a class of n students, it seems reasonable to expect that writing a free-response exam would be a O(1) time commitment, proctoring would also be a O(1) time commitment, while grading would be a O(n) time commitment.  Maybe grading speeds up as you get familiar with the particular set of mistakes your class makes, so I might believe something like O(\frac{n}{\log n}) instead of O(n).  In any case, asymptotically, grading takes a much greater share of time consumed than the other portions of the process combined.

With this in mind, I propose the following as a way of reducing the amount of necessary time spent in the grading process, while still giving a reasonable estimate of student understanding.  At least for a game theory class.

Game theory final exam (proposed)

(1) (100 points) Your score on this exam will be based entirely on this one question.  If x is your answer and no one in this class answers with a greater number, 100-x will be your score.  Otherwise, x will be your score.
x = _____

It is interesting to think about the possible strategies one may devise, which, for the student taking the exam, may depend on the class size and/or the student’s grade going into the final exam.  While it is fun to ponder, I will leave that as food for thought for the time being save that discussion for the comments.  Hopefully it will be clear that this isn’t entirely a serious proposal for an actual basis for grading a class, as forcing the entire class into a final exam that somewhat resembles the prisoner’s dilemma would be somewhat cruel.  There are some variations that may or may not be as evil:

Variation 1 (Unlimited collaboration)

The students are told in advance what the exam problem will be.  Thus, they are free to collaborate with as many of their classmates as they would like in order to come up with a strategy.  However, they cannot see what answers are actually submitted by their classmates.  This variation might be more evil.

Variation 2 (Limited collaboration)

The students are paired up and are allowed to discuss strategy only with their partner.  They may be able to see what their partner submits, but they have no information about what anyone else does.

Variation 3 (Multitrack)

The students are given the option to take one of two exams.  The first choice is the proposed exam above.  The wording could be modified slightly to restrict from the entire class to the subset of the class that chooses that choice.  The second choice is a standard final exam.  For the student who thinks, “I just need 50% on the exam to get the grade I want.”, it seems plausible for that student to take the first choice and write x=50 for a guaranteed score and a personal 3 hour savings.

Other thoughts

I thought about this game while grading some calc exams.  It turned out there were quite a few people who incorrectly solved the problem I was grading, yet according to my rubric, their score was exactly the answer they wrote down.  Purely a coincidence, but it sparked the thought that led me to this game.  I showed this game to a few people yesterday, and the first two variations directly came from some of those discussions.  One person I showed immediately declared, “I don’t want to play this game!”.  Between the game and the variations, the multitrack variation seems the most reasonable to even consider actually giving.  Concerning the title of this post – computer printouts (in some contexts) used to display ^H when the user hit the backspace key.  Thus, it is meant to indicate the previous characters are being stricken out.

Alternative Transportation

After finding out I could take advantage of public transportation to get to and from school, I decided to enroll in Alternative Transportation, a program offered by Transportation & Parking Services (TAPS) that provides incentives for people to substitute alternate means of transport over driving.  I ran into a few glitches along the way, but ultimately I succeeded at obtaining a night/weekend parking permit free of charge.  There are other incentives available, some of which depend on what mode of transportation you opt to use.  For instance, people who walk or ride their bicycles to school can take advantage of locker and shower facilities; people who carpool or vanpool can get an emergency ride home when an emergency prevents them from going home on the carpool/vanpool.

Since I had just started riding the bus to school, I decided to apply for Alternative Transportation as a public transportation commuter.  This choice ended up causing me a bit of grief.  TAPS accepts this kind of application for people who have had at least five rides on the bus.  However, TAPS only receives monthly reports from Riverside Transit Agency (RTA).  I had already made twenty trips to/from school by bus when my application was denied — when TAPS got the September report in mid-October, only two of those twenty rides would have appeared.

Instead of waiting until mid-November to reapply, I asked about applying as a walker.  By then I had already walked home from school twice, so I knew it was not an implausible option.  That application was accepted in about four hours.

Another choice I made that caused some grief was the method of delivery of the award:  I asked to pick it up instead of just having it mailed to me.  In the application approval e-mail, TAPS told me they would send another e-mail when the permit was available for pick up.  I waited weeks and never received that e-mail.  Finally, on Monday (ereyesterday) I sent TAPS an e-mail asking when I could pick up my permit.  The response was almost immediate.  With half an hour to my next class, I walked to the TAPS building and picked up my night/weekend parking permit.

My expectation was a permit for the quarter, but my expectations were exceeded.  The permit is a full year permit, valid from 1 July 2015 to 30 June 2016.  Long story short, Alternative Transportation seems to be a decent program when it works.  Getting it to work was a little bit of a headache for me, though the delays were at least as much from me waiting to deal with minor hassles as from TAPS’ side of the process.

Jokes

I’ve been holding onto these jokes for a little while now — one a variation on a well-worn theme, the other something I don’t think I’ve quite seen before.  Naturally, if you’ve seen something similar to the second one before, I would appreciate comments that can point to previous incarnations.

1.  A variation on a theme
There are 10 types of people in the world:
• Those who think this is a binary joke.
• Those who think this is a ternary joke.
• Those who think this is a quaternary joke.

• Those who realize this is a base 10 joke.
• Those who just don’t get this joke.

2.  Something else
I have discovered a truly marvelous punchline, which this joke is too small to contain.

Every day I get in the queue…

… to get on the bus that takes me to U…CR.

It looks like my schedule this quarter will allow for me to take public transportation to and from school.  That means I can avoid purchasing a parking permit.  While it is nice to be making a step towards `going green’, my motivations for taking this step are admittedly more financial than environmental.  As a student at UCR, I can ride on the RTA busses free, so I can save a bit more than 2% of my net income by not buying a parking permit (almost 3%, counting the little bit of gas I save).

It may not be much, but with the uncertainty of what happens after the end of this quarter, I’m happy to be able to cut corners financially, especially since it does not stress my schedule.  I might even be able to take advantage of routes other than the home ⇒ school ⇒ home circuit at some point.  _

QPL 2015

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.

CSUSB colloquium

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.

While driving

Out on the road today I saw some things that made me think. While none of those things were Deadhead stickers on Cadillacs, one thing I did see was a license plate, 4GIV2##. I think it was 258 at the end, but that was not the part I was focused on. I thought, “How cool to have an ID plate in the 4GIV series. And how even more cool would it be to have the plate 4GIV490.” (Matt. 18:21-22). Actually, I thought 4GIV539 would be cool because I misremembered the quote as seventy seven times seven. I’m glad I looked up the reference before writing this. * whistles *

Another thing I saw was a truck for a company called Shipping Experts Consumer Handling Transportation (SECH Transportation). Unfortunately I do not read Hangul, so I can’t really tell what their website says, but you’ve gotta appreciate their enthusiasm. Not having the benefit of seeing their website while on the road, my mind naturally went to the obvious mathematical connection – hyperbolic secant, the reciprocal to a perhaps more familiar shape: hyperbolic cosine, a.k.a. catenary. The catenary is the shape a chain or cord will naturally fall into if its ends are held fixed in a constant gravitational field. It is also the shape (inverted) of a famous US landmark.

Teaching as a Primary Instructor. Day 1.

Today was designed to be an introduction / icebreaker sort of day. I passed around a prop, and had the students introduce themselves to the class when the prop got to them. Once I got the prop back, I asked them if they knew what it was, knowing that none of them would. It was a slide rule. I told them it was a calculator, watching for a moment as the blank looks on their faces turn into dumbfounded looks before telling them a little bit about how slide rules were the calculators used before electronic calculators came on the scene in the 1970s. Of course I brought half a dozen slide rules with me today as part of what was probably the geekiest icebreaker ever. I showed them how to use the C and D scales to multiply 2 by 3, and split the class into groups with a slide rule in each group and gave them the task of multiplying 4 by 3. That was a trick, of course, because the opposite 1 is used for those two multiplications. Then I had them look at the A and D scales, using the hairline to line up with the 5 on the D scale. And I had them do the same with 3 on the D scale. They were pretty quick to notice that the operation there is squaring, despite the initial appearance that 5 takes you to 2.5. After asking what else they could expect to be able to do with a slide rule, given that you can multiply and square numbers, I asked them to find the square root of 2, using the slide rules I handed out. I hope they had fun with that. I know I did.

Still in their groups, I had them talk to each other for about ten minutes to round out the icebreaking in a more… normal? way. I wandered around for a bit, answering questions and addressing concerns, and after the ten minutes was up, I handed them their Problem of the Day. I wanted to make it very light for a very light day, so the questions were: What is my name? What are the names of the people in your group? and What is the name of the prop we used today? I had my name and the words “slide rules” written on the board, so it was mostly a matter of, “Are you paying any attention?” After I collected the PotDs, I pointed out the reason I decided to use slide rules as a prop for the icebreakers – logarithms are what make slide rules work, and they will be learning about logarithms later on in the Precalc class portion of their Summer Bridge ensemble. The point of the geeky icebreaker: I whetted their appetites for things to come. I also explained that the multiplication on a slide rule would be addition if the scales were linear instead of logarithmic, which means logarithms effectively turn a multiplication problem into an addition problem.

By then, they started asking questions, like why I like mathematics. I answered that “mathematics is fun (interruption: frustrating) – yes, it can be frustrating at times, but it is still fun – and beautiful. No, I’m serious.” I think any subject, if you go deep enough into it, has some kind of beauty. Math has lots of really cool things you can look at. Some examples I gave were that we can show \sqrt{2} is irrational… primes are 0% of all whole numbers, but there are just as many primes as whole numbers… just some snippets of cool stuff that would be accessible at the precalc level.

There was another question about what’s the hardest math I’ve dealt with. I told them the most abstract I’ve dealt with is Category theory, but the one with most nitpicky detaily nuissances that I’ve dealt with is Analysis. I don’t think I explained very well. I mean, how do you start to outline what’s so cool about Categories or any branch of Analysis when they’re taking precalc? For the sake of having somewhere to start, I reduced Categories down to manipulating arrows, and I reduced Analysis down to the details for why you can do what you will do in calculus. That is a gross oversimplification, and I don’t think I did a very good job of explaining that much. They seemed kind of bored with my answer to that question. Oh well. The remainder of the time I spent with them was answering questions about course materials and the like.

I suppose I should also note some notable things that happened last week.

Last week the other class I was supposed to teach today got cancelled because there was insufficient enrollment. That kind of sucks because it means my paycheck will be half as much. In anticipation of something like that, along with finances already being tight, I filed the FAFSA the weekend before so I could be eligible for student loans. I’ve never taken out a student loan before, so that’s kind of new territory for me. Don’t ask me how I managed to get to the beginning of my fourth year in a Ph.D. program without taking out any student loans; I would just say it’s a miracle. Turns out, even if I get offered student loans, the earliest they could possibly be disbursed would be somewhere in the middle of September. I was able to get an emergency loan through Grad Division, though. That should at least help tide me over. Basically, finances are stressful.

About the same time I found out about the class cancellation, I also found out the professor I plan to have as my advisor is going to be on sabbatical from January to the end of summer 2014. I was expecting to take my Oral qualifier exam somewhere toward May or June 2014, but the advisor is required to be there in person for the Oral. If I don’t take the Oral by the end of the academic year, the system is set to put an academic hold on registerring for Fall 2014. Effectively, that means I need to take my Oral by December or else get a petition going by December to waive the academic hold for a quarter. My intended advisor thinks I can be ready for the Oral by December, so I just need to put 110% focus on making sure I actually AM ready for it by then. Not to mention the 110% focus I need to put on making sure I pass my last written Qual, which will take place in November. The irony there is that I am going to be the last person in my class year to finish the written Quals, but I just might end up being the first in my class year to do the Oral qual (and thus the first to advance to candidacy). Basically, two major exams in the Fall quarter are stressful, but promise to be rewarding.

I’m still working on learning Hebrew, and I’m supposed to get a new apartmentmate some time this week. Things feel like they are falling apart at times, but other times I know they are falling into place. Where things land may not be comfortable all the time, but there are definite opportunities for growth from the experiences. Also! A few little notes. The Associator animation is not yet in its final state. It works, but there are things that could be better. The other little note – I finally found the spamomatic filter and noticed it deleted and removed 45 comments already. I don’t know what those comments said, so if you made some comments and they never posted, I now know one more place I have to look, when approving comments.

Associator

After more delays that I would like to admit, I finally got around to making this. Perhaps some explanation of what’s going on in the picture is in order. We begin with a category with (all) pushouts, starting with some objects and morphisms arranged as chained spans. A pushout is used to compose the two spans on the left. This gives a new span, L \rightarrow \Gamma'' \Gamma' \leftarrow J, which in turn can be composed with the span to its right via pushout. This is just one way that we can compose the chain of three spans. If we had built up from right to left instead of left to right, we would have \Gamma'' (\Gamma' \Gamma) at the top instead of (\Gamma'' \Gamma') \Gamma.

The associative property is really nice, so it would be good to show those two ways of composing spans are “the same” in some appropriate sense. Spoiler: that sense is “up to isomorphism.” Instead of directly finding them isomorphic to each other, it is easier to show each is isomorphic to some other thing – the colimit of the entire diagram we started with. This colimit will here be called an associator; it is something that both compositions of all three spans will be isomorphic to. I’m just going to sweep under the rug the question of whether that colimit will exist and just assume it does in this post.

So we’ve got the groundwork set, now we need to take advantage of some universal properties to give unique maps that make sub-diagrams commute. The first universal property taken advantage of is that of the first pushout. The associator has arrows going to it from \Gamma'' and \Gamma', so there will exist a unique morphism, \Gamma'' \Gamma' \rightarrow \Gamma'' \Gamma' \Gamma. such that everything commutes. Going up a step and using the arrow we just made, we can play the same game to get a unique morphism (\Gamma'' \Gamma') \Gamma \rightarrow \Gamma'' \Gamma' \Gamma that still makes everything commute.

To go in the reverse direction, we need morphisms from each of the Gammas to (\Gamma'' \Gamma') \Gamma. We already have one from \Gamma, and the other two can be built by composing morphisms. We can join two arrows together, tail to tip, to get a single arrow regardless of what category we are in. Now the universal property of the associator kicks in, giving a unique morphism \Gamma'' \Gamma' \Gamma \rightarrow (\Gamma'' \Gamma') \Gamma, but there’s a catch. We only know it makes some of the diagram commute. We have to put in a bit more effort to show the arrows involved in the pushouts will commute, too. The first step of this isn’t too bad — the three arrows that don’t point directly at \Gamma'' \Gamma' \Gamma will commute because the morphisms from \Gamma'' and \Gamma' to (\Gamma'' \Gamma') \Gamma commute with them by construction.

The universal pushout maps are a bit trickier, but since both legs of the pushouts commute with everything else, the universal maps will commute as well. Without the picture, that sentence might be a bit hard to understand. Look at the picture. They say those things are worth a thousand words. As parts of the diagram are shown to commute with the blue arrow on top, those part turn dark green. The arrows that already commute with everything get highlighted when they are used to show other arrows commute with everything. That should clarify what the picture is doing. Once everything commutes, the arrows turn black again, and we are left with unique maps in both directions at the top of the diagram that commute with each other. Long story short, this gives an isomorphism between the composition of spans by pushouts and the composition of spans “all at once” by associator. Watching the animation in a mirror gives the other way of composing spans by pushouts as isomorphic to this as well.

So this picture together with its mirror image shows (\Gamma'' \Gamma') \Gamma \simeq \Gamma'' (\Gamma' \Gamma), which is really the best we could possibly hope for when pushouts themselves are only unique up to isomorphism.