Vancouver 2008

So I've been getting a steady trickle of rejection letters from places where I applied for a postdoc, but yesterday morning I got a bit of much more positive correspondence. It sounds like the next place I'll be heading after TIFR will be the University of British Columbia. The position won't start until January, and funding is potentially on somewhat shaky ground after the first six months, but I think it will work out well.

Besides, everyone loves Canada, eh.

Inaugurating the treadmill

This morning provided one of these interesting moments when the curtain parts just a bit and I get to remind myself that I am in fact, somewhere else entirely.

My latest means of countering the fact that my office, my apartment, and the places where I eat are all within three minutes of each other, and everything outside that bubble involves lots of honking traffic, hence activities other than sitting in a chair or walking back and forth along the seaface or in the open area between the atrium and the auditorium tend to be limited has been (appropriately enough for India) some yoga in the morning. At first I was doing this on my own up on the roof of Jagdish, but then I learned from Costis that the sign on the bulletin board proclaiming "Gents Yoga Class, 7:30-8:30 AM" did in fact refer to something extant. It isn't really a class at all, but there is a guy who comes in every other week to provide a list of poses and some guidance. Other than Costis and I there is only one other young guy there, and he's on some totally other scheme of exercises. There are two other guys who come in, but they are older and seem to be following specifically prescribed regimens such as staring at a candle for twenty minutes and then doing some yogic breathing/snorting for a while.

So this morning, our yoga-wallah came in and asked us to come join him in the other room where the weight machines are for an "inauguration". What was to be inaugurated was a new treadmill. In contrast to the existing treadmill, this one is set up looking out the window, so that those who prefer to walk in place using a machine can stand on the second floor and look out and see what they are missing rather than be tucked into a corner under the stairs and stare at a wall. I guess that during the rainy season it can be nice to take a walk indoors, but I generally don't see the point. Anyway, when we came into the room, the Star-Trek-esque control panel was draped with a garland of orange flowers, and our friend had a couple sticks of incense burning. One of the men present was given a dry coconut, which he cracked open on a dumbbell. Some sweets were passed out and the guy who cracked the coconut tried out the new machine.

(The sweets were of the extremely sweetened pistachio flavored compressed milk product variety, and however good it may sound, it isn't, and I had to cut out of yoga early because the thought of suddenly ejecting that little blob of sugar with last night's dinner in tow while standing on my head was all too present.)

Apparently this equipment inauguration thing happens every so often. On telling this story I heard about a little bit of drama that came about when someone in the (string) theory group got a new piece of office furniture and deemed it necessary to acknowledge this fact. It seems that some more secular members of the department found the sanctifying of an office chair or filing cabinet rather offensive and there was a little bit of tension for a while. I'm certainly ignorant of much of the associated baggage, but as a ritual to acknowledge gratitude for getting a new thing, I thought it was pretty neat. I suppose it isn't all that unlike cutting a ribbon on a new bridge, just on a much smaller scale.

Solar eclipse

(picture links to a 1.6mb movie)

One rare occasions, the smog comes in useful.

As I was about done taking this a shoe shine man with no legs by the name of Sunil Kumar manualed his hand crank tricycle up to me, and inquired about my nationality, marital status, and lack of a rear brake.

Adventures

I sat with Nirtin Nitsure, current Dean of the school of Maths, during lunch today. Nirtin is a loquacious and lively character, by turns interesting and obnoxious conversation, drawing heavily on the desire to always play Devil's advocate to the greatest possible effect. A few days ago, we were having lunch and Yogi (who was Nirtin's student) sat down to join us. Seamlessly integrated into the conversation was Niritn's immediate and insistent question to Yogi, "Would you rather have an elephant's trunk or no nose at all?" A few seconds earlier we had been talking about Western and Indian attitudes regarding historical preservation, but anyway...

The point of this digression is that today Nirtin asked me what sorts of adventures I had been on recently or had planned.

"Only mathematical ones," I replied.

Which is pretty much true. So far, I don't feel like I have produced that which would very unambiguously indicate a successful career as a research mathematician, and this suspicion is confirmed by the paucity of job offers that present themselves to me. In May, I'll be back in the US for a conference on rational curves, a thing central to my research. So my plan is to do as much as possible between now and then to ensure that I have something to talk about with the other experts, and to use my experience at the conference as a sort of touchstone to help me figure out what the hell to do with a PhD in algebraic geometry. Clearly there are plenty of options, but the obvious one, academia, seems to have the feature that the exit is a one way door. I already promised Sándor that I wouldn't be a taxi driver with a PhD, so that old fall-back is out.

On a less personal but still meta-mathematical note (or meta-meta-mathematical note for those who keep track), a couple of interesting articles showed up on the arXiv recently. The first, cross listed in math.AG and hence noticed promptly, is math.HO/0703427 is an article titled Mathematical knowledge: internal, social and cultural aspects by Yuri Manin, one of the few who has the impressive distinction of a mathematical object denoted by his name hyphenated with that of Gauss. (Incidentally, I have a hard time mentioning Gauss without being annoyed that he gets his name attached to inventing hyperbolic geometry along with Bolyai and Lobatchevsky - clearly he worked out some of the important formulae, but it isn't so clear that he understood what he was looking at - and this was the essential feature of that discovery in my opinion. I feel that when Gauss, who was clearly one of the best mathematicians ever, gets his name tacked on to the discovery it belittles the work of the other two, and also the other work of Gauss himself.) Anyway, Manin's article rambles a bit, but in the rambling style of someone who has a lot of interesting things to say and wants to say all of them. It seems interesting and probably worth looking at if you are interested in what mathematicians think about mathematics. The second article came out last month; math.HO/0702396 by Terrence Tao is a discussion of what characterizes good mathematics. Just the list that he starts with makes an interesting prompt for thinking about this question.

More math movies

I made a couple more math movies with surf today.

The first is a family of curves that is only mildly more complicated than the Mobius strip.



And the second is related to the Hopf fibration on the three sphere.

Holi Technicolor Batman!

Sunday was Holi, the festival of color, a holiday of ambiguous origination, a bit like a Hindu version of Easter. There are elements of a spring festival in the picture, and then there are some dubious religious stories pasted on top. The TIFR colony's version of the holiday begins the night of the full moon with a brief bonfire and passing out of sweets. A brief trip to Gokul for dinner and we missed all but a few embers surrounded by a lot of empty chairs, and someone using another fire to burn some leaves and make a lot of smoke.

The festivities began the next morning at around 10am with a few people milling around near the cricket pitch, smearing powdered color on anyone within reach. I wandered out and soon found myself tinted. There wasn't much to the coloring ritual beyond a "Happy Holi" and a smear on the face. After a little while, a hose was passed up to a first story window and a mud puddle was constructed. People gave up color for brown with varying degrees of enthusiasm. Sreekar made a good show of resisting, but he too joined with the mud people. I lingered until I had to fend some of the revellers off lest they pick me up and carry me into the mud. Instead, I went and took my bike out for a spin around town.



I started with my usual route out to Marine Drive and up to Malabar Hill. On the way past the slum near Cuffe Parade a bucket of lavender colored water was thrown on me, and riding through the city as it evaporated off my shirt was quite pleasant. I figured I'd check out Banganga tank while I was on Malabar Hill, and found the standard cricket game going on, except that everyone was painted purple. A couple of times I stopped and let someone give my face a bit more paint.

Here's the view from my handlebars:



From Malabar Hill I cut east and North across the city and then turned left and headed north, perhaps as far as Dadar, I'm not sure. Traffic was really light, even for Sunday. Occasional groups of people milled about in chai shops, restaurants and doorways with varying ammounts of paint. A few of the sidewalk dwellers were taking baths as I rode by. It was nice to almost blend in for a change. Being painted purple partially hid the fact that I was ferengi. On the other hand, I was clearly one of the only ones exerting myself on this afternoon. It seems that in Mumbai, everyone takes the afternoon of Holi even easier than a normal Sunday, which is dead slow by Bombay standards. What that is by rest of the world standards I may have forgotten. Eventually, I decided that I'd had enough and turned around. It took a while to get the paint off my face, and the shirt is not going to be white again. That's okay.

In other news, this is an interesting article about the place of the slums in Mumbai. Specifiically, it talks about how all of the recycling that is going on as under the radar business.

And not related to Mumbai at all is this article on one of the more interesting biofuels on the horizon. It is lots better than ethanol.

A universal curve

What follows is a rambling example attempting to explain some mathematics. I didn't really have a well defined audience in mind when I wrote it (or rather, I had a number of well defined audiences in mind and all of them pulled in different directions) but hopefully it is interesting and intelligible to some anyway. I clearly need to draw some more pictures and insert them to break up the ramble, but until I get around to it, I'll leave that bit of work as an exercise for the reader.

I'll probably try to work this idea into a general audience talk, perhaps one introducing the minimal model program, so if anyone has any thoughts, I'd be glad to hear them.

Families of curves are rather interesting objects. Consider the set of all curves in a given geometric space Z and call this set Curves(Z). Then we can consider the universal curve, which is the subset of the Cartesian product Z × Curves(Z) containing pairs of a point in Z and a curve containing that point, namely

UCurve(Z)={(z,[C]) ε Z × Curves(Z) | z is in C}.

It turns out that with some reasonable restrictions on the type of thing that we want to consider as a curve, that Curves(Z) and UCurve(Z) have the structure needed to be considered as geometric objects in their own rights. A family of curves in Z is a subset of Curves(Z) that fits into the same language of describing geometric things that we use to describe Z, the curves on Z and Curves(Z).

A simple example of a family of curves is the set of lines through a given point in the Euclidean plane, E. Call this family H. It is a subset of Curves(E). H is by definition the real projective line, i.e. the circle, parametrized by slope, where a vertical line is said to have a slope of ∞. The universal curve, which will have the name F and sit inside UCurve(E), is relatively easy to see - it is quite similar to the plane itself. Using polar coordinates centered at our given point to describe the plane is almost the same as using the universal curve. The problem with polar coordinates is that when the radial coordinate is zero, the angular coordinate is meaningless. And standard polar coordinates use angle instead of slope, which forgets that we are interested in lines through a point, not rays coming out of that point. To get lines, we need to remind ourselves that x denotes the same line as x+180, which is a bit ad hoc.

So what we are looking at is a parametrization of the plane E by a slope coordinate and a line coordinate. For non vertical lines, we might as well use the symbols y/x for slope and the r for the line coordinate, reminding ourselves that y/x=∞ and negative values for r are okay. When we use these coordinates for the plane, there is ambiguity at the points where r=0. But the points (0,y/x) are on different curves, so they are not the same point on the universal curve. This difference between the universal curve and the plane is the key to what is going on. The universal curve coordinates work fine to describe the plane, except for the point where all the curves that are collected into the universal curve come together. On the plane, this is one point, but in the universal curve, there is a different point for each curve in the family - that is, there is a copy of H sitting inside of F such that the whole thing gets contracted to a point. This phenomenon is called blowing up a point, and F is called the blowup of E at a point. The copy of H that gets collapsed is called the exceptional curve.



F is topologically a Mobius strip. There is a picture of this in Hartshorne, conveniently reproduced on Davis's backpack for all you Seattle people without a copy or two lying around. But better than looking at a picture, draw your own. Draw a disk around the point that gets blown up instead of the whole plane and make a Mobius strip with a stapler or some tape. Draw the circle that follows the middle of the strip of your Mobius band (this whole circle is going to correspond to the point) and and a few lines across the strip perpendicular to the central circle, in both cases, press hard enough or use dark enough ink or use tracing paper so that you can see the line on both sides. You should think of the line that you draw being equally visible on both "sides" of the strip, so when you draw it, the ends will join through the paper. Right now this is just the way we'll do things. We aren't doing noncommutative geometry yet. Label these lines with slopes starting with ∞, then move into big negative numbers, increase up to 0 and then up to positive slopes and back to ∞. Trace your way around the central circle and mark the correspondingly sloped lines on the flat plane to convince yourself of what I'm rambling on about.

The Klein bottle shows up if we look at the family of complete geodesics through a fixed point on a sphere, say the south pole on the earth, in which case we get the meridians. The universal family is again very similar to the space on which the curves live, in this case the sphere. In this case, there are two places where we need to resolve the indeterminacy about which line we are on at the north and south poles all the lines come together. We do this by by blowing up these two points.

This corresponds to what is going on topologically when we look at a world map, for example with a Mercator or cylindrical projection. Latitude and longitude provide the location at any point other than the poles, but at the poles, the longitude is irrelevant information. But it is still a coordinate that we can read off of our map, so it must parameterize something. These somethings are the exceptional curves, one middle circle for each of the Mobius bands of the Klein bottle. At first glance, this comparison should seem sketchy, since we all know that a map makes the upper edge of the map collapse to one point - the north pole, the sides stitch together, and the bottom edge collapses to the south pole. So why bother with this intermediate step that seems to needlessly complicate things by introducing a Klein bottle? The advantage of the intermediary step is that one keeps an intrinsic picture of the geometry at the poles.

Without a labeling indicating that the meridian at x extends through the pole as the meridian at 180+x, there is no way to tell what the geometry is at the poles. This labeling is exactly the labeling that identifies the map as a Klein bottle.

So what is the point of this example? I thought of it in the context of two connected ideas. The first is the minimal model program (MMP) in algebraic geometry, and it relates to the interpretation of a map that appears to be a Klein bottle as in fact depicting a sphere. One way to understand the goal of the MMP is that it seeks to understand the right geometric object that corresponds to a given map. In this example, running an appropriate version of the MMP would tell us to take our Klein bottle map and blow down the two exceptional curves and recover a sphere with all the geometric data at the poles and elsewhere described as the mapmaker intended.

The second idea is the construction of the map as a universal curve. First of all, this has nice philosophical implications, because the way that a map is drawn is by first exploring the territory along paths. Perhaps a more vivid example comes from the knowledge we have that comes from astronomy. All this information has essentially coming straight at us along some sort of geodesic through spacetime. Sitting here on Earth can do little more than collect the information as it converges at our south pole of the universe. But philosophy aside, this idea of using a universal curve for a family of relatively simple curves through a fixed point as a map of some object to be studied is central to the proofs of many results in algebraic geometry. In particular this idea can be used to prove a result due to Shigefumi Mori, one of the key architects of the minimal model program, namely that a variety with ample tangent bundle is projective space.

It is here that what I am researching makes connection to this example. I believe that families of (rational) curves through some fixed point that are as simple as possible, given that I want my families to contain a curve pointing in every possible direction, should make a good enough map to reconstruct what sort of space these curves are living in. My particular tack is to try to simplify the problem even further by working with approximations to the family of curves, approximations that I call minimally indicating rational arcs. The trouble is, I'm not sure if these approximations actually make anything any simpler, but hopefully we'll see.

Gas!

For dinner last night I had four of the most delicious grilled cheese sandwiches ever created.