the best “real-life” use of geometry I saw this year

On May 3, 2003, the Old Man of the Mountain—a rock formation that had been known for at least two centuries as one of the natural wonders of New Hampshire—collapsed. No one saw it happen; that morning two park rangers looked up and realized he was gone. It had been expected that this day would arrive. The Old Man’s face was a remnant of ancient glacial movements, and it was not stable, thanks to erosion and freezing; it had already been repaired multiple times since the 1920s. In 2007, a project was begun to memorialize the Old Man, and in 2011 the “Profiler Plaza” was dedicated.

Over fall break this year, my wife and I made a trip to the western edge of the White Mountains, where the Old Man of the Mountain used to reside. We stopped by the memorial to the Old Man that is now located on the edge of “Profile Lake”, where I was astounded by the ingenuity of the project that had been created. Not content with photographs or descriptive plaques, the Old Man of the Mountain Legacy Fund sought to recreate the experience of viewing the famous visage.

This optical illusion is created by looking along any of several different steel structures, called “profilers”.

Each profiler has an array of raised features that, when viewed from an appropriate angle, line up to recreate the face on the mountain from the viewer’s perspective.

The distance from the Profiler Plaza to the Old Man’s former location is about half a mile, but for the profile effect to work requires careful placement of the viewer’s eyes. Thus each steel profiler comes equipped with three spots, marked according to the viewer’s height, so that they will be in the proper alignment. (Below is a picture of my wife looking at one of the profilers.)

I found this application of geometry to a memorial not only ingenious, but also quite stirring. The Old Man of the Mountain inspired several artistic works, including Nathaniel Hawthorne’s short story “The Great Stone Face”. When I was in high school, my mom directed a theatrical adaptation of this story, in which I played the role of the visiting poet who appears near the end of the tale. So I felt a special connection to this place as I visited it for the first time.

It seems this could make a useful cross-disciplinary lesson in school, say between English, geometry, and U.S. history. Students could study the stories of the Great Stone Face and the monument’s demise in 2003. Then they might be asked to choose a location and design the memorial, working out the necessary measurements. For instance, here is a link to a map with the face’s former location marked: 44.1606° N, 71.6834° W. The actual location of the profilers is on the north shore of Profile Lake. If anyone carries this out, I’d love to know how it goes!

Thanks for reading, and Happy New Year!

what I did last week: math in Mexico

Last Saturday I returned from a 10-day visit to the Centro de Ciencias Matemáticas on the campus of Universidad Nacional Autónoma de México in Morelia, where we held the International conference and workshop on surfaces of infinite type, bringing together about 45 participants for lectures, mini-courses, and collaboration.

It was a great success; all of the talks were good, and several people expressed gratitude that we had organized a conference on this particular topic. Our goal was to gather people interested in dynamical or geometric aspects of infinite-type surfaces, so we structured the conference around two mini-course series. The first, on infinite-dimensional Teichmüller spaces, was taught by Alastair Fletcher. The second, on a particular dynamical system known as the wind-tree model, was taught by Vincent Delecroix and Samuel Lelièvre. The reason that this conference was so interesting for us is that infinite-type surfaces are generally outliers in the conferences we attend. A bit of background.

Surfaces are spaces that, on a small scale, look just like the plane. Some familiar examples of surfaces are the plane itself, the sphere, and the torus. These are all examples of finite-type surfaces. The sphere and the torus are both compact, and the plane is homeomorphic to (i.e., “just like” in an appropriate sense) the sphere with one point removed (as can be seen via stereographic projection). Compact surfaces were classified in the late 19th and early 20th centuries, during the early development of the field of topology. In the orientable case, which is the main situation we’re interested in, every type of compact surface can be obtained essentially by “adding handles” to a sphere: the number of handles added is called the genus of the surface. So a sphere has genus zero, a torus has genus one, a two-holed torus has genus two, and so on. Labeling a surface by its genus is a convenient way to describe it, because any two compact surfaces with the same genus can be continuously deformed from one to the other. Finite-type surfaces are obtained from compact surfaces by removing finitely many points, called punctures. They are therefore labeled by a pair (g,n), where g is the genus of the starting surface, and n is the number of punctures.

Infinite-type surfaces, by contrast, have either infinite genus or infinitely many punctures (or both). Think of the plane with all points of the form (a,b) removed, where a and b are integers, or of an infinite chain of tori, attached one to the next. Such surfaces were classified by Kerékjártó in the 1920s, although later improvements and corrections to his proof were necessary. Essentially, any non-compact surface can be characterized (up to continuous deformation, i.e., homeomorphism) by its genus (which may be finite or infinite) and its set of ends. Roughly speaking, the ends of a surface describe the ways that a sequence of points can “escape to infinity” while remaining on the surface. (“Escaping to infinity” means leaving every compact subset of the surface and not returning.) In the case of finite-type surfaces, the ends correspond to the set of punctures; a compact surface has zero ends. More generally, the set of ends of a surface has a nice structure so that it can be considered as a subset of a Cantor set. To get a surface whose ends form a Cantor set, for instance, imagine starting with a Y-shaped pipe, which has a “base” and two “arms”.

Add two other Y-shaped pipes by attaching their bases onto the arms of the first one. Now there are four “free” arms, to which four more Y-shaped pipes may be added. Continue this process ad infinitum. The result is shown here (credit to Spivak’s Differential Geometry):

If you’re familiar with the Cantor set construction, you can see that the ends of the surface—each of which (except for the base of the original pipe we started with) corresponds to following a particular path through an infinite set of pipes—line up with a Cantor set. This surface has genus zero, however. A surface with infinite genus must have at least one end; the unique (up to homeomorphism) surface with infinite genus and one end has been humorously named the Loch Ness Monster by Étienne Ghys. While surfaces can be constructed with any number of ends (as long as they form a closed subset of the Cantor set), the Loch Ness Monster seems to show up naturally in our field most often whenever infinite-type surfaces appear.

Now let’s return to the topics of the conference mini-courses. Although there is only one topological orientable surface of genus g, such a surface can be given many distinct geometric structures. Think, for example, of a long, skinny torus as opposed to a short, fat one:

These are distinguished in their geometry, for instance, by the lengths of curves going around the two surfaces. When the genus is greater than 1 (or, more generally, when 2×(genus of the surface)+(number of punctures) is at least three), any geometric structure on the surface which is complete (meaning the ends are infinitely far away) must be hyperbolic—the total curvature (which measures whether the surface is more like a saddle or more like a sphere near each point) must be negative. Teichmüller space describes all the ways that a surface of a fixed type can have hyperbolic geometry in which the surface looks the same near every point (in more technical terms, the curvature is constant). When a surface has finite type, its Teichmüller space is finite dimensional; the required number of parameters is 6×(genus)+2×(number of punctures)-6. When the surface has infinite type, as one might expect, the Teichmüller space also becomes infinite-dimensional, and some care is needed to describe it. Our course on Teichmüller spaces dealt with some properties that are common to both finite- and infinite-type cases, as well as some of the peculiarities of infinite-type.

The second course, on wind-tree models, addressed a family of surfaces with a specific geometric construction. The name refers not to actual wind blowing through trees, but to a roughly analogous process that occurs in so-called Lorentz gas, which consists of particles striking obstacles following billiard dynamics. Various forms of this model have appeared since the early 20th century (starting with Paul and Tatiana Ehrenfest, after whom the model is sometimes named). In the version this course dealt with, the obstacles are identical rectangles with vertical and horizontal sides, centered at integer lattice points in the place. The particles are modeled by straight-line trajectories that reflect off the obstacles in the usual way, “angle of incidence equals angle of reflection”.

One might ask several questions about this model: Which particles escape to infinity? Which remain bounded? Which return close to their starting position infinitely often? How do these answers depend on the angle at which a particle is traveling? How do they depend on the shape of the obstacles? This system has an associated surface of infinite type which facilitates the study of its behavior. By a clever trick, much of the study can be related to a finite-type surface (in fact, of genus 2), where much is already known. In order to use the theory of the finite-type surface to draw conclusions about the infinite-type surface (and thus the wind-tree model itself), several new applications of dynamical and topological tools have been needed, with quite beautiful results.

In my own work, I tend to study surfaces where a connection with finite-type surfaces is not as apparent. The field of infinite-type surfaces in a dynamical setting is relatively new, and growing, which is why we felt the time was appropriate to hold this conference. I am grateful to my fellow organizers, the staff at UNAM, and all the participants for making it a great week.

Spargel, spargel, spargel

It means “asparagus”. Bonn is in high spargel season, and it's everywhere. My strongest first impression of Bonn comes from walking through the vegetable market hearing gravelly voices calling, “Spargel, spargel, zwei kilos drei euros!” The spargel that's available, by the way, is fat white asparagus, unlike any I've seen. Hannah claims that in the US white asparagus is generally considered a luxury. All of the restaurants are serving it; some are even announcing a “Spargelfest” and have their entire placards advertising dishes based on spargel (usually with hollandaise sauce). Hannah and I have invested in our own spargel and plan to cook it tomorrow. More on that when it happens.

First things first (or second, since I seem to have allotted first place to the supreme pleasure of the word “spargel”, spoken either aloud or internally). So, first things second, and we left for Germany late Thursday afternoon. All aboard Aer Lingus, where the uniforms are the color of clover and the planes are named after Irish saints (Hannah noticed this in Dublin). It was a relatively short flight from JFK to Dublin—only about five hours of air time. Once the taxiing and waiting on the runway time was factored in, about seven hours on the plane. So there wasn't a lot of time for sleeping. I watched Shutter Island up through dinnertime. (It is a good movie, which I would recommend for anyone who likes drama with a dose of suspense and just a tinge of horror, without really being horror-genre. Also for anyone who thinks that Leo DiCaprio and Ben Kingsley sounds like a good pairing. Once again Martin Scorcese proves his love for Leo.) After that, I tried sleeping for a while but was awake again after about a half-hour. Tuned in to “Big Bang Theory” (which I only ever watch on planes, but do so whenever possible) and “Mad Men” (which I had heard many good things about, but not seen before). Enough about in-flight entertainment…

How could we tell we were in Ireland from inside the Dublin airport? All the Jameson whiskey in all of the stores. One entire hallway was covered with it, the way one might see a string of windows showcasing diamond necklaces in the chic part of the mall. Also the fact that all of the clothes in the shops were green. Bright green. The same color as Aer Lingus uniforms. And we were reminded that Ireland uses the euro instead of the pound for currency, which was fortunate since we had euros on hand. Got some proper coffee. That was the reminder that we were in Europe, and happily were going to be in Europe for some time.

It wasn't a long layover in Dublin, and we were shortly on the way to Frankfurt. A couple of days ago, when I was checking the weather for packing purposes, I learned that this region gets a lot of rainfall in the summer, so it wasn't too much of a surprise when we got on the plane that it was announced thunderstorms in Frankfurt were delaying our departure. The pilot came on several times and explained how she was arguing with the Frankfurt airport control to push up our push off time as much as possible. This was great for two reasons. One, our pilot was a woman! Neither Hannah nor I recalled that happening in some time. (Additionally, I think I heard her say the co-pilot was a “Sir”. Sometimes it seems too bad we don't have knights in the US. Or royalty, for that matter, as I read a opinion columnist recently suggesting we have a titular king and queen who could take care of the photo-ops while the president takes care of actually fixing things. But I digress…) Two, our pilot was totally taking on the entire Frankfurt air system! Well, that's what it sounded like to us, anyway. We might have been a bit punchy at that point. I spent most of the time between landing and taking off in Dublin pestering Hannah with my chipperness, and promptly slept over all of England and France (and Belgium, perhaps?).

The Frankfurt airport has a train station attached, so after some wandering about and getting (English) directions from a friendly official, we ended up at the Fernbahnhof with tickets to Bonn. The train was punctual, and once we got on, we discovered it was stuffy and un-airconditioned. This would have been much less tolerable had there not been a team of young, drunk, male athletes who sang the entire trip. The times they sang in English was to wish happy birthday (twice) to their teammate Markos, and when they broke into John Denver's “Country Roads, Take Me Home” (which was a little surreal). Most of their singing was words they made up to short tunes they knew. Most memorable was when a women's soccer team boarded the train and the guys sang “Jetzt kommen die Frauen! Oh oh-oh-oh-oh!” The two teams proceeded to have singing competitions at each other. The men's team won, I think, but they also failed to capture as much attention from the women as they would have liked.

Once we got off the train, we only took a couple of wrong turns in getting to the apartment where we're staying. Our apartment is just north of the Marktplatz, which is full of fruit and vegetable stands. There is a large pedestrian section of town with cobbled roads lined with all sorts of shops and biergartens. There will be more to tell about the town once we've seen more and gotten more familiar with it. But I just want you to know, we have great access to a lot of spargel merchants.

deep river

During today’s tour of Mammoth Cave, the discussion turned to the formation of the caves and where the water sits now. As Hannah and I learned on a trip to the Pink Palace Museum in Memphis a couple of months ago, there was a time when most of North America was covered in a shallow ocean (as I recall, the main exception was the Appalachian Mountains, because the Rockies hadn’t formed yet). The tour guide, Joe, said it’s believed that’s when the cave was formed. There are five (explored) levels to the cave, and our tour descended to the third. The water table sits about 360 feet below ground level; during floods it can rise about fifty feet about that. We got down over 200 feet below the ground, and someone asked if the cave ever floods. After explaining the levels and depths, Joe said, “We have a book we really respect, and it says that the rest of these caves aren’t ever going to flood again.” That was one of my favorite examples of the guides’ blend of humor and frankness. We also saw a cave cricket, and some amazing canyon-shaped passages, and the “flowstone” of “frozen Niagara”, which looked in some ways like a rank of organ pipes cascading to the ground and in others like a fleet of fins of some gorgeous aquatic creature. It was a stunning trip, if exhausting (over four miles of walking up and down, after already being somewhat tired from short nights of sleep). Tomorrow we’ll take the lantern-lit tour.

who in the lord's name comest

It is now less than a week until Easter dawn. Six days until the bells ring and rejoicing begins afresh. All the world waited expectantly over the ages for the birth of Jesus; it did not even know to hope for his resurrection. Yet in that morning is all of our hope for more renewal, more glory, more joy and peace.

These past few days I’ve been spending time playing hymns in the evening rather than writing. There are so many depths to plunge in meditation, but as I found the ordinary work days more and more tiring, I needed to spend time in simpler, more prayerful worship—music revives me, while writing draws a lot of energy out of me. It has been a good week.

This week Hannah and I are in Kentucky, visiting Mammoth Cave National Park with both sets of parents. It is Cornell’s spring break, and a good chance for us all to come together. I will try to write at least briefly each evening; it is Holy Week, after all, and holiness comes from devotion.

cum sancto spiritu, part 1

This past weekend was my first stay at a convent (community? ecclesiastical farm? it’s a little hard to describe succinctly). Hannah and I went with a couple of friends to the Community of the Holy Spirit in Brewster, NY, where we spent two very restful days and nights, in part to visit Suzanne, a priest we knew through Cornell. The Melrose Convent is an outgrowth of St. Hilda’s House in NYC and comprises six or eight women who work, farm, pray, and occasionally teach in a lovely, somewhat secluded spot. There is a school, the Melrose School, on the grounds, but no classes were going on, even on Monday because of Columbus Day. We very much enjoyed our time there, and over a few posts I plan to give a relatively thorough account of the weekend.

We set out from Ithaca late Saturday morning. The first couple of hours in the car we spent singing (what delight!), sharing favorite hymns and songs from our youth. Although Hannah didn’t grow up in the church, the rest of us had very varied backgrounds and could cross-pollinate a lot. Jessie teaches Sunday school and looks for good songs to teach kids. I presented my perennial favorite, “The Fruit of the Spirit”:

The fruit of the spirit’s not a cantaloupe (nope!)
The fruit of the spirit’s not a cantaloupe (nope!),
So if you want to be a cantaloupe,
you might as well hear it,
You can‘t be a fruit of the spirit, ’cause the fruit is:
Love, joy, peace, patience, kindness, goodness, faithfulness, gentleness, and self-control.

Jessie was raised Quaker, and taught us a song about Lucretia Mott to the tune of “The Battle-Hymn of the Republic”. Try it:

Thank thee kindly, friend Lucretia, (repeat twice)
Thy light still shines for me!

Did that refrain work O.K. for you? Then try making this scan to the tune of the verse:

Throughout the town of Philadelphia she hid the fleeing slaves;
For the freedom of her sisters she did cross the ocean waves,
And she asked Ulysses S. Grant to grant a pardon for the brave.
Her light still shines for me!

We worked on that over and over, marching around the kitchen as we made meals at the convent.

We stopped twice on the way to the convent. The first was for lunch. There was some debate as to whether we should stop at a familiar fast food outlet or press a little deeper to find a local restaurant. Realize that this was a car full of people who really like eating fresh food and supporting local business, and so the latter option was preferred. We ended up at Last Licks in Liberty (exit 101 off of highway 17). The friendly manager welcomed us and explained that everything was served on sandwiches—subs, hoagies, paninis, and the like. We all had a bottle of one of Boylan’s products (I had my first birch beer, which was a lot like a root beer, but independently very good); I recommend these, when you can’t get hold of Ithaca Sodas. There were quaint posters of olden-days Pepsi advertising (“Worth a dime, costs a nickel!”). The manager told us how he had just flown in from Florida and was just stopping by the store to check how things were going, when he learned that they were understaffed, and so he was working. When we told him we were from Ithaca, he said they had some regular customers from Cornell who would stop by on the way to Yankees games.

The second stop was at a fruit and vegetable stand. Hannah had made the proclamation, “It’s not really fall unless we do something to a pumpkin,” and so we had been watching for pumpkin stands. A hand-written sign on the side of the highway alerted us to the appropriate exit, and we pulled into the driveway of a house with an elderly man sitting under a tarp. He had a piece of corn in his hand, ready for us to sample (I have not been getting nearly enough corn this season), and told us a smattering of dirty jokes. He tried to convince us to buy his book of jokes. We managed to get away with just the pumpkin we were seeking and some corn, carrots, and broccoli (which became dinner the next night).

Once we arrived at the convent, we were promptly greeted with a wealth of treats and an invitation to Vespers. In fact, there wasn’t much time at all before the prayers began. We went into the small round wooden chapel, and we visitors spread ourselves out among the residents who knew what was going on. (We’d all been to evening prayers before, but the peculiarities of the prayer books at the convent required some guidance.) Much of the Vespers service is occupied with reciting psalms on a chant tone. One verse is chanted by the officiant to establish the tone, and thereafter the two sides of the chapel alternate chanting verses. For the first few psalms, I sat quietly, listening and not quite ready to break into the chant myself. There was only one other man there besides myself, and he had decided not to chant along. So my voice would have been the one to break the sound of the women. Now I love the sound of women’s voices (usually in harmony, which chant is not) but that’s not quite what was keeping me from singing. A true unison has a certain kind of purity. Splitting the octaves creates an essentially different sound (and I was not about to try to sing in the women’s range, particular since I’d been sick all week). So just as some people relax by popping in a “Chant” CD to hear the soothing baritone of monks, I was enjoying the piping clarity of women chanting together.

But within a few moments, I developed a strong sense that something was wrong—not just that I was isolating myself from my fellow congregants, or holding back from singing. I’ve studied chant, in the context of music history. I’ve listened to plenty of recordings, made available for the benefit of scholars and consumers. I’ve read about how it tried to strike the balance between emotion (rejoicing, mourning, awe, or penitence as necessary) and removal from the secular realm. About how composers expanded chants to reflect the text (even on the simple word “Alleluia”), then moved them from the meditative to the musical realm. Once I joined in, I realized what had felt wrong. Chant is not meant to be listened to. It has elements of song and elements of speech, but it isn’t either. It’s meditative, certainly, for both those chanting and those “listening”, but it’s nearly meaningless until you participate. It’s deliberate, ancient, immediate (be glad I’m not going off into another Kierkegaard digression here on the immediacy of music), transcendent, spiritual, physical, perplexing, focusing, and holy (in the sense of being set apart from other things in the world).

I’m making this point a bit strongly. There is real musical merit in, and real “outside” appreciation possible for, the hymns, antiphons, and sequences of the liturgy. But a psalm is mostly just intoned; from a melodic standpoint, it’s almost entirely a single repeated note, with a concluding burble up or down. Listening to it may put you in a trance, but I can’t imagine how it would help you worship. I don’t think I can explain more than that how important it is to join in chanting the mass, or service, or whatever, whenever it is possible to do so in place of just listening.

After dinner, we got set up in the guest house, known there as the “longhouse”. We had our own kitchen, stocked with fresh organic goodies (from eggs to raw milk to cookies), a sunny sitting room (directly adjacent to the school; apparently previous residents could hear homeroom or French class going on), and several private rooms to disperse to. Suzanne provided us with her Eddie Izzard DVD collection (which we didn’t manage to watch, however; as a first inkling of what the community was like, if you’ve never met nuns before, the sister who was showing us around the house also declared herself a big fan of Izzard).

We returned to the main house and looked around as dinner was being prepared. Hanging in the hallway was the following prayer. Now, the deliberate vagueness of the authorship, along with the style of writing makes the purported source of this text somewhat suspect. In fact, I searched the internet once I got back and could get no more information; the prayer was always presented in a vacuum (or list of other inspirational texts), except on a couple of occasions when attention was drawn to some anachronisms in the writing. So I include it as an example of how the nuns at the Melrose Convent see themselves and their calling, not any sort of historical document. It’s also still full of useful things for all of us to think about:

17th Century Nun’s Prayer

Lord, Thou knowest better than I know myself that I am growing older and will someday be old. Keep me from the fatal habit of thinking I must say something on every subject and on every occasion. Release me from craving to straighten out everybody’s affairs. Make me thoughtful but not moody: helpful but not bossy. With my vast store of wisdom, it seems a pity not to use it all, but Thou knowest Lord that I want a few friends at the end.

Keep my mind free from the recital of endless details; give me wings to get to the point. Seal my lips on my aches and pains. They are increasing, and love of rehearsing them is becoming sweeter as the years go by. I dare not ask for grace enough to enjoy the tales of others’ pains, but help me to endure them with patience.

I dare not ask for improved memory, but for a growing humility and a lessing cocksureness when my memory seems to clack with the memories of others. Teach me the glorious lesson that occasionally I may be mistaken.

Keep me reasonably sweet; I do not want to be a Saint—some of them are so hard to live with—but a sour old person is one of the crowning works of the devil. Give me the ability to see good things in unexpected places, and talents in unexpected people. And, give me, O Lord, the grace to tell them so.

AMEN

Dinner was a veritable feast. Apparently excited about having us as guests, the nuns had prepared several courses, and had sent out specially for hot dogs (they don’t usually eat meat). Hannah and I were, as expected, exceeding happy about the presence of kale. (Over the summer, as we were picking up vegetables from our CSA farm, we would ask with increasing intensity, “Are we getting kale this week? We really hope we are, because we really like having kale.” The first time we did this, the farmers responded in perplexity, “We’ve never actually had a reaction like that from anyone about getting kale.”) And here before us was a new yummy way to prepare it! (They call it “killer kale”.) We also had homemade ketchup, mustard, and tomatillo salsa; “apple leather”, the real version of what Fruit Roll-Ups always dreamed of being; fresh grape juice and grape jelly made from wild grapes in the area; a mix of beans from the garden; and other courses. Apparently when you make all your food fresh from the garden, living with a vow of poverty isn’t so bad.

Lots more I could share about the humor of the sisters and the stories that came out about convent life (such as tales of mass hysteria, or the three nuns who got in a fist-fight), but this is getting a bit long. Moving on to the evening…

Back at the longhouse, we sat around reading for a while. Hannah was reading poetry by Elizabeth Bishop. I was reading hymns (because there was no piano in the house for me to play them on). For some reason, “Lead On, O King Eternal” was going through my head:

Lead on, O King eternal,
The day of march has come;
Henceforth in fields of conquest
Thy tents shall be our home.
Through days of preparation
Thy grace has made us strong;
And now, O King eternal,
We lift our battle song.

I was reading from the Pilgrim Hymnal, of which I have four copies but hadn’t examined too carefully yet. Christian hymnody has plenty of texts with battle imagery (with good Scriptural reason), and many of them are collected, along with other exhortations to fortitude, in the Pilgrim Hymnal’s section on Courage. But I was struck by the end of the second verse, which I’ll use to close this entry (as always, the whole text of the hymn is available at Cyberhymnal). In a way quite consistent with the verses I linked to above (q.v.), it emphasizes the means of winning the battle to be our acting in love:

For not with swords’ loud clashing,
Nor roll of stirring drums;
With deeds of love and mercy
The heavenly kingdom comes.

Zacatecas

No, you probably haven’t heard of this town. It does have a Wikipedia entry, however, and it is a UNESCO world heritage site, settled right in the center of Mexico, about 400 miles northwest of Mexico City. It’s also the site of the 7th joint meeting of the American Mathematical Society and the Sociedad Matemática Mexicana, running from Wednesday through Saturday of this week.

This is my first time to Mexico. And it’s beautiful. Zacatecas is in a valley, and it’s astonishingly clean. As you drive along the road surrounding the valley, you look out and see a wash of colors. About half of the buildings are standard brick color, but the rest are brightly painted blue, red, yellow, orange, so that the dominant impression is of jubilant pigmentation. The Cathedral is just outside our apartment, and although it’s not as grandiose as what one might encounter elsewhere, it sets a tone for churches all around the area: its dome and towers seem mimicked everywhere. There’s a cable car between two of the hills nearby, and we plan on trying it out at some point.

There is apparently some kind of festival going on around town. Bands—generally composed of trumpets of some variety, valve trombones, and a battery of percussion—wander the streets, with crowds of youths trailing behind, until they stop in some open place (in a local square, an arcade, or even just an intersection) and everyone dances. The attendees range from the clean-cut to the punk-ass, and they wear shotglass-sized clay cups on ribbons around their necks, from which they drink Mezcal when a supplier arrives. Occasionally the liquor rides in on a donkey.

Earlier this evening two bands, one clad in gray, the other in red, were dueling in the plaza next to the cathedral. I followed various bands around, hoping at some point to get up the nerve to ask for a dance partner (either in really broken Spanish, or maybe even in English, hoping they’d find it cute). As the evening continued, however, the proportion of peripheral males to females grew substantially, so I just watched. This is an awful time not to have Hannah here.

A concert stage is currently being set up in the plaza by the Cathedral, which is just outside our window. What are we in for tomorrow night?

The conference is going well. My particular crowd isn’t really represented here, but I’ve picked up some things in the conformal dynamics and differential geometry sessions. Meeting this city is well worth the trip.

marseillais once more

I’ve just returned from my travels to the States for Christmas and New Year’s. I spent the last three weeks with Hannah, which is the longest we’ve seen each other since the end of classes last spring. (Of course, that’s mostly due to the fact that we didn’t start dating until the beginning of classes last fall, which was five days before I left Ithaca.) We shuttled between our two families, danced and talked and kissed a lot. More details later. But now my clock is set back to CET, my verbal communication will be done via Skype rather than cell phone (I was having to use Hannah’s phone anyway, because I suspended my service for the school year while I’m gone), my coffee is espresso from a cafetière italienne rather than brewed in a French press (which, despite its name, I’ve only seen used in the States and the Copenhagen airport), and it’s time to get back to work. I’m not yet sure what I’ll be doing this spring, since a calculation I made before leaving in December reduced our hopes of obtaining the result we were looking for to essentially nothing. I’m preparing a talk for the Teichmüller seminar on other topic, however, and questions abound that require addressing.

Happy New Year to all.

enneadecahebdoversary

Hannah and I had our third date tonight. We went to the Memphis Pizza Cafe and got a medium vegetable supreme (olives, mushrooms, green peppers, onions, tomatoes, spinach, broccoli) with pepperoni, then drove down to the riverfront. We looked at the Pyramid with the replica of a Rameses II statue out front and walked through Tom Lee park, named for a levee worker who saved the lives of 32 passengers when a steamboat capsized. There was a superb metal sculpture of him riding out in his dinghy “Zev” and reaching to a man holding to a broken piece of wood. Before heading home to watch an episode of “Firefly”, we stopped by Perkins for some pie. It was a great date, very relaxing, but different in character from our previous dates, which were also lovely but involved dressing up and going out for a nice dinner. Even though it’s only our third time to go out, I think I can affirm that we like each other and this’ll work out. Tonight’s events were pleasant in themselves, but almost more importantly they formed an unobtrusive backdrop to several hours of conversation alternating between light and philosophical, as any good date conversation should. We’re looking forward to the time when we can have dates on more than a bimonthly basis.

faire du ciel part 2

Just arrived in Memphis last night for the Christmas holidays. Travel again was not so bad: I got here about 23 hours after I woke up in Marseille, which was only a couple hours later than originally planned. The fog in London was extremely thick, and I’m grateful we weren’t delayed further. The ciel is still a pretty nice place to be. Despite the delays, the ground was, too.

London’s Gatwick airport gets pretty crowded during these times of travel trouble. The waiting lounge (full of duty-free shops of all kinds and a few restaurants) had nary a seat open. I had breakfast at Garfunkle’s, which advertised itself as “legendary and loved.” I was impressed with how they manage such a hectic time. Service was prompt; food was good. I had the vegetarian British breakfast. Two things struck me about this meal: first, it’s hard to find places in Europe that specifically make options available for vegetarians, so that was nice to see. Second, I think this is the first restaurant I’ve ever seen to declare on their menu that their eggs are free-range. Excellent. It was pricey, of course. Just the previous night at dinner I had been hearing from one of the Hubbards’ children, who lives in London, just how expensive life there is. Apparently for $10 one can usually only expect a mediocre meal, and if one wants to eat well, one has to consider the $50 range. I spent $14 on eggs, beans, toast, hash browms, mushrooms, a tomato, and a cappuccino, which in light of the dinner discussion seems entirely reasonable. I’m just glad I didn’t have to stay there longer.

I was intrigued by a poster I saw, advertising all the destinations you could reach from the airport via rail. “The easiest way to get to and from Gatwick is by train,” it declared. Unfortunate, I thought, that it’s an airport and not a train station, then.

world-travelers, we

Sarah just left this morning for the States. I was going to drive her to the airport, but the car we were borrowing wouldn’t start. She took a taxi because her flight was at 6:45 and the first bus from the train station doesn’t arrive at the airport until 6:00. I’ll have this same problem next Thursday when I go home. I’m not sure I’ll take the same solution, but we’ll see.

I tried to go back to sleep after she left, but didn’t quite manage it… partly I was thinking about things I need to write and post here. So I signed on, and found that Blogger has been updated (it’s better integrated with other Google services now), and discovered that there’s a blog being written by a current Peace Corps Volunteer in Togo. Excellent! I thought. It’s even better than I thought, because he has videos! Here’s the latest one, of the market in his town.



From YouTube, you can look at “More from this user” to see other videos. I highly recommend watching the school visit. The conditions of the schools in West Africa are often hard to describe to Americans. Somehow learning gets done, despite the remarkable challenges and lack of resources.

I haven’t read through all the archives yet, because he’s been there for almost a year and a half. But I’m always in favor of resources that can give people a better idea of what it’s like to live there. This (the ability to blog easily) would indeed have been awesome six years ago when I was there. He mentions in a few early entries that there’s far too much going on to effectively choose what to write about. It’s true; every single week, every single day of Peace Corps is charged with experience and activity. For the first several months of our service, Annie and I would meet for dinner every night to debrief and decompress. I’m looking forward to reliving some memories through Aaron’s online records.

I do have things I want to write. All in good time. I hope the Advent season (Christmas shopping season, end-of-fall-semester season, whatever applies) is treating you all well.

Adrien's funeral

We returned Thursday night from Paris, where we spent all day visiting with others who knew Douady, sharing stories, and trying to process the events. The day began at the Institut Henri Poincaré, where approximately 25 mathematicians, mostly from the field of holomorphic dynamics, gathered. Roland, Sarah, and I took turns explaining what had happened. Two editions of Le Monde were produced, from last Wednesday and Thursday, which between them held five announcements of Douady’s death: they were placed by the family, l’Ecole Normale Supérieure, l’Université de Paris-Sud (where Adrien was a professor), SUNY Stony Brook, and Nicolas Bourbaki (the famously secretive French mathematical society, of which Douady was a member). Someone brought a great number of photos from the past couple or three decades, mostly amusing, and all capturing the conviviality of Adrien and his friends and students.

The funeral ceremony was held in the crematorium of the Père Lachaise cemetery, perhaps the most famous cemetery of Paris. (Their website has a virtual tour. You’ll find the crematorium towards the north-west corner; it’s the largest building on the site.) I’m not usually good at estimating numbers of people, but having discussed it with a few others, I can claim that there were about 250 people in attendance, nearly half of whom were standing in the back of the hall.

In my mind, the ceremony had four major pieces. First, the response of the mathematical community. This included childhood friends, collaborators, and students of Douady, as well as some who had simply felt his influence. Roland was the first to speak, and related the story of our visit to Les Arcs. He found a beautiful way to express the final hours. Adrien had told us, as we were contemplating going swimming that day, that it was for each person to decide if they would find it pleasing, and if so, then they should plunge in and enjoy it. Roland said perhaps Adrien was talking about all of life, too; one doesn’t hold back from doing what one loves. And Adrien loved to swim.

After the words of the mathematicians, the casket was carried to the top of the stairs at the front of the hall and placed in the furnace. There was a long respectful pause, during which was played music from Don Giovanni. (Hubbard has told me Douady liked singing the part of the Commendatore.)

Next the family members and other friends spoke. Adrien had three children, eight grandchildren (of whom four gave brief speeches), and four siblings (one of whom spoke on behalf of them all).

The final piece, which occurred somewhat in the middle of the family’s words, was the presentation of four brief film clips Adrien had been part of. Two were from the biographical film Adrien Douady, mathématicien, and the other two were from films intended to popularize mathematics. In all, Adrien’s humor and love for the sea were well-represented. One scene showed him on a staircase, singing heartily his setting of Baudelaire’s “Le vin de l’assassin”. Another watched him dive into the water, and subsequently to arrive on the boundary of the Mandelbrot set, which he clambered along barefoot just as he would the rocks of l’Esterel. A recurring image was the rabbit fractal, a Julia set given its name by Douady.

In the evening, Régine hosted all those interested at her and Adrien’s house. We were there briefly before we had to catch the TGV back to Marseille. Work has continued since Adrien’s death, even if it has slowed a bit to accomodate the necessary arrangements, travel, and memorial. Hubbard gave his class lectures both of the last two Fridays, and so we had to be back Thursday night. He says he believes continuing to do math even at this time is completely in the spirit of Adrien. Plans for conferences in Douady’s honor have already been born. He was loved, and he will be missed, but I believe those who worked with him and knew his passions will pay him due respect by carrying out those passions to the benefit of future generations.

faire du ciel

(Composed Saturday, while in flight)

It’s easy to get jaded about flying, particularly when one decides that all the inconveniences of screening, walking endless hallways, and sitting in cramped coach seats nearly outweigh the tremendous benefits of this mode of transport. I don’t quite do it often enough to get completely numb to the whole process, but I’ve flown a lot in my life—mostly while I was at St. Olaf, going back and forth between Minneapolis and Memphis. But every once in a while you take a beautiful flight, and flying is a treat. My brother and I took such a flight from Malaga to Barcelona a few years ago, when we marveled (we might even have gasped) at the enchanting countryside of eastern Spain. I’ve had a couple nice flights on Air France, whose slogan is “Faire du ciel le plus bel endroit de la terre” (“Make the sky the most beautiful place on earth”).

This morning, my first leg on my way back to Ithaca, with British Airlines from Marseille to London, was such a flight. There were a number of small pleasures. First was the congenial, energetic crew. One of the flight attendants was making her opening remarks about welcome and safety, when the captain completely and exuberantly overrode her (clearly, he didn’t know she was talking). He sounded like he loved what he was doing and who he got to work with; for I think my first time on such a large plane the pilot announced the names of the attendants with what was more and more clearly his characteristic aplomb. At the end of his monologue, the gentleman sitting next to me (who had been reacting to the excitement by singing circus tunes) broke into applause. Something about the British accents (and vocabulary) of the attendants I found delightful. And finally, on our entry into London, I was captivated by the cloudscape. (Of course there were clouds. I don’t think I’ve ever flown over England without it being completely covered in clouds.) I think of how until the 20th century no one could see the textures of the tops of clouds. As we began dipping into the puffs and whiffs I felt a wave of relaxation. It was midmorning. Once we were below the clouds, there was a layer of mist deepening into what I presume had been fog an hour earlier. The sunlight shone through the small chinks in the cover, turning the whole world into a dusty cathedral; but what was most striking was that, as we flew at the level between the clouds and the had-been-fog, one saw the shafts of light were suspended between the two. When we were closer to the ground, the gentleman next to me pointed out to his wife the visible vortex streaming behind the airilon of the wing; it was a constant shape, twisted like a corkscrew, and formed out of concentrated mist.

It’s amazing when technology can bring us new experiences, not just of technology itself, but of the natural world, as well.

context

So, it’s important to know for a while that I’m back in Ithaca for the week. I was going to post something about it earlier, but the entry got a bit long and unwieldy. I may give a summary of it at some point. But this trip to Ithaca is the context for the next few posts.

Øieblikket part 4

I finally made it to Copenhagen, and I didn’t see a single Kierkegaard-related building, statue, historical marker, or souvenir. I simply wasn’t ready for this trip.

Of course, that begs the question of why I’m in Denmark at all. There’s a dynamics conference going on at an out-of-the-way guest house belonging to one of the universities here. The topics are currents and bifurcation loci—respectively, those terms refer to very nice, but very general topological/geometric objects and to sets that describe how systems with parameters change as the parameters change. I can’t possibly convey the notion of currents in this medium, but I can at least give an example of bifurcation: say you have a system of one-way paths in the plane, with two distinguished points, one of which absorbs paths (called a sink), and one of which emits paths (called a source), and somehow you’re varying the distance between these two points. One can imagine reducing the distance to zero, so that the points coalesce—maybe then all the paths just flow straight through (the source and sink annihilate each other), or maybe they start orbiting a single point (the source and sink together form a center). The change in the number and kinds of distinguished points is called a bifurcation.

This isn’t really my area of mathematics, at all. I know a bit about currents, from geometric studies, but I’ve never put in time studying bifurcation of dynamical systems. Yet I’m actually excited to be here. I’m looking forward to learning a lot. It’s not a large or a long conference; it just runs two and a half days, and there are perhaps thirty participants. There are dynamical aspects of the areas I do usually study, but I’ve always considered myself a geometer rather than a dynamicist. This is a good chance to learn about other interesting fields—areas in which Dr. Hubbard is very active.

We’ve gotten through the first day of the conference, but I’ve said about all I’m going to for now on that topic. Yesterday Sarah and I had the day to spend wandering about Copenhagen. Mostly we split up and indulged our respective interests. I quickly ended up at the Christiansborg Palace, home of the royal, legislative, and judicial branches of government (the foundation stone reads “Rex, Lex, Jus”—I may get a picture of it posted in the next few days). It is the fifth castle or palace to stand on that site. The first was built in 1167 by Bishop Absalon, and you can go underneath to see ruins from the walls of Bishop Absalon’s castle and the second castle. I like ruins; it’s amazing to see the way buildings were laid out, how homes were shaped, how wells and drainage systems were placed in the community, and all that. I once saw in Geneva a baptismal font that had been used since the 8th century or so; you could see how its size had decreased as practice moved from adult baptism to infant baptism. At the Christiansborg Palace, you could see where walls from the first castle had fallen and been used as foundations for later generations of buildings.

After the palace, I wandered down to the waterfront. I ended up in the Royal Library, rather unintentially. On the ground floor is a café called Øieblikket. I was seriously tempted to buy some coffee there, just because it was the only thing remotely resembling a Kierkegaard connection that I found, but it was very expensive. Instead I wandered the stacks (as much as I could; they were mostly closed except to folk with research purposes). At 1 pm, music began playing throughout the main hall: it was a mixture of massive chords, tinkling bells, and light drumming (I think it was supposed to be a drum; it sounded like claves at first). It lasted three or four minutes and was marvelously relaxing.

Next I starting walking back to the train station, thinking I’d find a cheap cup of coffee and an internet connection there. Instead I was sidetracked by a sign indicating the National Museum was having a special exhibit on the life and times of Tycho Brahe. There were period furniture and costumes, including a full suit of armor. They also had reproductions of some beautiful plates from a 1603 book on astronomy, and instruments like those Brahe used (his tools have apparently been lost). I just browsed for a bit and moved on.

Copenhagen is a neat city. It seems like the bicycles almost match the cars in number. The buildings are a mixture of grand old architecture and chic modern flair. Next time I go, however, I’ll be ready for my Kierkegaard pilgrimage.

brief update

I’m writing this from Nashville, having just completed my usual 16 hours of driving + 4 hours of sleeping at a rest stop to get here from Ithaca. I’ll be here until tomorrow, then in Memphis until September 7, when I fly to Marseille. At some point I really must take a shower.

Some days are just too full of motion and emotion for one to properly sit down and write about it in the midst of everything. That’s what the last three weeks have been like. For now, suffice it to say that I’m happier than I think I’ve ever been, I miss Ithaca and the people there tremendously, I’m much more settled about leaving for France than I was three weeks ago, and it’s taken a lot of work to get to this point. Details will come in later essays.