No Bigons
Corpus Hypercubus
Petra – the Treasury building was not a treasury.
June 2, 2019 nobigons
I will be leaving Haifa in August this year and returning to the UK. After three years living in Israel, and being in no particular rush to do and see everything, we suddenly realized that if there was anything we still wanted to see or do, then we had better get on with it. There remain a bunch of museums in Jerusalem I'd still like to visit, but there was one major thing we had conspicuously failed to do: visit the ancient city of Petra in neighboring Jordan.
Due to its appearance at the climax of Indiana Jones and the Last Crusade, the treasury building is Petra's most iconic sight . The entire ediface is cut out of the rock face and is as impressive in reality as it appears on film. Despite the name, the building is not a treasury — in fact it is believed to have been a mausoleum for the Nabatean King Aretas IV. The name derives from the Arabic name Al Khazneh which is rooted in folk tales of treasures hidden inside.
But before we could admire the Treasury Building, we actually had to get to Petra. Due to relatively healthy relations between the Kingdom of Jordan and Israel, it is possible to cross the border between the two countries, albeit at one of a handful of crossing points. Most tourists book with a tour group, but it is entirely possible to go solo, so we opted to plan our own adventure. This meant an early start in Haifa to catch one of the sporadically scheduled buses that run from Nazareth into Amman, the capital of Jordan.
Amman is far less tourism oriented than Petra (or Aqaba where we would exit Jordan after visiting Petra), but it was worth visiting just to see the huge Roman amphitheater which dominates the downtown area.
Although I do not have any similar photographs to document the experience, eating the local kanafeh in Amman was a similarly worthwhile experience. In case you don't know what kanafeh is, allow me to explain. Imagine a cheese cake. Now make that cheesecake thinner and serve it hot and upside down. And instead of shortcrust pastry you have a base (now a lid since you turned it upside down) of shredded filo pastry with pistachio nuts sprinkled over it. And the cheese is a local Palestinian variety — a cross between something like mozzarella and ricotta. It is served with hot, sweet, sugary rosewater is poured over it. Kanafeh is a killer desert. Even as I type this I'm grieving the fact that I haven't eaten more of it. I may have to try making some after I have left. I won't be the first person to have tried, as this youtube video I stumbled over bears witness:
From Amman it was another early start and long bus journey to arrive at Petra where we were disgorged from our bus into the throng of tourists. Tourism is a major part of the Jordanian economy and Petra is a central to that. But until 1812 the capital of the Nabataean Kingdom, once a major regional trade center, was essentially lost to the world, the location a secret known only to Bedouin tribesmen intent on preventing the remains of the city from being plundered.
Johann Ludwig Burckhardt, was born in 1817 to a Swiss family of merchants. After studying Arabic at Cambridge, and was tasked by the African Association with finding the source of the Niger River, but instead decided to improve his Arabic first by traveling around the Middle East. Soon he was adopting the persona of Sheikh Ibrahim ibn Abdallah an Indian Muslim whose strange accent could easily be explained away so long as he wasn't actually in India. After hearing talk of what he realized must be the ruins of a city known to classicists only through passages of ancient text, he convinced a Bedouin to guide him there on the pretense of completing a pilgrimage.
I hired a guide at Eldjy, to conduct me to Harouns tomb, and paid him with a pair of old horse-shoes. He carried the goat, and gave me a skin of water to carry, as he knew that there was no water in the Wady below.
In following the rivulet of Eldjy westwards the valley soon narrows again; and it is here that the antiquities of Wady Mousa begin. Of these I regret that I am not able to give a very complete account: but I knew well the character of the people around me; I was without protection in the midst of a desert where no traveller had ever before been seen; and a close examination of these works of the infidels, as they are called, would have excited suspicions that I was a magician in search of treasures; I should at least have been detained and prevented from prosecuting my journey to Egypt, and in all probability should have been stripped of the little money which I possessed, and what was infinitely more valuable to me, of my journal book. Future travellers may visit the spot under the protection of an armed force; the inhabitants will become more accustomed to the researches of strangers; and the antiquities of Wady Mousa will then be found to rank amongst the most curious remains of ancient art.
Travels in Syria and the Holy Land, by John Burckhardt
At the time you could easily imagine Burckhardt fancying himself a brave and impetuous explorer. I think most of us today would, at the very least, would consider betraying the trust of your guide a major ethical lapse. Not that Burckhardt had long to reflect on the choices he made. He died five years later of dysentery.
Visiting Petra today is hazard free, to an extent Burckhardt himself could have imagined. There is still some legwork involved, however. Walking through the entire site is a good 8K hike. But it is an amazing 8K. You start by wandering through the outer limits of the city where you can still see graves cut into the rock face of the sounding valley. Then you hike through the Siq — a norrow passage between two rockfaces that legend tells was originally formed by the water that gushed from the stone that Moses struck with his staff. Eventually you arrive at the Treasury building, and then into the valley where you find the rest of the city.
The length of the hike combined with the heat seemed to take plenty of visitors by surprise. There is a brisk trade in camel, horse, and donkey rides to take the overwhelmed back and forth. You can be forgiven for not anticipating how much there is to see (there is an entire city) if all you were expecting was the Treasury Building. There is a huge amphitheater, temple, and — if you are ready to go to the site's furthest extent — the impressive Monastery which easily matches the Treasury for grandeur:
We also made a friend on the way up to the Monastery.
Finally, after spending 24 hours in Petra we took a taxi down to Aqaba where we were able to enjoy more kanafe before leaving Jordan and crossing into Eilat the next morning.
The view from Aqaba into Eilat.
The Hula valley, scarecrows, and malaria.
January 26, 2019 nobigons
In December we took a trip to the Hula valley, a wetland reserve located in northern Israel and a few hours out from Haifa by bus. The site is a major destination for ornithologists as it lies in the intersection of two major migration flyways. An estimated 500 million birds of over 400 different species pass through the valley each year, although I imagine that current environmental situation may mean those figures are out of date. Climate change notwithstanding we were able to see an awful lot of birds — mostly Cranes — taking a break from the long a perilous journey between breeding and wintering.
If we had visited the valley a hundred years earlier, it would have been a very different experience. We would have run the serious risk of catching Malaria for one thing, because the entire valley was swampland. Only in the 1950s was the process of draining these swamps undertaken. Canals were dug out canals and the river Jordan was deepened. The project was ostensibly a successful; the swamp was gone and the land could be used for agricultural purposes. As is often the case with such human enterprises, however, there were some unintended consequences. These swamps were full of peat — semi-decomposed vegetable matter — that when dried out tends to be seriously combustible. Sure enough the drained Hula valley was struck disastrous underground peat fires.
Unlike the forest fires that were sweeping California last year, which caught headlines and could be seen by satellites in orbit, peat fires can be incredibly hard to detect. They are a deeper, smoldering, slow burning kind of fire that takes place underground. They also happen on enormous scales. If they aren't caught early enough the only thing that can put them out is torrential rain. Peat fires are, globally speaking, the largest fires on earth and are estimated to make up 15% of mankind's contribution to greenhouse gases.
In the Hula valley the damage caused by the peat fires was extensive. In some places the damaged caused to the soil caused depressions as deep as 6-7 meters. This would go on to cause environmental damage to the Sea of Galilee and deterred migrating birds from stopping in the valley. In the 90s the decision was made to flood the Hula valley again. The idea was to develop a nature reserve by controlling the underground water levels. While the valley would remain drained during the winter, it wouldn't dry out during the summer. And as they restored the valley, the birds began return.
The summer in Israel, for me at least, are months of enduring the ordeal of endless mosquito bites. I am however grateful that I'm not running the risk of malaria, a disease that thrives in the swamps such as the one that existed in the Hula valley. So I can't say I disapprove of the original desire to transform the land. The individual who we can really credit of the elimination of the disease from Israel however is Gideon Mer. Visiting Rosh Pina a number of years ago during Passover I was taken to his old house and told that this was the man who had cured Malaria. Given that most diseases aren't eradicated by the miracle drug or vaccine, but rather through public health policy, I think it is fair to say this is close enough to the truth, in Israel at least. Here is his obit in the British Medical Journal:
On the bus back to the Technion, we could see the fields still used for agriculture, less than a kilometer away from the wetland reserve. It seems amazing that you could maintain a bird sanctuary while growing crops next door. If birds like to do anything, or so I've been led to believe, it is to fly into a farmers field and gobble up all the seeds. Indeed, we could see guards on patrol, circling the fields in what looked like little golf carts, ready to scare off any hungry cranes.
It was enough to make me wonder: do scarecrows actually work? I'm more familiar with scarecrows as a trope in popular culture. And bizarrely scarecrows are usually depicted as being almost entirely ineffective in repelling birds, while being a very scary staple of the horror genre. If you visit the Wikipedia page for scarecrows, you'll find there is far more page space devoted to scarecrows in fiction than to their alleged real world function.
I'm certainly not the first person to wonder this, or indeed to take to the internet to find an answer. Steve Coll in his New Yorker Diary back in 2009 describes being struck by the same question, but finds nothing helpful in his search results. It is a good example of an area of knowledge where the internet feels incredibly non-definitive. Since Coll's own inter-quest for an answer something closer to a definitive answer appears close to the top of googles listings: a Mental Floss article linking to scientific studies, government webpages, and the Daily Mail. If you are also beginning to wonder if the ostensible legitimacy of areas of the web are giving way to a wave of jankiness then this article — the most informative survey of they literature on scarecrow efficacy I could find — is only going to stoke your fears.
Confronting the Kugelmugel.
December 31, 2018 January 7, 2019 nobigons
Whatever Christmas spirit I may have lacked for being in Israel, I was able to suitably redress by spending the first week of December in Vienna. The whole city feels like it was built to be decorated with gaudy winter lights.
I arrived on a Sunday afternoon, and after finding my hotel and getting dinner I decided I needed to go off on a bloody long walk. Walking is my principal source of exercise when I am taking these short trips, and I certainly wasn't going to let the inclement weather get the better of me.
Rain will eventually threaten to get the best of you if your waterproof jacket doesn't live up to it's name so finding hot chocolate became a pressing concern. Vienna is world renowned for its cafe culture, but this led google maps to send me towards fancy coffee houses with startlingly long queues for admission. Whatever chain outlets existed weren't open on a cold and wet Sunday evening. Fortunately, while most of the Christmas market had packed up and left, there was still one stall left open offering hot punch (spiced wine and rum) that was able to fortify me until I got back to the hotel.
Vienna is renowned for it's cultural history and visitors can behold the incredible architecture, visit one of the many museums, or experience the live music the city's many music conservatories cultivate. I was only around for a week and most of the day time I was busy hallucinating about line patterns inside the free group. I did manage to spend one morning walking across the city. Given my limited window of opportunity I had to prioritize. So that meant the Museum of Art Fakes, and then the Republic of Kugelmugel.
In 1971 the Austrian artist Edwin Lipburger, with help from his son, built a spherical house for himself out of wood. As you can see from the picture above, the house isn't just vaguely spherical. It is literally a sphere. Lipburger was apparently obsessed with the "cosmic harmony" of spheres. He christened his new home the Kugelmugel. The municipality of Vienna was less enamored with ball shaped houses, and as it turned out they had some kind of rule or regulation forbidding the construction of such a dwelling. This was the beginning of an ongoing dispute between Lipburger and the city that led to him declaring the Kugelmugel an independent state and would escalate to Lipburger going to jail for ten weeks — specifically for erecting unauthorized street signs.
Eventually, some kind of arrangement was reached when in 1982 the Kugelmugel was moved to Prater park, where it can today be found by the fun fair. Lipburger was apparently induced with the promise of electricity and running water, neither of which were ultimately provided. He was the Republic of Kugelmugel's only citizen, and after his death in 2015, the Kugelmugel remains in Prader park as a strange tourist attraction.
Today the Kugelmugel stands among the growing ranks of the worlds micronations — the outsider artists in the world of statecraft. There have been many motivations for claiming your own independent ministate. In the UK the Principality of Sealand existed as a platform for pirate radio in the 60s and 70s. Lipburger's own declaration of independence was made partly in protest but also as some kind of artistic statement. Although stamps and passports were issued, it is hard not to read his endeavor as a means of challenging the idea of a modern state, rather than a credible attempt to create one.
Whatever Lipburger's intention, the impression it gave to this curious visitor was perhaps far from what he would have wanted. It is surrounded by an eight foot fence with barbed wire running across the top, and a sign above the gate give the impression of a former Soviet bloc country that doesn't care for visitors. The sign marking the street Antifaschismus-Platz rings with an irony similar to the likes of the Democratic People's Republic of Korea.
Feral Cats and Cyprus
Last month we decided to take advantage of the off-season and spent a long weekend in Cyprus; one night in Nicosia and then two in the Troodos mountains. The flight is an hour, so if you forget about the hassle of getting to and from the airport it is the perfect short getaway from Israel.
Cyprus and Israel share a number of striking similarities: while Cyrpus is an island, Israel certainly seems like one metaphorically; both are stuck on protracted disputes over sovereignty and the status of certain minorities. But a more immediate similarity was finding ourselves surrounded by large numbers of feral cats. We had only to go as far as the airport bus stop before we encountered a cluster of skittish felines just off the grassy verge. It would be the first of many we'd encounter.
Cyprus now has around 1.5 million stray cats — that exceeds the human population of Cyprus of just under 1.2 million. For perspective, the UK has a population of 66 million, and the latest cat census puts the total stray and feral population at 10 million. (Israel has roughly 2 million cats out there and a population of 8 million.) Both Israel and Cyprus have half-hearted spay and neuter programs that attempt to curb the population growth. According to one article I found Cyprus and Israel share a public perception problem:
She added that the most important thing to do to reduce cat numbers is to educate owners on the benefits of spaying and neutering.
"Unfortunately, a lot of animal owners don't use this service," Foote said. Some even cite religious beliefs, that animals should be left "as God intended them to be".
More cats than humans: population out of control – Cyprus mail.
Back in 2015, Uri Ariel, the Israeli minister of Agriculture and Rural Development, proposed deporting either all male or female cats from Israel as an alternative to having them spayed or neutered. His objection to the otherwise effective program was that it violated Jewish law, in particular the injunction to "be fruitful and multiply." His suggestion was met with widespread scorn in many quarters.
The cats of Cyprus, however, have a claim to a special place in history. All the cat-themed relics that archeologists had been digging up in Egypt, had many to assume that cats were first domesticated under the dynastic rule of the pharaohs. But a cat grave has been found in Cyprus, dated to be roughly 9500 years old, predating the Egyptian claim. Since cats are not native to the island, it can be presumed they were first brought over to Cyprus by Turkish farmers along with all their other domesticated animals when they settled on the island.
There is also the later story of Saint Helena of Constantine (mother of the Emperor Constantine) sending over a boat load of cats from Egypt or Palestine to deal with a snake problem at Saint Nicolas' monastery. We actually took a minor detour to check out the monastery of the cats, knowing nothing about the history, but vaguely hoping, based on the name in the guidebook, that we could find a feline-themed monastic order. Maybe little cats dressed up as monks. Or a painting somewhere of Saint Nicholas being led to the site by the Holy Spirit in the form of a cat. I was to be disappointed: aside from the presence of a fair number of strays, there was nothing evident in the monastery itself to earn the title. Cyprus is covered with a great many Greek orthodox churches, but even during our brief stop I noticed this particular monastery was picking up an unusual amount of traffic — presumably curious cat lovers enticed by the name.
A Greek orthodox church in the Troodos mountains.
Cyprus is a divided island. It was divided by a British army officer in 1964 who drew a line in a map with a green chinagraph pencil. While the situation in Israel/Palestine sits atop of the list of contentious, intractable ethno-national impasses, the situation in Cyprus is not so far behind. While the island sits just off the coast of Turkey, it is demographically dominated by Greek-Cypriots and the Turkish Cypriots remain a minority. Historically, the island passed through many hands: the Venetians, Ottomans, and then the British being the most recent. It was during the British rule that the cause of Enosis — unification with Greece — came to the fore, with the Greek orthodox church being a major agitator for the cause.
In 1950, in the face of British refusal to hold a referendum, the church organized their own. It is hard to take the 95.71% approval on face value as only Greek Cypriots could vote, and those votes had to be publicly registered with signatures in books provided by the Greek orthodox church. Dissenters would effectively mark themselves out for later reprisals. I've been looking through the internet to see if I could find an image of one of these ledgers from the referendum. The following image appears to be attached in some way to the associated Wikipedia page, but I cannot vouch for its authenticity:
I've just finished reading Lawrence Durrell's memoir Bitter Lemons of Cyprus, which recounts three years he spent on the island in the 50s when the situation boiled over into protests, riots, and terrorism. He arrived as a sometime civil servant hoping to be able to live cheaply while starting a full-time writing career. He acquired a cottage in the north of Cyprus after a comic/alarming/farcical property deal with a local family, and settled down into village life. His ability to speak Greek allowed him to befriend the locals and enjoy their goodwill, but he soon becomes increasingly aware that the British administration is blind to the building desire for independence and the potential for unrest.
The perspective Durrell offers is an interesting one. While the Cypriots have no autonomy over their own affairs, he maintains that there was a great deal of goodwill, or "amity", between the locals and the British. He notes that he never once heard accusations of corruption against any governing officials. At the same time he understands that the Greek Cypriots desire to be taken seriously and have their right to self determination respected. But the British are in a bind: declaring Cyprus an independent state is certainly a reasonable demand, but an immediate rush to unify with Greece, subsequent to independence, would alarm the Turkish Cypriots and possibly Turkey itself. Durrell also notes that a Cyprus unified with Greece might not actually be so great for Cypriots. A delicate situation, but the real problem, as Durrell recalls it, was that the British administration didn't really believe that the Cypriots ever had it in them to fight, and thus ignored the issue.
Durrell meanwhile runs short of money and starts teaching in a Nicosia high school before taking a job as a British civil servant on the island. When the troubles begin he sees first hand how badly prepared the administration are to deal with rapid developments, both conceptually and practically. He also witnesses many of his former student enlist in EOKA and their guerilla campaign. In the following passage he confronts one of his former students, caught in the act of bomb-throwing:
He was not far from tears, but the face that he turned to me tried to be composed, impassive. He did not speak but stared at me with a look of furious anguish – as if indeed a wolf were gnawing at his vitals. 'He had a bomb too,' said Foster wearily. 'Bloody little fools! What do they think they gain by it? He threw it in the churchyard by the cross-roads. I suppose he thought he'd scare us all out of our wits.'
'Are you in EOKA?' I asked.
'We are all EOKA. All Cyprus,' he said in a low controlled voice. 'If he wants to know why I threw it in the churchyard tell him because I was a coward. I am unworthy. But the others are not like me. They are not afraid.' I saw suddenly that what I had mistaken for hatred of my presence, my person, was really something else – shame. 'Why are you a coward?' He moved a whole step nearer to tears and swallowed quickly. 'I was supposed to throw it in a house but there were small children playing in the garden. I could not. I threw it in the churchyard.'
Superb egotism of youth! He had been worried about his own inability to obey orders. It is, of course, not easy for youths raised in a Christian society, to turn themselves into terrorists overnight – and in a sense his problem was the problem of all the Cypriot Greeks. If Frangos had been given a pistol to shoot me I am convinced that he would not have been able to pull the trigger. 'So you are sorry because you didn't kill two children?' I said. 'What a twisted brain, what a twisted stick you must be as well as a fool!' He winced and his eyes flashed. 'War is war,' he said. I left him without another word."
"Bitter Lemons of Cyprus" by Lawrence Durrell
In 1960 Cyprus finally became an independent state. In 1974 after a coup backed by Greece's military junta, Turkey intervened, invading northern Cyprus. The coup failed, but so did negotiations between Cyprus and Turkey and after the violent displacement of both Greek and Turkish Cypriots, the north of the island was captured.
We crossed the green line in Nicosia into the Turkish Republic of Northern Cyprus (as of the moment, only recognized by Turkey). While the green line was once impassable after the Turkish invasion, today the most you have to worry about is the queues. We didn't hang around for long, but if we had, we might have seen a whirling Dervishes performance.
My surprise highlight of the weekend was visiting one of the ten painted churches in the Troodos mountains. These are small Byzantine churches with vivid murals on the inside (and recognized UNESCO world heritage sites). I've been in a bunch of churches, and seen many cathedrals with impressive architecture, but something about the interior of the church Pedoulas was impressive in a different way. I recommend checking out the Wikipedia page.
On finite covers of surfaces with boundary…
November 28, 2018 January 19, 2019 nobigons
I have a new preprint on the arxiv, joint with Emily Stark. We provide the first known examples of one-ended hyperbolic groups which are not abstractly coHopfian. That means that there is a one ended hyperbolic group \(G\) which contains a finite index subgroup \(G' \leq G \) that embeds \( G' \rightarrow G\) as an infinite index subgroup. I encourage you to look at the paper for details. The main example and proof can be drawn out on a single side of A4 — it's a simple surface amalgam and we exploit the tremendous flexibility you have when you take a finite cover of a surface with boundary.
We use the following Lemma extensively. It's from Walter Neumann's 2001 paper Immersed and virtually embedded \(\pi_1\)-injective surfaces in 3-manifolds, although, as he says, it is apparently "well known".
The utility of this Lemma is that it reassures you that if you can imagine your desired cover — such as the following I've drawn below — and it satisfies a basic necessary Euler characteristic computations, then the cover does in fact exist.
The surface \(F'\) is a degree 5 cover. The collection of degrees in this example are 1+3+1=5. Note that if the 3 was replaced with a 2 and I tried to find a degree 4 cover with the specified boundary it would be impossible.
For our main example you can compute your desired covers by hand, but it is worth knowing what kind of covers of a surface with boundary you can take. This lemma tells you exactly how much control you have. And mathematical research, like all forms of insecurity, is really all about control.
The proof given above is brief, to say the least, so I think it is worth expanding on the details.
First, we remind ourselves how you might construct such a cover by hand. Take a surface with genus one and a single boundary component. From a group theoretic point of view this is just the free group generated by two element \( \mathbb{F}_2 = \langle x, y \rangle \) bundled together with the conjugacy class of the commutator \( [x,y] \). For me at least, finding finite index subgroups of the free group boils down to futzing around with graphs. Thus, we let \(X\) be a bouquet of two circles and let \(\langle x,y \rangle = \pi_1X\).
I actually drew it by hand.
On the right I drew the surfaces with boundary and on the left I drew the corresponding graphs with the loop corresponding to the boundary. Once you have drawn the graphs out it's easy to verify that you have the boundary components you want. The trick is knowing you can find the desired finite covers. The key insight is that \(\alpha\)-sheeted covers of a graph \(X\) are in a correspondence with representations into the permutation group on \(n\) elements: \( \pi_1 X \rightarrow \textrm{Sym}(\alpha) \)
We can see how this correspondence works in practice in the example I just drew:
Giving each of vertex a number we can see that the edges labelled by a given generator of our free group gives a permutation. In this example the generator \(x\) gives the permutation \((1,2)(3)\) (see the red edges on the right), while the generator \(y\) gives the permutation \((1,2,3)\) (see the blue edges on the left). Thus we have a homomorphism determined by mapping the generators to the corresponding permutation.
At this point we need to be careful because there are some left-right issues hidden here. When I multiply group elements \(xy\) I am composing paths in the fundamental group. That means I concatenate the corresponding paths, starting with the \(x\) path and then following it with the \(y\) path. I'm reading the composition from left to right. In contrast, when I usually compose a pair of permutations \(\sigma_1 \sigma_2\) I compute the composition by reading them from right to left. But in order to be able to interpret my homomorphism correctly I'm going to have to compose my permutations in reverse order, from left to right.
Now we can compute the image of the element corresponding to the boundary curve: $$ xyx^{-1}y^{-1} \mapsto (1,2) \circ (1,2,3) \circ (1,2) \circ (3,2,1) = (1,2,3).$$ Tracing out how the boundary curve lifts is equivalent to computing this permutation element. This makes it clear that there is a single boundary component covering the previous with degree 3. (In this example it doesn't matter in which direction we composed the permutations).
Conversely, choosing pair of permutions, say, $$(1,2)(3,4)\textrm{, and } (2,4,3) \in \textrm{Sym(4)}$$ to be the images of \(x\) and \(y\) we can construct a corresponding cover by taking 4 vertices and adding the appropriate labelled edges:
Now when we compute (remembering to compose our permutations from left to right) the image of our commutator element we get $$ xyx^{-1}y^{-1} \mapsto (1,2)(3,4) \circ (2,4,3) \circ (1,2)(3,4) \circ (3,4,2) = (1,4)(2,3).$$ Thus the surface has two boundary components, each covering the boundary in the base surface with degree two.
The take away from this discussion is that finding suitable covers corresponds to finding a suitable homomorphism $$\phi : \pi_1 X \rightarrow \textrm{Sym($\alpha$)}$$ such that the image of the elements corresponding to boundary curves are permutaions with the desired decomposition into cycles. Our weapon of choice is the fact that any even permutation can be written as the commutator of an \(\alpha\)-cycle and an involution:
Un résultat extrémal en théorie des permutations. Jacques, Alain; Lenormand, Claude; Lentin, André; Perrot, Jean-François, C. R. Acad. Sci. Paris Sér. A-B 266 1968 A446–A448
First we consider the case where \(\Sigma\) has a single boundary component. So \(\Sigma\) is a surface with genus \(g\), Euler characteristic $$ \chi(\Sigma) = 2 – 2g – |\partial \Sigma | $$ and a single boundary component, so \(|\partial \Sigma | = 1\). This corresponds to the free group generated by \(2g\) elements and the group element corresponding to the boundary, which is the product of commutators: $$( \langle x_1, y_1, \ldots x_g, y_g \rangle, [x_1,y_1]\cdots [x_g, y_g] ).$$
Suppose we wish to construct a cover of degree \(\alpha\) with the boundary components of degrees \(\alpha_1, \ldots, \alpha_k\). Then apply the above Theorem to the permutation $$\sigma = (1, \ldots, \alpha_1)(\alpha_1 +1, \ldots, \alpha_1 + \alpha_2) \cdots (\alpha_1 + \cdots + \alpha_{k-1} +1, \ldots, \alpha).$$ The theorem only applies if \(\sigma\) is an even permutation, which we compute to be equivalent to $$ \sum_i (\alpha_i -1) = \alpha – k$$ being even. As \(\chi(\Sigma) = 1-2g\) this is equivalent to \(k\) having the same parity as \(\alpha\chi(\Sigma)\), the sufficient condition given in the statement of our theorem.
Thus there exists permutations \( \sigma_x, \sigma_y \in \textrm{Sym}(\alpha)\) such that \( [\sigma_x, \sigma_y] = \sigma \). The homomorphism $$ \phi: \pi_1 \Sigma \rightarrow \textrm{Sym($\alpha$)}$$ given by mapping \(x\) to \(\sigma_x\) and \(y\) to \(\sigma_y\) therefore corresponds to a cover with the desired boundary.
Now we consider the slightly trickier general case where we have multiple boundary components, which is to say \(|\partial \Sigma| = b\). In which case the pair \((\Sigma, \partial \Sigma)\) corresponds to $$ (\langle x_1, y_1, \ldots, x_g, y_g, t_1, \ldots t_{b-1} \rangle , \{t_1,\ldots, t_{b-1}, t_{b-1}\cdots t_{1}[x_1,y_1] \cdots[x_n,y_n] \} ).$$
Now suppose we desire that the \(i\)-th boundary component is covered with degrees \(\alpha_1^i, \ldots, \alpha_{k_i}^i\), then let $$ \sigma_i = (1, \ldots, \alpha_1^i)(\alpha_1^i +1, \ldots, \alpha_1^i + \alpha_2^i) \cdots (\alpha_1^i + \cdots + \alpha_{k_i-1}^i +1, \ldots, \alpha)$$ for \(1 \leq i \leq b\). Now we wish to find \(\sigma_x, \sigma_y\) such that $$[\sigma_x, \sigma_y] = \sigma_1 \cdots \sigma_b.$$ This requires that the product of the \(\sigma_i\) is even. This means that the sum $$ \sum_i \sum_j (\alpha_j^i – 1) = \sum_i (\alpha – k_i) =\alpha b – \sum_i k_i = \alpha (\chi(\Sigma) -2 + 2g) – \sum_i k_i$$ should be even, which is true precisely when the total number of prescribed boundary components \(\sum_i k_i\) has the same parity as \(\alpha \chi(\Sigma)\).
Given that our parity condition is satisfied, we define our homormorphism \(\pi_1 \Sigma \rightarrow \textrm{Sym}(\alpha)\) as follows:
$$\begin{align} x_1 \mapsto & \sigma_x \\ y_1 \mapsto & \sigma_y \\ x_2 \mapsto & 1 \\ \vdots \\ y_g \mapsto & 1 \\ t_1 \mapsto & \sigma_1^{-1} \\ t_2 \mapsto & \sigma_2^{-1} \\ \vdots \\ t_{b-1} \mapsto & \sigma_{b-1}^{-1} \end{align} $$
Then it only remains to verify that $$ t_{b-1}\cdots t_{1} [x_1,x_2] \cdots [x_g, y_g] \mapsto \sigma_{b-1}^{-1} \cdots \sigma_1^{-1} [\sigma_x, \sigma_y] = \sigma_{b-1}^{-1} \cdots \sigma_1^{-1} \sigma_1 \cdots \sigma_{b-1} \sigma_b = \sigma_b $$ and conclude this gives us our desired cover.
(I'd like to thank Emily for informing me about Neumann's Lemma, and Nir for various discussions related to this.)
Chicago and back to Haifa
November 4, 2018 January 9, 2019 nobigons
I finally managed to capture the chalkboard on camera. The diagram is the central focus and remains up through the entire talk, with various additions and modifications made along the way.
I like to commit to the diagram.
Giving the same talk several times over the space of a month allows you to appreciate a few things. Minor adjustments can make a big difference, especially to pacing. Having already given a talk once really boosts your confidence. Having given a talk four times already doesn't give you complete confidence. Responses can vary dramatically as audiences latch onto different aspects of what you are doing, and as you invariably emphasize specific things. At Boston College they wanted to hear more about special cube complexes, something many of them had heard about but had little exposure to, while at UIC I had audience members who had themselves considered the specific conjecture I am working on and were curious to hear about the obstructions I had encountered.
After roaming all over Manhattan I had a far more limited Chicago experience. I don't think I strayed any further than a mile from the hotel and campus and most of my time was spent talking mathematics.
Now I'm back in Haifa, recovering from jet-lag. Despite the rain that marked my departure a month ago, I've returned to amazing weather.
Boston and then New York
October 30, 2018 May 4, 2019 nobigons
After a week in Boston I'm now in New York. I was out there just over a week and gave seminar talks at Boston College and Tufts. I also saw a free performance of Hamlet at Wellesley College given by Actors From the London Stage. I studied Hamlet for AS-level English literature, so at least ten years ago I had more than a passing familiarity with the play. Although I watched multiple film adaptions at the time (including Hamlet 2000 which I believe was widely considered disappointing, but has a stellar cast and may be worth revisiting), I don't believe I ever actually saw it on stage.
It was a pleasant suprise to discover that Hamlet is great. The initial palace intrigue really does work, once you have taken a ten-year-step-back from studying it on the page. It made me yearn for ghosts to return from their horror fiction exile and claim their rightful place in contemporary drama.
I also went for a walk in the woods and met this fellow:
Now I am in New York. The Big Apple, where everything is Big expensive, and I've been going on Big walks, and now my Big legs feel like they're going to fall off. Oh. And everything is Big expensive. It's like I never even left the airport.
It's like London with a grid system, less rain, and museums that aren't free.
I've been having a blast here.
On top of my list of things I was excited to do in NY was visit The Strand bookstore. I miss the joys of English language bookstores while living in Israel, and The Strand is the epitome of everything I wanted. Extensive and well curated stacks brimming with wonderful literature. The store's reputation came to me partly through word of mouth, but also through youtube videos I subsequently stumbled upon. For example:
Infuriatingly, the video doesn't seriously set out to answer the question posed. I still found it engrossing.
(The individual featured in this video, Michael Orthofer, is a fascinating figure. He is the editor-and-contributor-in-chief of The Complete Review, an internet passion project dedicated to reading, reviewing and documenting the most significant works of contemporary literature. And I don't mean just English literature. I mean world literature. The guy is reading upwards of five books a week and reads in multiple languages. He writes on the blog how much of a relief it was that no Nobel prize for literature was awarded this year, leaving him free to dedicate more time to the actual business of reading and writing. The New Yorker ran a profile on him and his obsession. If you are bookishly inclined you may have already stumbled across his website in the past. It is old school, homegrown html at its finest. Take a look at it. It probably won't surprise you that Orthofer wrote the website in the previous century. I ran into it while trying track down a translation of The Pornographers by Nozaka Akiyuki [I don't even remember why]. It is hard not to already look on the website with a kind of nostalgia. It seems so far removed from what the internet has become. The whole concept is so insane, non-commercial, personal, and unscalable that it feels like a product of a bygone era. Even our academic webpages are now becoming slick, Web 2.0, bland, templated monstrosities. It is interesting to note, however, that The Complete Review outlasted many of its contemporaries that were more commercially inclined. Like Pets.com.)
Although I am determined to travel light on my north American excursion, I went a little nuts and bought a piles of books. As with all the most dreadfully insane acts, it was of a deeply premeditated nature, with it's own internal logic and self justification. I bought a bunch of books, but only short books.
The Scarlet Letter I bought in Boston, the volumes above in Montreal, and everything below in New York.
I also spent a day in the Met, went to see an adaption of Vonnegut's Mother Night at 59E59, dropped by New York Public Library to see Winnie the Pooh, walked across the Brooklyn bridge, and visited ground zero.
My seminar talk at Columbia went fantastically well. I've failed to take a picture of the chalkboard after any of the seminar talks I've given so far, but there is still UIC. And I'll probably scan or photograph my notes as well.
Columbia's campus is amazing.
As my trip progresses and my final return flight approaches, so Halloween also approaches.
Some Halloween treats.
Returning to Montreal
October 24, 2018 February 25, 2019 nobigons
The first overcast sky I'd seen in quite a while.
I set off from Haifa just as the first rain since the start of the summer started to fall. On arrival in Montreal the shorts and T-shirt I had set off in became insufficient to the demands of the local climate, so I had to switch into warmer clothes and revive myself with Tim Horton's hot chocolate.
Two years had passed since I was last in Montreal to defend my thesis — the morning after the 2016 election. It was interesting to see what had changed. Cannabis was about to be legalized. The building work on McGill/Sherbrooke had been completed. The math department was full of unfamiliar faces. Many restaurants had closed (no more SmartBurger). Murals commemorating the late Leonard Cohen had appeared across the city. Poutine was pretty much as I remember it, though.
Montreal is now also home to the world's largest permanent barbie exhibit. Which is completely free to access.
One among many of the Barbies on display.
Canadian Thanksgiving took us all by surprise, so I gave my McGill seminar talk earlier than expected. It was the first outing of the talk I'm touring while I'm out here in North America, presenting my new proof (and generalization) of Leighton's graph covering theorem. The seminar went well, but I feel like I'm going to be experimenting a little to find the best way to explicate the coset summation argument I use. Piotr appreciated the fact that I was able to give the entire proof within the hour.
I was also able to do quite a bit of math. A fair chunk of Thanksgiving was spent discussing math with Dani and scrawling our ideas all over my notebook.
Don't read too much into anything going on here…
I also took pictures of new murals I spotted around this city:
Gromov, cheese, pretending to quit mathematics, and French.
September 26, 2018 January 19, 2019 nobigons
In December last year, the Notices of the AMS ran a collection of reminiscences in memory of Marcel Burger (1927-2016), the late French differential geometer. He was also a former director of the Institut des Haute Etudes Scientific and, according to the Wikipedia, played a major role in getting Gromov positions in Paris and at the IHES in the 80s. Gromov contributed to the article, listing Berger's mathematical achievements, before sharing a more personal anecdote:
Within my own field Gromov has had a profound influence. His essay Hyperbolic Groups led to the term "combinatorial group theory" being more or less abandoned and replaced with "geometric group theory". As a graduate student I found the monograph frequently cited as the origin for an astounding range of ideas. At some point I had trouble finding a copy of the paper online and for a brief moment wondered if the paper itself were just an urban myth or elaborate hoax.
Gromov's foray into group theory is just one episode in a long career. His first major breakthrough was in partial differential equations; the "h-principal" which, according to Larry Guth, was analogous to observing that you don't need to give an explicit description of how to put a wool sweater into a box in order to know that you can actually put it into the box. It is actually a little hard, at least for myself, to get a full grasp on Gromov's other contributions as they span unfamiliar fields of mathematics, but I recommend this nice What is… article, written by the late Marcel Berger, describing Gromov's contribution to the understanding of isosystolic inequalities.
There is the perception about great mathematicians that, while they are no doubt very clever, somehow they have lost a little of the common sense that the rest of us possess. I'm naturally inclined to discount such thinking; I am far happier believing that we are all fool enough to take absence of common sense, on certain occasions. However, it is hard to dismiss the idea entirely given the following admission by Gromov, in his personal autobiographical recollections he wrote on receipt of the Abel prize in 2009:
The passage speaks for itself, but I wish to emphasize that Gromov's discovery of the correct pronunciation of French verb endings came after ten years of living in Paris. You don't have to have extensive experience learning foreign languages to appreciate how remarkable an oversight this is. It certainly puts his remarks in other interviews about dedicating one's life to mathematical pursuits in a rather strong light.
If you read the entire autobiographical essay you will find it rather short on biography. The best biographical details I have found came from this La Monde article written on his being awarded the Abel prize. Even then the details seems to be coming second hand. The most interesting parts concern his leaving the Soviet Union:
I wanted to leave the Soviet Union from the age of 14. […] I could not stand the country. The political pressure there was very unpleasant, and it did not come only from the top. […] The professors had to teach in such a way as to show respect for the regime. We felt the pressure of always having to express our submission to the system. One could not do that without deforming one's personality and each mathematician that I knew ended up, at a certain age, developing a neurosis accompanied by severe disorders. In my opinion, they had become sick. I did not want to reach that point.
Gromov, according to Georges Ripka, via La Monde (apologies for my translation)
As the article goes on to explain, Gromov decided his best chance to escape was to hide his mathematical talent. He quit math, quit his university, and burned all his academic bridges. He stopped producing mathematics. Or at least writing it. He joined some meteorological institute and did research on paper pulp. Eventually, he was granted permission to emigrate to Israel, but on landing in Rome in 1974 he set off for the States instead, where Jim Simons secured him a position at Stony Brook.
(As a side note, this is Jim Simons of Renaissance Technology and Simons foundations fame. Simons, aside from his mathematical contributions, is probably one of the most important mathematicians alive in terms of funding, supporting, and propagandizing for mathematics. He is considered influential enough for the New Yorker to profile. Alongside Gromov, he is one of the names that every mathematicians should know.)
Yom Kippur, 2018
Yom Kippur, or the Day of Atonement, is the tenth day in the Jewish new year. It precedes the start of the new academic year here in Israel. Due to all the High Holidays — Rosh Hashanah, Yom Kippur, and Sukkot — new arrivals at the Technion, eager to start postdoctoral fellowships, find themselves sitting things out for a month until university life commences. For those of us already here, this is usually time spent getting job application material together. So it isn't the most exciting month of the year for us.
There is a silver lining: witnessing first hand the one day (from sunset to sunset) that the streets of Israel are empty of cars. Because Yom Kippur is more than just a legal holiday. Everything is shut down: TV, radio, airports, shops, and public transport. I've even heard a third hand story of someone with a minor injury being told to wait until the next day before going in to hospital.
I was in Israel two years ago for Yom Kippur. I went out for an evening jog and, to my pleasant surprise, found the streets full of children tearing downhill on their push bikes being chased by their parents. While observant Israeli's fast, a lot of secular Israelis — kids in particular — take the opportunity to do some serious cycling. You need to appreciate how unfriendly Israeli roads are to cyclists every other day of the year to understand how magical a time this must be.
This year we went out for a walk around the Carmel center just after sunset to enjoy the empty streets. We spotted this car sitting in the middle of the street, apparently abandoned as the final seconds of the day had passed by.
You can find plenty of videos on youtube of people out cycling on the empty streets, but here is just one of them:
As you can see from the video, it is inaccurate to say that there are absolutely no cars on the road. In this video, at least, the reason is that their route passes through Daliyat al-Karmel , a Druze town, where few people observe the holiday.
Even within Haifa, however, we were still able to spot a few cars on the streets. One in particular had a gang of skateboarders clinging to it. I was just quick enough to take the following picture:
This year I wanted to see the highway deserted, so in the afternoon we walked down from the mountain to the beach, crossing the main road along the coast on the way.
One way or another, this was my last Yom Kippur in Israel. I think it might be the one facet of Israeli life — a day when even the cars stop — that I will miss the most.