Hence, alchemy.

[This post is something of a sequel to my previous post.]

William Chaloner was born sometime around 1650, making him maybe a decade older than Isaac Newton. He did not receive the schooling Newton did, and he certainly didn’t have a chance at Cambridge. He had the misfortune to be apprenticed into a trade with little future: making nails. A machine — the slitting mill — had arrived that readily produced rods of steel that could easily be cut up and hammed into nails, rendering a previously skilled trade an unskilled one. With protective guilds unwilling to admit him into a more lucrative trade, and arriving in London with no obvious means to support himself, he turned to criminal enterprises.

That is, Chaloner’s first attempt to rise above mere subsistence turned him into a purveyor of sex toys. London in the 1690s was as famous, or perhaps notorious, for its spirit of sexual innovation as Berlin would be in the 1920s. Prostitution was ubiquitous, as much a part of the life of the wealthy as it was that of the poor, who supplied most of the trade’s worker’s. The best brothels vied to outdo each other in their range of offerings — so much so that Dr. John Arbuthnot, a man about town in the early eighteenth century apparently spoke for many when he told a madam at one of the better houses, “A little of your plain fucking for me if you please!”

Newton and the Counterfeiter by Thomas Levenson, pg 57

Chaloner soon moved on to various forms of con-artistry: quack medical advice, divination, and “thief-taking”. The latter involved informing on criminals or political subversives in order to collect financial reward. The Metropolitan police would not be formed until the 19th century, so such people were often the only avenue available for bringing criminals to justice. But the thief-takers often played both sides, exploiting whatever opportunities they could get, and often goaded people into committing crimes so that they could be “caught”. Chaloner made it his business to play both sides.

However much Chaloner made from such rackets — and it would not have been inconsiderable — he wanted more. And the biggest racket in all of England was going on in plain sight, with evidence everywhere to be seen. The racket in question was counterfeiting the King’s coin.

It was not an especially sophisticated game. The low production quality of hammered coins meant that an enterprising fellow could clip the edge of a coin and the coin would still be a coin, but you also had a fingernail of precious gold or silver. And in England it was silver that was of interest. The state of the commodity markets in Europe meant that you could take your pile of silver clippings to continental Europe, buy their cheaper gold, then return to England and convert it all back to silver at a profit and start clipping all over again. Classic arbitrage.

As a consequence, the silver coins of England were beginning to look somewhat diminished. Many of them weren’t even silver at all. Many were outright counterfeits made of baser metals. This caused all kinds of problems, not least of which was the ability of King William III to pay his own troops to fight his war in France. Foreign bankers were unwilling to accept English currency at a good price, and silver was vanishing from England for mainland Europe.

The solution was the Great Recoinage of 1696. The old coins were to be replaced with new machine-struck coins that bore milled edges to prevent clipping and render counterfeiting extremely difficult. This British state at this point in history was rife with corruption, sinecures, and cronyism, so initially at least this whole project was chaotic and in real danger of disaster. It was during this financial turmoil that Chaloner seized the opportunity and set up sophisticated counterfeiting operations that managed to produce high quality fakes of the new coins.

Reading Newton and the Counterfeiter by Thomas Levenson, it is unclear if counterfeiting really was so great a scam. Certainly there was no effective law enforcement in England at this time. And while Chaloner was committing a capital offense, juries were unwilling to sentence men to death on the contradictory hearsay that actually arrived in court. That said, the kind of operation Chaloner ran required the cooperation of a great many people. Not only the skilled craftsmen required to make the dies used to cast the fake coins, and the crew to actually run the production line, but also the actual buyers for the knock-off coins. All these people could potentially betray you. Even if you did not face the executioner, you might have to endure a brief stay in London’s hellish Newgate jail.

The jail used in 1696 was almost brand new, constructed on top of the ruins left by the Great Fire of 1666. The facade of the rebuilt prison was given a hint of the elegance with which its architect, Sir Christoper Wren, hoped to endow the whole city. But such graces did nothing to alter the essential character of a place that was, as Daniel Defoe’s Moll Flanders put it, not “the emblem of hell itself” but a kind of entrance to it” too. Defoe wrote from personal experience: he had been imprisoned there briefly, for debt. Other celebrated inmates confirmed Defoe’s judgement. Casanova, imprisoned at Newgate under accusation of child rape, called it “this abode of misery and despair,” and infernal place “such as Dante might have conceived.”

pg 151

Chaloner would be pursued, with unusual diligence, for his crimes by the recently appointed Warden of the Mint who he had been provoking with the particular flagrancy of his crimes and deceits. In what was an act of considerable bravado Chaloner, who had already been caught for counterfeiting activities, conducted a political campaign to gain access to the Royal Mint, ostensibly to offer his “expertise”, but in reality to take whatever advantage he could. This political campaign involved impugning the newly arrived Warden. As perhaps the title of Levenson’s book has given away, this Warden was Isaac Newton, the celebrated natural philosopher.

Given the absence of anything remotely like a rigorous understanding of economics, soliciting Newton’s views on the currency crisis in England was a pretty reasonable thing to do. That said, everyone seemed to have a view on potential solutions. Newton’s own views would be borne out — not just his understanding that re-coinage was necessary, but also the inevitable failure of having a currency simultaneously based on both gold and silver, and his prescient views on the potential of fiat currency. But his duties as Warden of the Mint were simply to oversee the re-coining, and prosecute clippers and counterfeiters.

The first of these tasks Newton was eminently suited, given his facility with quantitative reasoning. He also had the virtue of considering his position as more than a mere sinecure. Having tired of life in Cambridge, he had been seeking some eminent position in London with which to apply his talents. He made the entire process the object of his attention, from the amount of coal consumed each day, to the rate at which the crews could, and reasonably should, hammer our the coins. Under his oversight the re-coining was completed ahead of schedule. (And to the standards of the day, far more safely than it would have otherwise been done).

The second of his tasks — prosecuting counterfeiters — he abhorred. Nevertheless Newton proved himself to be utterly ruthless. The full details of the lengths he went to have been lost — in that the paperwork was deliberately destroyed in part of what was likely a cover-up.

Conduitt chose not to explain why Newton wanted to destroy the papers, but one inference is that Newton enjoyed the role of inquisitor too much. In this view, Newton proved willing, perhaps eager, to terrorize his captives in pursuit of the necessary confessions and betrayals with a viciousness that even that strong-stomached time would tolerate. Formally, torture had not been used in England as an investigative tool for about half a century before Newton came to the Mint. Elizabeth I had face repeated rebellion, often animated by Catholic ambitions on her Protestant throne — and she was England’s most prolific torturing monarch …

But while official torture fell out of favor, interrogators still knew how to put the boot in as needed. Isaac Newton had plenty of ways to extract the information he wanted from reluctant prisoners and he made use of them. Most of them were within the customary bounds of police detection: trading in fear, not pain. He offered brief reprieves for information: he coerced husbands with threats and promised rewards to wives and lovers. But there is one — and only one– reference to his use of more brutal methods in the records he did not burn. In March 1698, Newton received a letter from Newgate written by Thomas Carter, one of Chaloner’s closes associates. The letter was one of a flurry of messages Carter had sent to confirm that he was eager to testify against his former co-conspirator, but this one had a postscript. “I shall have Irons put on me tomorrow,” he wrote, “if yo[ur] Worship not order to the contrary.” In other words: Don’t hurt me! Please. I’ll talk. I’m ready.

Newton and the Counterfeiter by Thomas Levenson pg 165

Ultimately, Newton was victorious. He was patient and methodical and able to rally his superior resources to hound his man, subjecting Chaloner to an extended stay in Newgate while he gathered witnesses and finally wrong-foot him in the trial. The trial itself being a brief and prejudiced affair, as characterized English justice at that time.

But beyond the torture and lack of due process, there was a central hypocrisy to Newton’s activities. Newton and his famous chums were themselves guilty of crimes quite reminiscent to the ones he was prosecuting. The main difference, I think, was that Newton was practicing the upper class equivalents, which were not concerned with actually making a pile of money, but of a more recreational nature. Here is a passage on his relationship with John Locke:

In part, he relished the opportunity to tutor so well regarded a man. He gave Locke a private, annotated edition of the Principia and composed for him a simplified version of the proof that gravity makes the planets travel elliptical orbits. But Newton’s intimacy with Locke seems to have extended well beyond such benevolent displays of mastery. From the beginning, Newton allowed himself to write openly about secret matters. Both men had subterranean interests — in alchemy, for one, the ancient study of processes of change in nature; and in questions of biblical interpretation and belief, which brought them to the edge of what the established English church would damn as heresy.

Newton and the Counterfeiter by Thomas Levenson pg 43

And more seriously, and quite parallel to the crime of counterfeiting Newton was a very active alchemist. Literally attempting to turn base metals into gold. A process if successfully performed at scale would have created unprecedented economic chaos. But like I say, that wasn’t his ultimate goal. Really he was looking to alchemy to settle the theological implications of his scientific endeavors. He saw performing alchemy as a means of proving the intervention of “God” (or rather Newton’s own notion of God) in the natural world:

He knew that all the theorizing, all the theological argument, all the indirect evidence from the perfect design of the solar system could not match the value of one actual, material demonstration of the divine spirit transforming one metal into another in the here and now. If Newton could discover the method God used to produce gold from base mixtures, then he would know — and not just believe — the the King of Kings would indeed reign triumphant, forever and ever.

Newton and the Countereiter by Thomas Levenson pg 85

It should be understood that once you set aside all the secrecy and strange codes that Newton cloaked his alchemist pursuits in, the experiments he performed were serious and rigorous. Even if he failed to make any progress or establish any new body of knowledge. In this enterprise at least he resembles quite closely many of his peers — making quite serious, but ultimately unsuccessful attempts at making a breakthrough.

William Chaloner was hanged from the neck until he was dead, on 22 March 1699. It was not the worse fate he could have met under English law. Newton was not in attendance. He would live on until 1727 when he died in his eighties and buried in Westminster Abbey.

Levenson has a recently released book that seems to pick up where this one left off, tracking the rise of modern finance and the influence the Scientific Revolution had on it. I’ve also stumbled on this podcast where Cambridge historian of science, Patricia Fara, discusses her own upcoming book which seems to have considerable overlap with Levenson’s. The first question she is asked is how Isaac Newton managed to die a wealthy man, which was actually a pretty good place to start. (Newton had invested in the East India Trading company, which means, among other things, slavery.)

There has been a murder in Gathertown

If you orient yourself temporally you may remember that back in August there was a online fracas involving mathematics. A teenage girl, doing her makeup before work, decided to take the opportunity to lay down for her TikTok followers her skepticism about the idea of math generally:

Who came up with this concept? “Pythagoras!” But how? How did he come up with this? He was living in the … well I don’t know when he was living, but it was not now, where you can have technology and stuff, you know?

Grace Cunningham, TikTok user.

As was keenly observed by the many keen observers out there, the initial response was a pile-on that combined general misogyny with gen-Z hatred; it was the latest installment in the long running complaint about kids these days. This reactionary abuse was soon countered by a more positive wave of responses that acknowledged that her questions were not only legitimate, but exactly the kind of questions our curriculum does little to answer.

I don’t think many mathematicians are particularly satisfied about the way our subject is generally taught. At the university level I find it hard to love force marching students through rote material, stripping centuries worth of mathematics of all its scientific and historical context along the way. So obviously I am happy to see any student kicking back at what we inflict on them. But for those who have made mathematical communication their vocation it was a solid gold opportunity to evangelize. Euginia Cheng wrote a pdf answering Grace Cunningham’s formalized list of questions, and Francis Su wrote a twitter thread.

Grace Cunningham was calling the bluff on the pretenses of her education. In particular, the pretense that you should obviously be learning whatever we are telling you. “Why are we even doing this?” is a legitimate question in a mathematics course, and “why on earth did anyone prove these theorems in the first place” is an even better one. “How did people know that they were right,” presents the awkward truth that people most often are certainly incorrect about many things. What makes these questions awkward is that the people teaching you mathematics will frequently know little to nothing about the history and context within which the theory was developed. Mathematicians are terrible, as a rule, at scholarship, and the history of ideas within mathematics is an essentially distinct field. Most of the context that I have for the mathematics I do is essentially gossip, urban myth, and pablum. Fortunately, while we might be terrible historians we remain excellent gossips, so at least we have plenty of stories to tell.

(I should also concede that it is impossible to generalize in any way about most of my peers. Many of them are tremendously knowledgeable about all kinds of things and wonderful educators. I am, at least to some extent, either projecting or talking about our very worst failings.)

I was dissatisfied by the responses I found to Cunningham’s questions. Not least of all because I don’t think they really answer the questions. No actual historical context was given. The answers more resemble the kind of general motivation and propaganda we give students to encourage them to listen in class and do their homework. I think a good answer would address the fact that the people who developed much of classroom mathematics had some pretty wild ideas about what they were doing. Their motivations would be pretty alien to us, and is a far cry from their homework, exams, or getting a well paid job.

Just to make this explicit: How many of us who have ever taught or taken calculus a calculus course have even done any astronomy? Just from doing a little reading, an obvious observation seems to be that when people sat down to first learn calculus from Newton’s Principia, the big incentive for them was the promise of a serious set of answers about the Sun, the Earth, the Moon, the stars, and even comets. A modern mathematician explaining their motivation for calculus today is a little like a 21st century Western evangelical Christian explaining what the “Old Testament” is all about to an orthodox rabbi.

My modest reading has focused on the life of Isaac Newton. I read Jame’s Gleick’s biography of Newton (highly recommended) and I have a few more on the shelf. I already had some understanding that aside from developing calculus Newton was a heretic, alchemist, and later in life warden of the royal mint. I knew he lived through times of plague, apocalypse, dictatorship, conspiracies, and his work was a major part of the scientific revolution. Particularly pertinent to Cunningham’s question is the fact that for centuries after Newton’s death there was a suppression of the full range of Newton’s intellectual activities. It was only when John Maynard Keynes acquired a substantial portion of Newton’s surviving papers at auction that the truth came out. For a long time Newton’s preoccupations would be considered intellectually inconvenient for all those trying to boost his posthumous reputation, and that of British science with it.

The idea of knowledge as cumulative — as a ladder, or a tower of stones, rising higher and higher — existed only as one possibility among many. For several hundred years, scholars of scholarship had considered that they might be like dwarfs seeing further on the shoulders of giants, but they tended to believe more in rediscovery than progress. Even now, when for the first time Western mathematics surpassed what had been known in Greece, many philosophers presumed they were merely uncovering ancient secrets, found in sunnier times and the lost or hidden.

Isaac Newton – James Gleick (pg 34-35)

Here is a not entirely fanciful reading of Newton’s life: starting his university career dissatisfied with the existing knowledge, and curious about the latest developments in astronomy, Newton develops his theory of calculus. But he is not yet really a scientist. He is still very much a wizard. A young man who has uncovered some profound secrets and is keen to discover more. He invests huge amounts of time and energy in alchemy and theology. The alchemy involved tracking down obscure texts that he hoped would contain the secret knowledge of transforming base metals into precious metals, and his notebooks from this period often amount to his copying out these texts. It also involved working with mercury, a poisonous metal known to drive the alchemists who used it to madness.

His theological interests were no less hazardous since they would have been viewed as clearly heretical to both the Protestant and Catholic religious authorities at the time. By studying the earliest Greek manuscripts he discovered that the concept of the Trinity — that the Godhead is three and one; Father, Son, and Holy Spirit — emerged late in the early church, and certainly couldn’t be considered part of the original Christian tradition. Newton concluded Jesus was not at the same level as God and had never claimed to be. At a time in England when having Catholic sympathies could land you in trouble, this was a dangerous view to have.

I would argue that Newton transformed from a wizard into a scientist the moment the German mathematician Leibniz independently derived his own theory of calculus. No longer had Newton uncovered a forgotten knowledge, but he had derived a theory that someone else could also derive. He was now entered into a race to establish the precedence for his own results — and this meant writing up.

For decades his tools of calculus had languished in notebooks and in his mind. Now he had to write them down, and he chose to present them in the style of Euclid’s Elements, with axioms, definitions, lemmas, theorems. And most intriguingly, in order to prove the correctness of his theory, he drew upon experimental data: astronomical observations from the newly establish Greenwich observatory and tidal charts. He was able to explain and predict natural phenomena that perplexed his contemporaries such as the sudden appearance of comets, and their unusual paths across the night sky. We can recognize this now as a prototype of the modern scientific method, but back then it was controversial, becoming part Newton’s dispute with Leibniz.

Newton wrote many private drafts about Leibniz, often the same ruthless polemic again and again, varying only by a few words. The priority dispute spilled over into the philosophical disputes, the Europeans sharpening their accusations that his theories resorted to miracles and occult qualities. What reasoning, what causes, should be permitted? In defending his claim to first invention of the calculus, Newton stated his rules for belief, proposing a framework by which his science — any science — out to be judged. Leibniz observed different rules. In arguing against the miraculous, the German argued theologically. By pure reason, for example, he argued from the perfection of God and the excellence of his workmanship to the impossibility of the vacuum and of atoms. He accused Newton — and this stung — of implying an imperfect God.

Newton had tied knowledge to experiments. Where experiments could not reach, he had left mysteries explicitly unsolved. This was only proper, yet the German threw it back in his face: ‘as if it were a Crime to content himself with Certainties and let Uncertainties alone.’

Isaac Newton – James Gleick (pg 176-177)

Data is now the recognized currency of modern science, and theology is, well, theology. The mathematical analysis that makes calculus rigorous didn’t come until much later. Newton had started using infinite series in his calculus, but it was understood that you had to be careful because sometimes you could get some bad results.

When Cunningham asks her TikTok followers how early mathematicians knew they were right, in Newton’s case at least, it seems that there are three answers. Newton first convinced himself with arguments we would not consider mathematically rigorous along with his his own empirical observations. Decades later he convinced his peers by publishing a full written account of his theory (in Latin) that provided supporting data. Then a century or so later the full theory of mathematical analysis was developed.

These questions have complicated answers for Newton, but they are really no less complicated for us today, even if they are quite different answers. We live in the age of the arxiv, computer assisted proofs, machine learning, and bodies of work that amount to many hundreds of pages. I’m not going to lie; I love the drama of it all. Some would like to present mathematical proof and progress as being an enterprise free from being sullied with the humanity of its practitioners. For my part I am of the belief that the reasons people commit themselves to mathematics are more complicated than just the aesthetic appreciation of equations.

On finite covers of surfaces with boundary…

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.)

Gromov, cheese, pretending to quit mathematics, and French.

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.)