“I believe that in all men’s lives at certain periods, and in many men’s lives at all periods between infancy and extreme old age, one of the most dominant elements is the desire to be inside the local Ring and the terror of being left outside.”
Read the whole essay here. Sorry for the pause in blogging -- I should start up again soon.
How many unlabeled graphs are there with \(n\) vertices and \(k\) edges? (More below the fold.)
Read MoreExciting news! Jie Liu has proven a conjecture of mine -- thereby resolving an old conjecture of Sommese. I'll briefly explain his result in this post.
Read MoreIn 1910, G.K. Chesterton (one of my favorite authors -- I highly recommend The Man Who Was Thursday and the Father Brown mysteries) diagnosed all the world's problems. You can read his famous diagnosis, What's Wrong With the World, at Project Gutenberg. And if you do so, you'll find that Chesterton was a bit of a monster.
Aaaaaaayyyyyyyy....
If you don't want to read the whole thing, you can read Scott Alexander's excellent review. Chesterton's ideas are wildly conservative, but they are beautifully written:
“What is wrong is that we do not ask what is right.
...
Now (to reiterate my title) this is what is wrong. This is the huge modern heresy of altering the human soul to fit its conditions, instead of altering human conditions to fit the human soul. If soap boiling is really inconsistent with brotherhood, so much the worst for soap-boiling, not for brotherhood. If civilization really cannot get on with democracy, so much the worse for civilization, not for democracy. Certainly, it would be far better to go back to village communes, if they really are communes. Certainly, it would be better to do without soap rather than to do without society. Certainly, we would sacrifice all our wires, wheels, systems, specialties, physical science and frenzied finance for one half-hour of happiness such as has often come to us with comrades in a common tavern. I do not say the sacrifice will be necessary; I only say it will be easy. ”
Chesterton goes on to explain why modern (liberal) values are wrongheaded; he is against feminism, he opposes educating the masses, he is appalled by socialism and against industry. His opposition is rooted in appeals to what he believes are universal values: the desire for order and prosperity, for equality, for each child to have a chance at happiness. Of course he twists these values in the service of (what I believe to be) awful ends.
It is heartening, though, to see how thoroughly his brand of conservatism has lost. Chesterton concludes his essay with a description of contemporary technocratic attempts to reduce the prevalence of lice among poor children:
“A little while ago certain doctors and other persons permitted by modern law to dictate to their shabbier fellow-citizens, sent out an order that all little girls should have their hair cut short. I mean, of course, all little girls whose parents were poor. Many very unhealthy habits are common among rich little girls, but it will be long before any doctors interfere forcibly with them. Now, the case for this particular interference was this, that the poor are pressed down from above into such stinking and suffocating underworlds of squalor, that poor people must not be allowed to have hair, because in their case it must mean lice in the hair. Therefore, the doctors propose to abolish the hair. It never seems to have occurred to them to abolish the lice. Yet it could be done.”
He ends with a beautiful appeal to what he believes must be a universal value, desired by all:
“Now the whole parable and purpose of these last pages, and indeed of all these pages, is this: to assert that we must instantly begin all over again, and begin at the other end. I begin with a little girl’s hair. That I know is a good thing at any rate. Whatever else is evil, the pride of a good mother in the beauty of her daughter is good. It is one of those adamantine tendernesses which are the touchstones of every age and race. If other things are against it, other things must go down. If landlords and laws and sciences are against it, landlords and laws and sciences must go down. With the red hair of one she-urchin in the gutter I will set fire to all modern civilization. Because a girl should have long hair, she should have clean hair; because she should have clean hair, she should not have an unclean home: because she should not have an unclean home, she should have a free and leisured mother; because she should have a free mother, she should not have an usurious landlord; because there should not be an usurious landlord, there should be a redistribution of property; because there should be a redistribution of property, there shall be a revolution. That little urchin with the gold-red hair, whom I have just watched toddling past my house, she shall not be lopped and lamed and altered; her hair shall not be cut short like a convict’s; no, all the kingdoms of the earth shall be hacked about and mutilated to suit her. She is the human and sacred image; all around her the social fabric shall sway and split and fall; the pillars of society shall be shaken, and the roofs of ages come rushing down, and not one hair of her head shall be harmed. ”
That is, he believes it incontrovertible that little girls should have beautiful long hair! If you ever doubt that we've come a long way in the last century, remember: the last, unquestionable value, the denominator of Chesterton's thought, is just unimportant today. That red-heard she-urchin should be able to cut her hair however she damn well pleases.
This is Biosphere 1:
Photo Credit: The NASA Deep Space Climate Observatory.
And this is Biosphere 2:
Photo Credit: Wikipedia User Johndedios. I want to go to there.
Biosphere 2 was an attempt by Space Biosphere Ventures (and the Institute for Ecotechnics) to develop an entirely closed, self-sustaining environment. In 1991, 8 people went in, intending to stay inside for two years. With the exception of one major injury, requiring medical intervention in the outside world, they succeeded -- despite declining oxygen levels and interpersonal conflict so severe that by the end of their mission, the group was barely on speaking terms. The second mission lasted only a few months before members from the first crew returned to sabotage the biosphere. In 1995, the $200 million building was sold to Columbia University, and then later ownership was transferred to the University of Arizona, where it serves as a research location and tourist attraction (which I hope to visit in a few months).
The biospherians themselves were a strange mixture of scientists and experimental theater performers, led by the charismatic John Allen and funded by billionaire (and, based on the design above, aspiring supervillain) Ed Bass. I've just purchased Rebecca Reider's Dreaming the Biosphere, which I'll probably review here soon. So stay posted if you'd like to know more.
Biosphere 2 is not the only attempt at a man-made closed ecological system:
(AFP Photo/Sian Proctor) Are lovers of geodesic domes more likely than average to shut themselves away in an isolated habitat for years on end?
From the Daily Mail (ugh). This man clearly loves his biosphere.
If anyone knows of other examples of closed ecological systems, especially ones in which people have lived (aside from e.g. spacecraft), I'd be thrilled to hear about them.
In June 2016, Neil deGrasse Tyson proposed (not entirely seriously, as this series of tweets should make obvious) the creation of a new country: Rationalia, governed by the dictum "All policy shall be based on the weight of evidence." To Tyson and the other citizens of Rationalia (including physicist Brian Greene, whose office I briefly occupied while mine was under construction this summer), this was obviously a good idea...
Read MoreWhen I was in graduate school, I came up with what I think is a nice proof of the fundamental theorem of algebra. At the time, I wrote it up here somewhat formally; I thought it might make a nice blog post, since the formal write-up obscures the very simple underlying ideas. The goal was to use the minimal amount of technology possible -- in the end I use just a little algebra and some elementary point-set topology, as well as the implicit function theorem...
Read MoreA dilemma is a difficult choice between two alternatives. I recently learned that there is a word for a choice between three alternatives: trilemma. But what if I have a hard choice between four options?
I was curious, so I did some googling. It turns out that there's some disagreement as to what a choice between four options should be called -- is it a quadrilemma or a tetralemma? (The Greek and Latin prefixes for "three" are both "tri-," so there's no conflict in that case.) I'd argue for the Greek tetralemma, as the suffix -lemma comes from the Greek word for premise, and Google seems to agree: there are 25,100 hits for tetralemma and only 6,940 for quadrilemma.
Here's a dilemma: should I be spending my time on this or on something more important?
I was curious as to how this played out for more -lemmas; the Greek "tetralemma" and "pentalemma" dominate the Latin "quadrilemma" and "quintilemma," respectively. However, the internet seems to have found the Latin-Greek mix "sexalemma" irresistible, for obvious reasons, and the Latin "septalemma" easily won out over the Greek "heptalemma." There are a huge number of "nonalemmas," apparently, and almost no "ennealemmas." And apparently people with 100 choices prefer the Latin-Greek creole "centilemma" to the pure Greek "hectalemma." I was unable to find either the Latin or the Greek prefix for 99, but I'm sure that for those linguists with 99 problems, this ain't one.
Some interesting multilemmas:
The Charleston Mercury argued, reported the New York Times in 1861, that the Confederate States were caught on the horns of a quadrilemma: their options were to negotiate, engage in privateering, fight on the sea, or use the economic power provided by the cotton trade to forward their interests.
On the other hand, the "non-classical logic of India" preferred the tetralemma, or catuskoti, which was the claim that a statement could be either true, false, neither, or both(!). This seems to be an originally Buddhist idea which has made its way into other parts of Indian philosophy.
Perhaps following Lewis, contemporary Christian apologists have come up with quintilemmas and other myrialemmas. The pentalemma seems to be largely of interest to online dictionaries.
Please don't google "sexalemma." On the other hand, there are several interesting hexalemmas -- for example, this paper by Campbell Brown argues that it is better to exist than not, rebutting previous depressing work of Benatar (David, unfortunately, not Pat). Brown postulates the existence of a person, named Jemima, who exists in many worlds, and compares those worlds to one in which she does not exists. He extracts six alternatives from this comparison -- if you are interested despite my description, feel free to read more.
The Buddhist tetralemma above has been extended to a septalemma by the Jains. Other literature refers to it as a heptalemma.
Apparently the problem of time in quantum gravity leads to an octalemma.
And most of the hits for "decalemma" are the result of Google generously interpreting my search as an interest in "decal Emma."
Let \(\mathscr{A}_g\) be the moduli space of principally polarized Abelian varieties of dimension \(g\). The complex-analytic space (stack) associated to \(\mathscr{A}_g\) is a \(K(\pi,1)\); that is, its only non-vanishing homotopy group is \(\pi_1\), which is \(Sp_{2g}(\mathbb{Z})\). In particular, the cohomology of \(\mathscr{A}_g\) is the same as the cohomlogy of \(Sp_{2g}(\mathbb{Z})\).
Jesse Silliman, a graduate student at Stanford, has told me an argument that shows that this is in some sense maximally untrue when one considers the "etale homotopy type" of \(\mathscr{A}_g\).
Read MoreI've just finished reading Cathy O'Neil's book Weapons of Math Destruction, which I highly recommend. (One notable feature of the book is that the skull and cross-bones on the cover is the second known example of mathematical piracy.)
Wea-puns of mass destruction?
The book is, as you might guess from the title, quite negative about the use of big data and mathematical models in government and the corporate world. This is a point of view that I felt some knee-jerk disagreement with; that said, Cathy is quite clear that her intent is only to discuss the negative features of big data and the use of mathematics in social and business planning:
“Big data has plenty of evangelists, but I’m not one of them. This book will focus sharply in the other direction, on the damage inflicted by WMDs [weapons of math destruction] and the injustices they perpetuate.”
And I think one should read the book with that comment in mind; of course the models the book complains about have some redeeming features and effects. But that complaint (which is prevalent in the Amazon reviews) misses the point -- to decide whether, on balance, they are a good thing, one has to have a careful accounting of their evils. This book is that accounting -- it makes no pretense at even-handedness and does not try to weigh the good against the bad, except in the most minimal way. I don't view that as a strike against it.
I do have some mild quibbles with the book -- I think that in some cases, the book is uncharitable to the users of the algorithms it objects to. For example, on page 110, Cathy discusses the use of personality tests in job applications. Certain answers on these tests reveal that the test-taker has "high narcissism." "Who wants a workforce peopled with narcissists?," the text asks. This section is at best misleading -- as the author probably knows, the narcissism these tests discuss is not necessarily pathological. Rather, narcissism in this setting is a technical term, which may in fact be healthy. And throughout, the book offers the potential for abuse of algorithms as a strike against them (or offers anecdotal cases of abuse). For example, in the discussion of car insurance companies' use of opt-in technology that tracks one's driving habits, Cathy suggests that soon this technology will be opt-out, at a significant cost. I'm not necessarily skeptical that this will happen, but I'd argue that we should wait for the abuse to occur before objecting to the technology.
In any case, I think this is an important (and excellent) book, and a necessary counterweight to the techno-utopianism to which I, and many in government (in particular in the current administration), business, and academia are often prone. I doubt the book will cure me completely of my faith in technocracy. But I think its real goal is likely to temper that faith with some skepticism. At bottom, the book advocates for rigor in modeling, and for internalizing negative externalities -- who can argue with that?
I've been thinking recently about typefaces -- the four to eight readers of this blog may have noticed that the font used in the body text of these posts has changed. I've also been thinking about best practices for mathematical typesetting, for my next paper. I might write a more serious post on this topic another time.
While researching the topic, I ran into a few interesting articles:
Hoefler and Frere-Jones in the film Helvetica. Credit: Helvetica (documentary) Directed by Gary Hustwit.
One of my favorite books growing up was Dougal Dixon's Man after man: an anthropology of the future, which imagines the development and speciation of humanity in the far future -- under the influence of both genetic engineering and apocalyptic disaster.
Somehow the genetically engineered humans of the far future have mullets.
I looked back on the book recently and was struck by how imaginative Dixon is, but also how his imagination is limited in some ways -- the future he imagines is visibly an '80s future (see: the haircuts of the "hivers" he imagines in the picture above). You can find a semi-legal copy of the book online here.
Let \(R\) be a ring -- it is well-known that the category \(R\text{-mod}\) of (left) \(R\)-modules does not determine \(R\). For example, the functor $$-\otimes R^n: R\text{-mod}\to \text{Mat}_{n\times n}(R)\text{-mod}$$ is an equivalence of categories.
That said, there is one additional piece of data that lets us recover the ring \(R\) from the category \(R\text{-mod}\).
Read MoreJohn Lesieutre has just sent me an exciting new preprint in which he constructs a smooth projective variety \(X\) such that
Whether or not such varieties existed has been an open question (asked by many people) for a long time.
Read MoreI'm teaching Calculus III at Columbia this semester, and am kind of amazed at the exploitative prices charged for Stewart's Calculus, 8th Edition, "Early Transcendentals". (Not that anyone knows what "Early Transcendentals" are). On the other hand, I now know how Stewart was able to build the famous "integral house":
Who says that math doesn't pay?
The man loved calculus...
Stewart's concert hall, in which he could, for a brief moment, forget the cries of the millions of calculus students whose textbook purchases paid for his mansion habit
Still at UGA, at the very enjoyable but weirdly named conference TAAAG. The conference featured a very interesting talk by Padmavathi Srinivasan, as well as mini-courses by Michel Raibaut, Ben Williams, and Arnav Tripathy (whose website I can't find).
Arnav's great talk on the integral Hodge and Tate conjecture, with lots of gestures
There were also many short talks, by many of the other participants.
Padma gave a great talk about her work comparing
As I understand it, the conductor only depends on the family of curves (it is more or less the difference in Euler characteristic between the general fiber and the special fiber of a minimal regular model), whereas the discriminant only depends on the covering of the base given by the family of Weierstrass points. In the case of a minimal regular model, these numbers are just the Euler characteristics of the vanishing cycles sheaf on the family of curves (resp. the family of Weierstrass points). It would be nice to enhance her inequality to a map of sheaves...
This is a picture of Alexander Grothendieck, taken near the end of his life:
You shall not pass!
I took this picture of a page in a book that was just lying around in Oberwolfach a couple of weeks ago (I think the book was published by the Fields Institute). Unfortunately, I failed to note down an exact attribution -- if anyone knows the name of this book/the photographer, please let me know.
EDIT: Commenter DH writes that the source of this photo is Masters of Abstraction, by photographer Peter Badge.
This is a continuation of a previous post. Recall that we wanted to prove the following claim:
Claim. Let \(\rho\) be a representation of \(SL_4\) such that no irreducible subrepresentation of \(\rho\) descends to \(SL_4/\mu_2\). Then if \(\rho\) is self-dual, we have that $$8\mid\dim \rho.$$
Read MoreStill at UGA -- I just saw a great talk by Jason van Zelm (a student of Nicola Pagani who apparently does not have a webpage), constructing non-tautological cycles on \(\overline{\mathscr{M}_g}\) for \(g\geq 12\)...
Read MoreI'm at UGA for the week, in between SWAG and TAAAG. Today Danny Krashen gave a great talk on this paper of Auel, First, and Williams. The paper is one of the latest in a long tradition of papers which construct counterexamples by making a topological computation and then approximating the relevant topological spaces by algebraic varieties. (To my knowledge, this technique began with Totaro, but it has been exploited to great effect by Antieau, Williams, and most recently, fellow Ravi student Arnav Tripathy.) ...
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