Read the whole essay here. Sorry for the pause in blogging -- I should start up again soon.

# Graph Theory and \(\mathfrak{sl}_2\)

How many unlabeled graphs are there with \(n\) vertices and \(k\) edges? (More below the fold.)

Read More# Jie Liu on projective space

Exciting 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 More# What's wrong with the world?

In 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.

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:

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:

He ends with a beautiful appeal to what he believes must be a universal value, desired by all:

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.

# Biospheres

This is Biosphere 1:

And this is Biosphere 2:

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:

- BIOS-3 was a similar project in the Soviet Union, begun in 1973. The longest mission inside was 180 days. I haven't been able to find much information on it -- if anyone knows the story, please let me know.
- The HI-SEAS project is an attempt to simulate life on Mars, in the habitat below. The latest mission lasted a year and ended in August 2016, and was apparently extremely boring, with the exception of a breakdown in the water treatment system -- a disaster that could have killed everyone on the mission, or at least forced them to open the door.

- There have been many attempts to create small closed ecological systems, including some by the original Biosphere 2 team. Indeed, they created some while inside Biosphere 2 -- a biosphere in a biosphere in a biosphere. These are now a pretty common elementary school activity, and some beautiful examples can be purchased on Amazon. Here's a particularly entertaining 40-year-old example:

- Biospheres are a staple in fiction. The widely panned Bio-Dome, in which two idiots are trapped in a habitat clearly modeled after Biosphere 2, is an early example. The Martian is a recent (and excellent) entry in the genre.

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.

# Rationalia, USA

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...

# A "minimal" proof of the fundamental theorem of algebra

When 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 More# -Lemmas

A 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.

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 Lewis Trilemma: the argument that Jesus was either "Lunatic, Liar, or Lord." The eponymous Lewis is C.S. Lewis of Narnia fame, who was also a famous Christian apologist.
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."

# Are Shimura Varieties \(K(\pi, 1)\)'s?

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 More# Weapons of Math Destruction

I'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.)

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:

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?

# The Typographical Equivalent of a Knife Fight

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:

- Adam Townsend has written a nice article about choosing a font for mathematics writing at Chalkdust Magazine.
- Dan Rhatigan wrote an interesting master's thesis about mathematical typesetting -- one of the pleasures of reading these sorts of documents is that they are almost invariably beautifully typeset -- Dan's thesis is no exception.
- In sadder news, I just found out that the venerable type foundry Hoefler & Frere-Jones (now Hoefler & Co.) has split up in what this article refers to as "the legal equivalent of a knife fight in the street." My CV is typeset in Hoefler Text; the Rhatigan thesis above is typeset in Whitney, also created by the firm. Frere-Jones alleged that Hoefler promised him a 50-50 partnership and then delayed giving him equity for
*13 years*, even after Frere-Jones transferred ownership of valuable typefaces to the firm for a nominal sum of $10. You can see Hoefler and Frere-Jones, enjoying happier times, in the clip below (from the hit documentary*Helvetica*).

# Man After Man

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.

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.

# Morita Theory, Tannaka Duality, and Approximate Tannaka Duality

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 More# Varieties with infinitely generated automorphism group

John Lesieutre has just sent me an exciting new preprint in which he constructs a smooth projective variety \(X\) such that

- \(\text{Aut}(X)\) is discrete, and
- \(\text{Aut}(X)\) is not finitely generated.

Whether or not such varieties existed has been an open question (asked by many people) for a long time.

Read More# Integral House

I'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":

# TAAAG

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).

There were also many short talks, by many of the other participants.

Padma gave a great talk about her work comparing

- The discriminant of a hyperelliptic curve, with
- The conductor of the curve.

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...

# My Hero

This is a picture of Alexander Grothendieck, taken near the end of his life:

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.

# \(SL_4/\mu_2\) and a mod \(8\) congruence

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 $$8\mid\dim \rho.$$

# More Tautological Classes?

Still 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 More# Krashen the party

I'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.) ...

Read More