One of my favorite questions is: for which \(g, n, p\) is the moduli space of \(n\)-pointed genus \(g\) curves \(\mathscr{M}_{g,n, \mathbb{F}_p}\) unirational/uniruled? Will Sawin has just posted a beautiful paper on the ArXiv answering this question in most cases, for \(g=1\). Indeed, he shows that for \(n\geq p\geq 11, \mathscr{M}_{1, n, \mathbb{F}_p}\) is not uniruled... (more below the fold)

Read More# Mumford at the Met

I was at the Met yesterday, where by chance I ran across the Concinnitas Project. (Pictures below the fold.)

Read More# C.S. Lewis on Commutative Algebra

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 graphs are there with \(n\) vertices and \(k\) edges? Below is a graph of the number of isomorphism classes of (unlabeled graphs) on \(7\) vertices.

## Graphs on 7 vertices

Observe that the number of graphs on \(7\) vertices with \(k\) edges is unimodal --- it has only one peak. Indeed, if \(g_{n,k}\) is the number of graphs with \(n\) vertices and \(k\) edges, the sequence $$(g_{n, k})_{0\leq k\leq {n\choose 2}}$$ is always unimodal for fixed \(n\). The remarkable thing is that one can prove this fact using the representation theory of the Lie algebra \(\mathfrak{sl}_2\).

Yesterday I gave a talk for the Columbia Undergraduate Math Society explaining this proof. The use of \(\mathfrak{sl}_2\) to prove unimodality is a well-known technique, which has been developed by Richard Stanley and others, but I find it quite difficult to understand the meat of these sorts of arguments when reading the combinatorics papers in which they appear, since they are often treated in the full generality of "posets with the Sperner property." So I figured I'd sketch the argument in this case. You can find full notes on my talk (which include a classification of the finite-dimensional irreducible representations of \(\mathfrak{sl}_2\)) here, or on my talks page.

Recall that an \(n\)-dimensional representation of \(\mathfrak{sl}_2\) is the same as a triple of \(n\times n\) matrices \(E, F, G\) such that $$[E,F]=H, [H,E]=2E, [H,F]=-2F.$$ The key fact we will need from the classification of \(\mathfrak{sl}_2\)-reps is the following:

**Theorem 1.** Let \(V\) be a finite-dimensional complex representation of \(\mathfrak{sl}_2(\mathbb{C})\), and let \(d_k(V)\) be the generalized eigenspace of \(H\) corresponding to the eigenvalue \(k\). Then the sequences $$(d_k(V))_{k\text{ odd}}, (d_k(V))_{k\text{ even}}$$ are unimodal.

This theorem follows trivially from the classification of finite-dimensional \(\mathfrak{sl}_2\)-representations. (In fact, \(H\) always acts diagonalizably, and its eigenvalues are always integers, but we won't use this.) We call the (generalized) eigenspace of \(H\) with eigenvalue \(k\) the \(k\)-*weight space. *We are now ready to prove:

**Theorem 2.** Fix a non-negative integer \(n\). Then the sequence $$(g_{n, k})_{0\leq k\leq {n \choose 2}}$$ is unimodal.

**Proof. **By the theorem, it suffices to construct an \(\mathfrak{sl}_2\)-rep \(W_n\) such that the weight spaces of \(W\) have dimension equal to \(g_{n,k}\), and so that the eigenvalues of \(H\) all have the same parity. We let \(G_{n,k}\) be the set of *labelled *graphs with \(n\) vertices \(\{1,\cdots, n\}\) and \(k\) edges, and set \(U_{n,k}=\mathbb{C}^{G_{n,k}}\). That is, \(U_{n,k}\) is a vector space with a basis consisting of labelled graphs on \(n\) vertices and with \(k\) edges. The symmetric group \(S_n\) acts on \(U_{n,k}\) by permuting the vertices. We set $$U_n=\bigoplus_k U_{n,k}, W_{n,k}=U_{n,k}^{S_n}, W_n=\bigoplus_k W_{n,k}.$$

**Proposition.** $$\dim(W_{n,k})=g_{n,k}$$

**Proof of Proposition.** Exercise.

In particular, it suffices to construct a \(\mathfrak{sl}_2\)-representation on \(W_n\) whose weight spaces are the \(W_{n,k}\), and all of whose eigenvalues have the same parity. In fact, we will construct an \(\mathfrak{sl}_2\)-representation on \(U_n\) which commutes with the \(S_n\)-action (and thus preserves \(W_n\)).

Let \(e_{i,j}: U_n\to U_n\) be the operator (defined on the basis of labelled graphs) $$e_{i,j}: g\mapsto \begin{cases} g\cup (i,j) & \text{ if }(i,j)\not\in g\\ 0 & \text{otherwise}\end{cases}$$ and \(f_{i,j}: U_n\to U_n\) the operator $$f_{i,j}: g \mapsto \begin{cases} g\setminus (i,j) & \text{ if }(i,j)\in g\\ 0 & \text{otherwise.}\end{cases}$$ That is, \(e_{i,j}\) adds an edge between vertices \(i\) and \(j\) if there isn't one there already, and sends a graph to \(0\) otherwise; similarly \(f_{i,j}\) removes the edge between \(i\) and \(j\) if there is one, and sends the graph to \(0\) otherwise. We set $$E=\sum_{i<j} e_{i,j}, F=\sum_{i<j} f_{i,j}, H=[E,F].$$ One may easily check that if \(g\) is a graph with \(k\) edges, then $$Hg=\left(2k-{n\choose 2}\right)g,$$ so the eigenspaces of \(H\) are precisely the \(U_{n,k}\). Moreover $$[H, E]=2E, [H,F]=-2F.$$ Thus we have constructed the desired \(\mathfrak{sl}_2\)-representation on \(U_n\); \(F, E\) and hence \(H\) commute with the \(S_n\)-action by definition, so we obtain the desired representation on \(W_n=U_n^{S_n}\). Now Theorem 2 follows immediately from Theorem 1. \(\square\)

### Notes

One observation is that we've actually proven a lot more than we set out to -- indeed, if \(\chi\) is any representation of the symmetric group, we've shown that the dimensions of the \(\chi\)-isotypic pieces of \(U_{n,k}\) have unimodal-dimensions. These pieces may be interpreted as spaces of functions on labelled graphs which transform in a certain way under relabelling.

The use of \(\mathfrak{sl}_2\) to prove unimodality of combinatorial sequences is a pretty venerable technique -- for example, if \(P_{k,\ell}(n)\) is the number of ways of partitioning \(\ell\) into \(k\) pieces of length at most \(\ell\), one may prove that the sequence \(P_{k, \ell}(n)\) for fixed \(k,\ell\) and varying \(n\) is unimodal using \(\mathfrak{sl}_2\)-representation theory. This proof has a reinterpretation -- one may think of the representation in question as coming from the Hard Lefschetz Theorem on the cohomology of the Grassmannian!

**Question.** Is there a smooth polarized variety with an \(S_n\)-action whose cohomology is isomorphic to \(U_n\) above as an \(S_n\times \mathfrak{sl}_2\)-representation? (Here the \(\mathfrak{sl}_2\)-representation comes from the polarization from Hard Lefschetz).

Richard Stanley has used the Hard Lefschetz theorem to prove amazing results about \(f\)-vectors of polytopes; more recently, June Huh, Eric Katz, and others have proven some really cool related combinatorial statements using algebraic geometry.

# 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.

The conjecture was:

**Conjecture** (L---, now proven by Jie Liu). Let \(X\) be a smooth projective variety over \(\mathbb{C}\), and \(\mathscr{E}\) an ample vector bundle on \(X\). Suppose that $$\text{Hom}(\mathscr{E}, T_X)\not=0.$$ Then \(X\) is isomorphic to \(\mathbb{P}^n\) for some \(n\).

This conjecture was known (due to Andreatta-Wisniewski) in the case that there was a map \(\mathscr{E}\to T_X\) of constant rank; Liu in fact proves something slightly stronger:

**Theorem (Liu).** Let \(X\) be a smooth projective variety over \(\mathbb{C}\), and suppose \(T_X\) contains an ample subsheaf \(\mathscr{F}\). Then \(X\simeq \mathbb{P}^n\) for some \(n\).

The proof is quite nice. Old results of Araujo, Druel, and Kovacs immediately show that there exists an open subscheme \(X_0\subset X\) and a map \(X_0\to T\) whose fibers are isomorphic to \(\mathbb{P}^r\) for some \(r\). We wish to show that \(T\) is a point. (Up to this point, this more or less follows Andreatta-Wisniewski). This is not particularly hard if the \(\mathbb{P}^r\)-bundle \(X_0\to T\) is defined away from codimension two; indeed, in this case, one may find a proper curve in \(T\) and obtain a contradiction using standard techniques originally due to Peternell-Campana in this case -- this is essentially how Andreatta-Wisniewski proceed. The key fact is that if \(f: Y\to C\) is a \(\mathbb{P}^r\)-bundle over a proper curve \(C\), the relative tangent sheaf \(T_f\) contains no ample subsheaves.

Unfortunately if \(\mathscr{F}\) is not locally free, I don't see how to extend the fibration to codimension two, so this approach seems to be sunk.

Liu cleverly gets around this issue by working on an *open* subset of \(X\) rather than a closed subset. Unfortunately the paper is quite densely written and I do not as yet understand all the details -- if someone would like to explain what is happening "morally," I would love to hear it.

In any case, the results of the paper in which I made this conjecture allow one to deduce from this result an old conjecture of Sommese, which classifies smooth varieties containing a projective bundle as an ample divisor. This conjecture has attracted a lot of interest over the last few decades, so it's very gratifying to see it put to rest.

# 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 \(\mathscr{A}_g\). (Jesse proposed it as an argument for why (compactifications of) \(\mathscr{A}_g\) and its level covers have trivial \(H^1\) for \(g>1\), but a slight enhancement actually shows much more). Moreover, the argument is anabelian, which always gets me excited. A similar argument should work for many Shimura varieties of rank \(>1\), but I'll just sketch it in the case of \(\mathscr{A}_g\) with \(g>1\).

Let \(G\) be the etale fundamental group of \(\mathscr{A}_g\), namely \(Sp_{2g}(\hat{\mathbb{Z}})\). This is a pretty pathological group (for example, its center is infinitely generated), but whatever. Given any continuous representation $$\rho: G\to GL_n(\mathbb{Z}_\ell),$$ there is a natural map $$\iota: H^*(G, \rho)\to H^*(\mathscr{A}_g, \mathscr{F}_\rho),$$ where \(\mathscr{F}_\rho\) is the lisse sheaf on \(\mathscr{A}_g\) associated to \(\rho\), and the left hand side is continuous cohomology. Now suppose that \(\rho\) actually comes from some representation of \(Sp_{2g}(\mathbb{Z})\) into \(GL_n(\mathbb{Z})\).

I claim that in this case \(\mathscr{F}_\rho\) shows up in the cohomology of some power of the universal Abelian variety, and is cut out by absolute Hodge cycles, so in particular \(\rho\) canonically extends to a representation of the arithmetic fundamental group of \(\mathscr{A}_g\) (possibly after some finite field extension). Moreover, at any \(K\)-point \(x\) of \(\mathscr{A}_g\), the action of the Galois group of \(K\) on \(\rho\) factors through its tautological action on the Tate module of the fiber of the universal Abelian variety at \(x\).

Let's choose a geometric point at some CM point \(x\in \mathscr{A}_g\), with associated CM field \(E\). Then the action of the Galois group of \(E\) on \(G\) factors through the Artin reciprocity map by the main theorem on CM; hence the action on \(H^*(G, \rho)\) is Abelian. But the action on the etale cohomology of \(\mathscr{A}_g\) is independent of basepoint, so running over all CM basepoints, we can see that anything in the image of \(\iota\) has to be of Tate type! In particular, none of the interesting cohomology of \(\mathscr{A}_g\) comes from its etale fundamental group. The same argument works for any level cover of \(\mathscr{A}_g\). Thus in some sense these things are as far from \(K(\pi, 1)\)'s as possible.

Of course, \(H^1(\mathscr{A}_g)=H^1(G)\) more or less by definition, so this shows that \(H^1(\mathscr{A}_g)\) (and any level cover thereof) is pure of Tate type --- in particular, any smooth compactification of (a level cover of) \(\mathscr{A}_g\) must have \(H^1=0\).

Exercise -- how did I use that \(g>1\)? (Obviously this last sentence is untrue when \(g=1\)). Hint: It's when I said that the action of Galois on \(G\) factors through the Artin reciprocity map; in this step, I used the solution to the congruence subgroup problem, which is of course false for \(g=1\).

# 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 \(R\text{-mod}\). Namely, if we let \(F\) be the functor $$F: R\text{-mod}\to \mathbb{Z}\text{-mod}$$ which sends an \(R\)-module \(M\) to its underlying abelian group, we have that $$R\simeq \text{End}(F).$$ This is a version of Tannaka duality for algebras; we call this functor \(F\) the *fiber functor*.

Let us consider a special case of this situation. Suppose \(R\) is a topological ring, complete with respect to some two-sided maximal ideal \(\mathfrak{m}\), and suppose that \(R/\mathfrak{m}=k\) is a (commutative) field. Then any continuous \(R\)-module is a (possibly infinite) iterated extension of \(k\). A little exercise shows that if \(S\) is another topological ring, complete with respect to \(\mathfrak{m}'\) and with commutative residue field \(k'\), then a (continuous) functor $$G: R\text{-mod}\to S\text{-mod}$$ sending \(k\) to \(k'\) and inducing an isomorphism $$\text{Ext}^*(k, k)\overset{\sim}{\to} \text{Ext}^*(k', k')$$ is an equivalence of categories. In particular, if such a functor commutes with the fiber functors, then $$R\simeq \text{End}(F_R)\simeq \text{End}(F_S)\simeq S.$$

In some of my work on fundamental groups, I've run into the following situation, which can be thought of as an "approximate" version of the remarks in the previous paragraph. Suppose that \(R, S\) are complete with respect to some ideals \(\mathscr{I}, \mathscr{I}'\) which are *not* maximal; rather, \(R/\mathscr{I}\simeq S/\mathscr{I}'\) are local rings whose only non-trivial ideal is the nilradical (i.e. they are direct limits of Artinian local rings). Moreover, we have a functor $$G: R\text{-mod}\to S\text{-mod}$$ which sends \(R/\mathscr{I}\) to \(S/\mathscr{I}'\). However, the induced map $$\text{Ext}^*(R/\mathscr{I}, R/\mathscr{I})\to \text{Ext}^*(S/\mathscr{I}', S/\mathscr{I}')$$ is not an isomorphism; rather, its kernel and cokernel are annihilated by some fixed element \(\beta\); moreover, there is a functor going the other way with similar properties, and both of these functors commute with fiber functors. Then one finds that \(R\), \(S\) are *approximately isomorphic*, in that there are maps \(R\to S, S\to R\) so that the induced maps $$R/\mathscr{I}^n\to S/{\mathscr{I}'}^n, S/{\mathscr{I}'}^n\to R/\mathscr{I}^n$$ have kernel and cokernel annihilated by some fixed power of \(\beta\) (in terms of \(n\).

Actually, this is not exactly the situation I am in, but a toy version which is close enough to the real thing. This kind of "approximate" Tannaka duality seems to arise naturally from certain constructions in integral \(p\)-adic Hodge theory; if someone else has thought about it, I would love to hear about it!

# 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. John's construction is characteristically awesome. He observes that the set of automorphisms of \(\mathbb{P}^1_{k}\) which have a unique fixed point \(\infty\), namely the group of upper triangular unipotent matrices in \(PGL_2(k)\) contains many non-finitely generated subgroups as long as \(k\) is infinite --- in particular, this group is isomorphic to the additive group of \(k\). For convenience, we assume \(k\) has characteristic zero (which seems to be necessary for his construction as written, though I'm sure it can be adapted to positive characteristic). John observes that we can construct infinitely generated subgroups as follows:

If \(z_1, z_2, z_3, z_4\) are points in \(\mathbb{P}^1_k\), there is a unique involution \(\iota: \mathbb{P}^1_k\to \mathbb{P}^1_k\) such that $$\iota(z_1)=z_2, \iota(z_3)=z_4.$$

If \(p_1, p_2, p_3, p_4, p_5\in \mathbb{P}^1_k(k)\), we let \(\iota_{ij, kl}\) be the involution such that

$$\iota_{ij, kl}(p_i)= p_j, \iota_{ij, kl}(p_k)=p_l$$

Now if we let

\(p_1=0, p_2=1, p_3=2, p_4=3, p_5=6\)

the subgroup \(\Gamma_P\) of \(PGL_2=\text{Aut}(\mathbb{P}^1_k)\) generated by the \(\iota_{ij, kl}\) contains the matrices

$$\begin{pmatrix} 3 & 0\\ 0 & 1\end{pmatrix}, \begin{pmatrix} 1 & 1\\ 0 & 1\end{pmatrix}$$

and thus the matrices

$$\begin{pmatrix} 1 & \frac{1}{3^n} \\ 0 & 1\end{pmatrix}$$

which generate a non-finitely generated abelian group.

Now, John constructs a surface \(S\) containing a smooth rational curve \(C\) such that the group of automorphisms of \(S\) which preserve \(C\)

- is finite index in \(\text{Aut}(S)\), and
- has image containing \(\Gamma_P\) in \(\text{Aut}(C)\).

By various tricky product and blowup constructions, John then constructs a \(6\)-fold whose automorphism group is isomorphic to the group of automorphisms of \(S\) that

- preserve \(C\), and
- have the unique fixed point \(\infty\) in \(C\).

In particular, this automorphism group has an abelian quotient (the automorphisms of \(C\) fixing \(\infty\)), which contains the non-finitely generated group generated by the matrices

$$\begin{pmatrix} 1 & \frac{1}{3^n} \\ 0 & 1\end{pmatrix}$$

from before. Hence it is not finitely generated.

If I understand correctly, it's not too hard to find blowups \(Y\) of \(X\) such that the group of automorphisms of \(Y\) which descend to \(X\) has some given structure -- the hard part is to arrange that \(Y\) does not obtain any new automorphisms. John does this by introducing the notion of a \(\mathbb{P}^r\)-averse variety, i.e. a variety \(X\) so that any map \(\mathbb{P}^r\to X\) is constant. The point is that if \(X\) is \(\mathbb{P}^r\)-averse, any blowup \(Y\) of \(X\) along a smooth center \(Z\) of codimension greater than \(r\) satisfies

$$\text{Aut}(Y)=\text{Aut}(X; Z)$$

since any automorphism must preserve the fibers of the blowup (which are isomorphic to \(\mathbb{P}^s\) for some \(s\geq r\).)

John modifies his surface \(S\) (by replacing it with the \(6\)-fold \(S\times S\times T\), where \(T\) is some surface of general type). The variety thus obtained is \(\mathbb{P}^2\)-averse. This lets him retain control of the automorphism group of his variety even after blowing up the variety in codimension at least \(3\). He cleverly blows up some more or less randomly chosen things in codimension at least \(3\) to make sure any automorphisms of the blowup descends to an automorphism of the form

$$\text{id} \times \phi\times \text{id} \in \text{Aut}(S×S×T).$$

Then a clever final blowup reduces the automorphism group to the desired one.

In a previous paper of ours, John and I studied the issue of how blowups affect automorphism groups -- we use some \(p\)-adic methods to gain some control on this question. In particular, one thing we show is that if \(X\) is a variety such that $$\pi_0(\text{Aut}(X))$$ is finite, then blowups in high codimension, or along simple loci, do not alter the finiteness of this (component group of the) automorphism group. One thing John's construction is telling us, I think, is that if the automorphism group of \(X\) is more complicated, blowups in high codimension can have a rather more drastic effect.

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

We first translate this into the language of weights, so we can use the Weyl dimension formula. Recall that irreducible representations of \(SL_4\) are indexed by \(4\)-tuples of non-negative integers $$\lambda_1\geq \lambda_2\geq \lambda_3\geq \lambda_4=0.$$ We let \(\lambda\) be the \(n\)-tuple \((\lambda_1, \lambda_2, \lambda_3, \lambda_4)\), and denote the representation corresponding to \(\lambda\) by \(S^\lambda\). So for example $$S^{(n, 0, 0, 0)}=\text{Sym}^n(V),$$ where \(V\) is the standard representation of \(SL_4\). We first show:

**Lemma 1.** Let \(\rho\) be an irreducible representation of \(SL_4\) which does not descend to \(SL_4/\mu_2\). Then \(4\mid \dim\rho\).

**Proof.** We have \(\rho=S^\lambda\) for some \(\lambda\); the hypothesis is equivalent to the statement that $$\sum_{i=1}^4 \lambda_i \equiv 1\bmod 2.$$ In other words, three of the \(\lambda_i\) have the same parity.

Now the Weyl dimension formula (or equivalently, standard results on specializations of Schur polynomials) tells us that $$\dim \rho =\prod_{1\leq i<j\leq 4}\frac{\lambda_j-\lambda_i+j-i}{j-i}.$$ The denominator of this product is \(12\), so we must show that the numerator is divisible by \(16\). Now of the four pairs \(i, j\) with \(j-i\) odd, at least two must have that \(\lambda_i-\lambda_j\) is odd as well. Thus $$4\mid\prod_{1\leq i<j\leq 4, j-i\text{ odd}} \lambda_j-\lambda_i+j-i.$$ For the two pairs with \(j-i=2\), at least one must have that \(\lambda_i-\lambda_j\) is even, so $$2\mid \prod_{1\leq i<j\leq 4, j-i\text{ even}} \lambda_j-\lambda_i+j-i.$$

Now some annoying casework lets us eke out one more factor of two; if I come up with a slick way of doing it, I will write it down...

Multiplying gives the desired result. \(\blacksquare\)

Now we analyze the case where \(\rho\) is self-dual.

**Lemma 2.** Let \(\rho\) be as in Lemma 1. Then \(\rho\) is not self-dual.

**Proof.** Let \(\lambda\) be the \(4\)-tuple of integers corresponding to \(\rho\). Then \(\rho\) is self-dual iff $$(\lambda_1, \lambda_2, \lambda_3, \lambda_4)=(-\lambda_4+\lambda_1, -\lambda_3+\lambda_1, -\lambda_2+\lambda_1, -\lambda_1+\lambda_1).$$ In other words, we have $$\lambda_2=\lambda_1-\lambda_3.$$ But recall that $$\lambda_1+\lambda_2+\lambda_3=2\lambda_1$$ had to be odd in the setting of Lemma 1. Contradiction. \(\blacksquare\)

**Proof of Claim. **Let \(\rho\) be as in the claim. Then it is a direct sum of irreducibles, none of which descend to \(SL_4/\mu_2\). Thus there exist a collection of irreducibles \(V_i\) (none of which are self-dual, by Lemma 2, such that $$\rho=\bigoplus_i V_i \oplus V_i^{\vee}.$$ But by Lemma 1, each \(V_i\) has dimension divisible by \(4\), so $$8\mid \dim \rho$$ as desired. \(\blacksquare\)

Applying the arguments from my previous post on this topic, we get a version of the Auel-First-Williams result for the stack \(B(SL_4/\mu_2)\) (as well as a version of their theorem for symplectic involutions of Azumaya algebras). In a sequel post, I might explain how to deduce a weak version of their actual results from this computation.