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## The man who knew elliptic integrals, prime number theorems, and black holes

I went to see The Man Who Knew Infinity yesterday. I have nothing much to say about the movie as a movie that wasn’t already said in Scott Aaronson‘s review, except that I learned a few fun facts during the Q&A session with writer/director Matthew Brown afterwards. Namely, it’s a little surprising the movie was able to get high-profile stars like Dev Patel and Jeremy Irons on board given that it was made on a relatively low budget. Apparently, Dev Patel signed on because he really wanted to popularize the story of Ramanujan, and Jeremy Irons signed on because he was hooked after being given a copy of Hardy’s A Mathematician’s Apology.

(Disclaimer: this blog does not endorse any of the opinions Hardy expresses in the Apology, e.g. the one about mathematics being a young man’s game, the one about pure math being better than applied math, or the one about exposition being an unfit activity for a real mathematician. The opinion of this blog is that the Apology should be read mostly for insight into Hardy’s psychology rather than for guidance about how to do mathematics.)

Anyway, since this is a movie about Ramanujan, let’s talk about some of the math that appears in the movie. It’s what he would have wanted, probably.

## Separable algebras

Let $k$ be a commutative ring and let $A$ be a $k$-algebra. In this post we’ll investigate a condition on $A$ which generalizes the condition that $A$ is a finite separable field extension (in the case that $k$ is a field). It can be stated in many equivalent ways, as follows. Below, “bimodule” always means “bimodule over $k$.”

Definition-Theorem: The following conditions on $A$ are all equivalent, and all define what it means for $A$ to be a separable $k$-algebra:

1. $A$ is projective as an $(A, A)$-bimodule (equivalently, as a left $A \otimes_k A^{op}$-module).
2. The multiplication map $A \otimes_k A^{op} \ni (a, b) \xrightarrow{m} ab \in A$ has a section as an $(A, A)$-bimodule map.
3. $A$ admits a separability idempotent: an element $p \in A \otimes_k A^{op}$ such that $m(p) = 1$ and $ap = pa$ for all $a \in A$ (which implies that $p^2 = p$).

(Edit, 3/27/16: Previously this definition included a condition involving Hochschild cohomology, but it’s debatable whether what I had in mind is the correct definition of Hochschild cohomology unless $k$ is a field or $A$ is projective over $k$. It’s been removed since it plays no role in the post anyway.)

When $k$ is a field, this condition turns out to be a natural strengthening of the condition that $A$ is semisimple. In general, loosely speaking, a separable $k$-algebra is like a “bundle of semisimple algebras” over $\text{Spec } k$.

## Coalgebraic geometry

Previously we suggested that if we think of commutative algebras as secretly being functions on some sort of spaces, we should correspondingly think of cocommutative coalgebras as secretly being distributions on some sort of spaces. In this post we’ll describe what these spaces are in the language of algebraic geometry.

Let $D$ be a cocommutative coalgebra over a commutative ring $k$. If we want to make sense of $D$ as defining an algebro-geometric object, it needs to have a functor of points on commutative $k$-algebras. Here it is:

$\displaystyle D(-) : \text{CAlg}(k) \ni R \mapsto |D \otimes_k R| \in \text{Set}$.

In words, the functor of points of a cocommutative coalgebra $D$ sends a commutative $k$-algebra $R$ to the set $|D \otimes_k R|$ of setlike elements of $D \otimes_k R$. In the rest of this post we’ll work through some examples.

## Coalgebras of distributions

Mathematicians are very fond of thinking about algebras. In particular, it’s common to think of commutative algebras as consisting of functions of some sort on spaces of some sort.

Less commonly, mathematicians sometimes think about coalgebras. In general it seems that mathematicians find these harder to think about, although it’s sometimes unavoidable, e.g. when discussing Hopf algebras. The goal of this post is to describe how to begin thinking about cocommutative coalgebras as consisting of distributions of some sort on spaces of some sort.

## Maximum entropy from Bayes’ theorem

The principle of maximum entropy asserts that when trying to determine an unknown probability distribution (for example, the distribution of possible results that occur when you toss a possibly unfair die), you should pick the distribution with maximum entropy consistent with your knowledge.

The goal of this post is to derive the principle of maximum entropy in the special case of probability distributions over finite sets from

• Bayes’ theorem and
• the principle of indifference: assign probability $\frac{1}{n}$ to each of $n$ possible outcomes if you have no additional knowledge. (The slogan in statistical mechanics is “all microstates are equally likely.”)

We’ll do this by deriving an arguably more fundamental principle of maximum relative entropy using only Bayes’ theorem.

## Monads are idempotents

It’s common to think of monads as generalized algebraic theories; the most familiar examples, such as the monads on $\text{Set}$ encoding groups, rings, and so forth, have this flavor. However, this intuition is really only appropriate for certain monads (e.g. finitary monads on $\text{Set}$, which are the same thing as Lawvere theories).

It’s also common to think of monads as generalized monoids; previously we discussed why this was a reasonable thing to do.

Today we’ll discuss a different intuition: monads are (loosely) categorifications of idempotents.

## Lie algebras are groups

Once upon a time I imagine people were very happy to think of Lie algebras as “infinitesimal groups,” but presumably when infinitesimals fell out of favor this interpretation did too. In this post I’ll record an observation that can justify thinking of Lie algebras as groups in a strong sense: they are group objects in a certain category which can be interpreted as a category of “infinitesimal spaces.”

Below we work throughout over a field of characteristic zero.

For starters, the universal enveloping algebra functor $\mathfrak{g} \mapsto U(\mathfrak{g})$, which a priori takes values in algebras (it’s left adjoint to the forgetful functor from algebras to Lie algebras), in fact takes values in Hopf algebras. This upgraded functor continues to be a left adjoint, although the forgetful functor is less obvious. Given a Hopf algebra $H$, its primitive elements are those elements $x \in H$ satisfying

$\Delta x = x \otimes 1 + 1 \otimes x$

where $\Delta$ is the comultiplication. The primitive elements of a Hopf algebra form a Lie algebra, and this gives a forgetful functor from Hopf algebras to Lie algebras whose left adjoint is the upgraded universal enveloping algebra functor.

The key observation is that this upgraded functor $\mathfrak{g} \to U(\mathfrak{g})$ is fully faithful; that is, there is a natural bijection between Lie algebra homomorphisms $\mathfrak{g} \to \mathfrak{h}$ and Hopf algebra homomorphisms $U(\mathfrak{g}) \to U(\mathfrak{h})$. This is more or less equivalent to the claim that the natural inclusion $\mathfrak{g} \to U(\mathfrak{g})$ induces an isomorphism from $\mathfrak{g}$ to the Lie algebra of primitive elements of $U(\mathfrak{g})$, which can be proven using the PBW theorem.

Hence Lie algebras embed as a full subcategory of Hopf algebras; that is, they can be thought of as Hopf algebras satisfying certain properties, rather than having extra structure (in the nLab sense). What are these properties? For starters, they are all cocommutative. This is important because cocommutative Hopf algebras are group objects in the category of cocommutative coalgebras (this is not true with “cocommutative” dropped!), which in turn can be interpreted as a category of infinitesimal spaces. (For example, this category is cartesian closed, and in particular distributive.)

Hence Lie algebras are group objects in cocommutative coalgebras satisfying some property (for example, “conilpotence”; see Theorem 3.8.1 here).

## Drawing subgroups of the modular group

Previously we learned how to count the finite index subgroups of the modular group $\Gamma = PSL_2(\mathbb{Z})$. The worst thing about that post was that it didn’t include any pictures of these subgroups. Today we’ll fix that.

The pictures in this post can be interpreted in at least two ways. On the one hand, they are graphs of groups in the sense of Bass-Serre theory, and on the other hand, they are also dessin d’enfants (for the rest of this post abbreviated to “dessins”) in the sense of Grothendieck. But you don’t need to know that to draw and appreciate them.

## Connected components in a distributive category

Previously we claimed that if you want to check whether a category $C$ “behaves like a category of spaces,” you can try checking whether it’s distributive. The goal of today’s post is to justify the assertion that objects in distributive categories behave like spaces by showing that they have a notion of “connected components.”

For starters, let $C$ be a distributive category with terminal object $1$, and let $2 = 1 + 1$ be the coproduct of two copies of $1$. For an object $X \in C$, what does $\text{Hom}(X, 2)$ look like? If $C = \text{Top}$ and $X$ is a sufficiently well-behaved topological space, morphisms $X \to 2$ correspond to subsets of the connected components of $X$, and $\text{Hom}(X, 2)$ naturally has have the structure of a Boolean algebra or Boolean ring whose elements can be interpreted as subsets of the connected components of $X$.

It turns out that $\text{Hom}(X, 2)$ naturally has the structure of a Boolean algebra or Boolean ring (more invariantly, the structure of a model of the Lawvere theory of Boolean functions) in any distributive category. Hence any distributive category naturally admits a contravariant functor into Boolean rings, or, via Stone duality, a covariant functor into profinite sets / Stone spaces. This is our “connected components” functor. When $C = \text{Aff}$ the object this functor outputs is known as the Pierce spectrum.

This construction can be thought of as trying to do for $\pi_0$ what the étale fundamental group does for $\pi_1$.

## Distributive categories

Among all of the standard algebraic structures that a student typically encounters in an introduction to abstract algebra (groups, rings, fields, modules), commutative rings are somehow special: the opposite category $\text{CRing}^{op}$ behaves like a category of spaces, so much so that an entire field of mathematics is dedicated to doing geometry based on it.

In general, suppose you find yourself in some category. What sort of behavior could you look for that might qualify as “behaving like a category of spaces”?

One thing to look for is distributivity. Recall that a distributive category is a category $C$ with finite products $\times$ and finite coproducts $+$ such that finite products distribute over finite coproducts; more explicitly, the natural maps

$X \times Y+ X \times Z \to X \times (Y + Z)$

should be isomorphisms, and also the natural maps $0 \to X \times 0$ should be isomorphisms, where $0$ denotes the initial object. (Curiously, distributive categories are themselves like categorified versions of commutative rings.)

This is a pretty good test. The following familiar categories are distributive:

• $\text{Set}$
• More generally, any bicartesian closed category, and in particular any topos
• $\text{Top}$
• $\text{Aff} = \text{CRing}^{op}$

These are all reasonable candidates for categories of “spaces.” On the other hand, the following familiar categories are not distributive:

• $\text{Grp}$
• More generally, any nontrivial category with a zero object, and in particular any abelian category

You might object that there is also an entire field of mathematics dedicated to treating groups as geometric objects. I contend that the geometric object a group describes is actually a groupoid, and $\text{Gpd}$ is distributive!