Feeds:
Posts

Higher linear algebra

Let $k$ be a commutative ring. A popular thing to do on this blog is to think about the Morita 2-category $\text{Mor}(k)$ of algebras, bimodules, and bimodule homomorphisms over $k$, but it might be unclear exactly what we’re doing when we do this. What are we studying when we study the Morita 2-category?

The answer is that we can think of the Morita 2-category as a 2-category of module categories over the symmetric monoidal category $\text{Mod}(k)$ of $k$-modules, equipped with the usual tensor product $\otimes_k$ over $k$. By the Eilenberg-Watts theorem, the Morita 2-category is equivalently the 2-category whose

• objects are the categories $\text{Mod}(A)$, where $A$ is a $k$-algebra,
• morphisms are cocontinuous $k$-linear functors $\text{Mod}(A) \to \text{Mod}(B)$, and
• 2-morphisms are natural transformations.

An equivalent way to describe the morphisms is that they are “$\text{Mod}(k)$-linear” in that they respect the natural action of $\text{Mod}(k)$ on $\text{Mod}(A)$ given by

$\displaystyle \text{Mod}(k) \times \text{Mod}(A) \ni (V, M) \mapsto V \otimes_k M \in \text{Mod}(A)$.

This action comes from taking the adjoint of the enrichment of $\text{Mod}(A)$ over $\text{Mod}(k)$, which gives a tensoring of $\text{Mod}(A)$ over $\text{Mod}(k)$. Since the two are related by an adjunction in this way, a functor respects one iff it respects the other.

So Morita theory can be thought of as a categorified version of module theory, where we study modules over $\text{Mod}(k)$ instead of over $k$. In the simplest cases, we can think of Morita theory as a categorified version of linear algebra, and in this post we’ll flesh out this analogy further.

What’s a fire, and why does it – what’s the word – burn?

I was staring at a bonfire on a beach the other day and realized that I didn’t understand anything about fire and how it works. (For example: what determines its color?) So I looked up some stuff, and here’s what I learned.

More on partition asymptotics

In the previous post we described a fairly straightforward argument, using generating functions and the saddle-point bound, for giving an upper bound

$\displaystyle p(n) \le \exp \left( \pi \sqrt{ \frac{2n}{3} } \right)$

on the partition function $p(n)$. In this post I’d like to record an elementary argument, making no use of generating functions, giving a lower bound of the form $\exp C \sqrt{n}$ for some $C > 0$, which might help explain intuitively why this exponential-of-a-square-root rate of growth makes sense.

The starting point is to think of a partition of $n$ as a Young diagram of size $n$, or equivalently (in French coordinates) as a lattice path from somewhere on the y-axis to somewhere on the x-axis, which only steps down or to the right, such that the area under the path is $n$. Heuristically, if the path takes a total of $L$ steps then there are about $2^L$ such paths, and if the area under the path is $n$ then the length of the path should be about $O(\sqrt{n})$, so this already goes a long way towards explaining the exponential-of-a-square-root behavior.

The man who knew partition asymptotics

(Part I of this post is here)

Let $p(n)$ denote the partition function, which describes the number of ways to write $n$ as a sum of positive integers, ignoring order. In 1918 Hardy and Ramanujan proved that $p(n)$ is given asymptotically by

$\displaystyle p(n) \approx \frac{1}{4n \sqrt{3}} \exp \left( \pi \sqrt{ \frac{2n}{3} } \right)$.

This is a major plot point in the new Ramanujan movie, where Ramanujan conjectures this result and MacMahon challenges him by agreeing to compute $p(200)$ and comparing it to what this approximation gives. In this post I’d like to describe how one might go about conjecturing this result up to a multiplicative constant; proving it is much harder.