Leinster - The many faces of magnitude

Abstract

The magnitude of a square matrix is the sum of all the entries of its inverse. This strange definition, suitably used, enables us to define the “magnitude” of many objects in different contexts across mathematics. All of them can be understood as measures of size. For example, the magnitude of a metric space combines classical quantities such volume, surface area, and dimension. The magnitude of a category is closely related to Euler characteristic. The magnitude of a graph is an invariant sharing features with the Tutte polynomial (but not a specialization of it). Magnitude also appears in the difficult problem of quantifying biological diversity: under certain circumstances, the greatest possible diversity of an ecosystem is exactly its magnitude. And there is now a theory of magnitude homology, which has the same relationship to magnitude as ordinary homology does to Euler characteristic. I will give an aerial view of this landscape.

1. The big plan

There are different notion of size (sets, volume, euler charakteristic for spaces with inclusion-exclusion and product). Can we categorify them? Yes, using Magnitude (angereicherte Kategorie)!

1.1. Matrixmagnitude

Definition Magnitude of a matrix

Magnitude is defined by summing the entries of the inverse matrix.

1.2. Enriched Categories

With the matrix magnitude we can define a magnitude for finite enriched categories.

Construction Enriched Category

• Take a monoidal category and enrich it.
• The morphisms become the objects of a more enriched structure (probably defined by new morphisms).

Enriched Categories are more general than categories, since they trivially contain categories. An important example are metric spaces.

Definition Magnitude for enriched categories

Let a magnitude on a finite category be defined as a function that maps objects to ring elements. Then this induces a function defined on $\text{Hom}(a,b)$. Thes can be described by a matrix. The magnitude of the matrix then defines the magnitude auf the enriched category.

Why is the magnitude interesting?

• The magnitude is the Euler characteristic of the Klassifizierender Raum (the space of the category if you draw the points and arrows and look from afar)
• The magnitude for a finite Group ist $|G|=1/order(G)$
• magnitude generalizes the Euler characteristic of a triangulated manifold/orbifold

There are also some new invariants which could be generated. In the following we will take a look at the magnitude of a metric space.

Metric space

What is the intuition behind a metric space as a enriched category? How to generate a metric space from an enriched category?

Category	Metric space
object $a$	points $a$
Morphisms $\text{Hom}(a,b)$	$d(a,b)$
Composition Operation $\text{Hom}(a,b)\circ \text{Hom}(b,c)\subseteq \text{Hom}(a,c)$	triangle inequality $d(a,b)+d(b,c)\ge d(a,c)$

I am not quite sure wheter the last thing is still correct.

If we apply the definition of magnitude do metric spaces, what magnitude do we get from specializing?

Construction metric magnitude

• Take finite metric space $A=\{a_1,...,a_n\}$
• Write the distance in a matrix as $e^{-d(a_i,a_j)}$
• Invert the matrix
• Add up all entries

What can the definition do?

Simple Properties

• Empty set has mag 0
• On point has mag 1
• Two points have mag 1 when they are close together and get to 2 when farther apart. (2 for ininity distance) ⇒ magnitude = effective number o points

We notice, that we chose the scaling $e^d$ in the definition above. This gives us the idea, to use a variable in the basis instead. This way, we get a magnitude function $|tA|$ of a metric space. Since this is a numbered valued function in one variable it carries a lot more information.

Until now we only looked at the magnitude of finite spaces. We want to generalize it to compact spaces We want to generalize the

On Generalizations to compact metric spaces

Meckes showed that all sensible generalizations are the same.

Sadly it is very hard to calculate the mgnitude for special sets. (e.g. disc). Still, some examples exist

Magnitude examples

• Line segment $|t[0,l]|=1+\frac{l}{2}\ast t$
• Euclidian Ball: Mag Function is a rational polynomial

Studying magnitude is very fruitful, there are a lot of connections to geometric and topological invariants.

• Meckes: Minkowski dimension
• Willerton: Connection to the volume and total scalar curvature
• Gimperlein, Goffeng and Louca: Contains the surface area of a subset of ${\mathbb{R}}^n$

Graph theory

The following studies an undirected finite graph with simple edges. We can calculate the magnitude as a finite metric space by the procedure given above. This allows us to calculate the magnitude of a graph.

Power Series

The magnitude is always a power series with integer coefficients. It becomes a rational function when compacted.

Simple Properties

• Behaves like cardinality
  • In the case of edge-less graphs it returns the number of diconnected points
  • Multiplicative under cartesian products of graphs
• Resembles the Tutte polynomial
• Invariant under so called whitney twists

Homology

If you where to describe Magnitude as a shadow connecting all branches of mathematics like a squid then magnitude homology is the actual body of that squid. Sadly it is not easy to define. Currently there exists a definition of finite metric spaces and recently one for compact spaces has been constructed.

Properties:

• Magnitude detects the magnitude of holes (see Magnitudenhomologie (Graphtheorie)).
• Can distinguish graphs with the same magnitude
• Magnitude homology of a convex ${\mathbb{R}}^n$-set is trivial
• Nontrivial if there is a closed geodesics
• Asao: “The more geodesic are unique the more the homology is trivial”

What else can we do with magnitude

• Connection to persistent homology (Otter, Cho)
• Application to networks (Meneara)
• Mag cohomo (Hepworth)
• connection to path homology (Asao)
• Spectral sequence (Asao, Hepworth, Roff)
• and much more!

Ecology

How to measure biodiversity? Usually we use the Entropie (Informationstheorie). But what happens if we have to decide between many similar species and a few diverse species? Luckily there is a formula, that interpolates between prioritizing the above two options.

However if we have two frogs and one fish species, we would intuitively assume that there needs to be proportionally more fish to raise biodiversity since the fish is taxonomically farthest apart. By this we’re now measuring diversity on a metric space.

This gives us the maximization problem. How to put fish in the lake to maximize diversity? It turns out, there is a very natural maximum diversity fish proportion. This is closely related to Magnitude!

This maximum diversity can be applied to more general metric space! (I guess, how to chose points to maximize “diversity”)

Referneces

• Leinster Entropy and Diversity (Viele Tiere!)
• https://www.maths.ed.ac.uk/~tl/magbib/