Matthias ? - Representations of An-type quivers and persistent homology

Abstract

From a dataset we will grow a set of balls and look at the homology of the induces spaces. From this, we will get a Persistence module. Then we create a Persistence Barcode. This will give us a description of stability. We will also try to adress Zig Zig Persistence

Talk

1. Persistence Modules

Definition

Data $P$ is understood as a finite set of points in ${\mathbb{R}}^d$.

We will endow this with a topology by placing balls with a given radius around the balls. This will approximate the topology of the points $P$. Consider the functor $U:{\mathbb{R}}_{>0}\to Top$ given by the union of balls of radius $\alpha \subseteq {\mathbb{R}}_{>0}$. It $\alpha \le \beta$ then there is a natural inclusion $U(\alpha )\subseteq U(\beta )$.

Definition

This $U$ is called the union of balls filtration.

We also have the homology functor $H_i$ mapping from the category of topologies to the categories of vector spaces. The composition then is $H_i\circ U:{\mathbb{R}}_{>0}\to vec{t}_K$.

Definition

A $T$-modules function $M:T\to vec{t}_K$ is calleda persistence module. A map into finite-dimensional vectors? Or vector-spaces?

To ease computations we replace $U$ by $K$ which sends a radius to a simplicial complex instead of a topology.

Definition

Let $\mathcal{P}$ be a collection of sets. The Nerve (Kategorientheorie) of $\mathcal{P}$ is an abstract Simplizialkomplex. The zero skeleton consists of all sets in $\mathcal{P}$. We have one abstract $k$-simplex if $k$ simplices intersect in one point.

Definition

Let $P\subseteq \mathbb{R}$ as above. The Cech-complex of $P$ with radius $\alpha$ is the Nerve of the set of balls with radius $\alpha$. the functor $C(P):{\mathbb{R}}_{>0}\to Simp$ is the functor $K$ we wanted to construct earlier. Once again, we have an inclusion maps for smaller radii.

Note: The word filtration means we have some kind of graded topologies.

Definition (Delaunay-filtration)

We define the Voronoi-cell of our points in the usual sense. Then we define the Voronoi-Diagramm.

Definition

An $\alpha$-cell of $p_i$ is the intersection of the Voronoi cell of $p_i$ with the ball of radius $\alpha$ around $p_i$

By doing this, we can find out the topological structure of the filtration while reducing intersections as much as possible.

Definition

The Delaunay complex consists of the nerve of the $\alpha$-cells. These complexes induce a functor into simplicial complexes. This is called the Delaunay filtration

Definition

The $\alpha$-Vietoris-Rips-complex is the complex where a set of points is included as a simplex if all points are at most $2\alpha$ distance away.
Similarly we define the Vietoris-Rips filtration as the functor from the radius to the Simplicial complexes.

Just now, we defined three different filtrations. The Nerve Theorem states that all of these gives us equivalent spaces (equivalent kind of in the sense of norms). Obacht: The nerve theorem is sometimes quoted wrongly. There is a paper by Ulrich Buyaer which tries to clean it up though

We classify the pros and cons of each filtration

	Pro	Cons
Cech	Relatively cheap	The size is also big but not as bad as Del
VR	Relatively cheap	Not capturing homology
Del	Homputes homology honestly	Bad in high dimension
There are some great applications in biology.

Now this is surprising: The map $H_i\circ K:[m]\to Vec$ is a representation of $A_n$.

Theorem

Every representation $M$ of $rep(A_n)$ can be written as a direct sum of $[b_i,d_i]$. For a given representation $M$ the set of $[b_i,d_i]$ interpreted as a real subset is called a Persistence Barcode.

2. Computation of Barcodes

Theorem

The category $Fun(\mathbb{N},vect)$ of persistence modules is isomorphic to the category of finitely generated graded modules over $k[x]$

We are led to believe that this correspondence is like the correspondence between path algebras and quiver representations.

3. Stability

4. Zig Zig Persistence