# Research activity

I did my PhD at the INRIA-Saclay, Palaiseau(France) located near Paris. My research focused on Persistence Modules.

You can directly access the manuscript. ## Research field :

I studied persistent homology.

I was interested by linear representations of a specific type of quiver usually constructed from partially ordered set.

One can consider the category $(\mathbb{R}, \leq)$ composed of the reals number with their ususal ordering. A functor $(\mathbb{R}, \leq) \rightarrow Vect$ with values in the category of vector spaces is then called a persistence module.

The index set (source category of this functors) can also be replaced by sub-sets of $\mathbb{R}$ like $\mathbb{N}$ or $\mathbb{Z}$.

This functors are naturaly modules over a certain ring (in the case of $\mathbb{N}$, the ring is $k[x]$).

If we allow more dimensions by taking as index set $(\mathbb{R}^n, \leq)$ with $(x_1, \dots, x_n) \leq (y_1, \dots, y_n) \Leftrightarrow x_1 \leq y_1, \dots$, we then speak of multipersistence.

If the order alternate, for example consider the quiver given by $0 \leq 1 \geq 2 \leq 3 \geq \dots$, it is called zigzag persistence.

This field is called persistent homology because this functor, with values in the category of vector spaces, is obtain by computing the homology of a filtered topological space. The simplest construction consist of taking a point cloud $P \subset \mathbb{R}^m$, and then computing the sub-levelset of the (euclidean for example) distance function to the point cloud: $X_k = d^{-1}_P(]-\infty, k])$. This collection of sets is called a filtration. We obtain our persistence module by applying the homology functor $\mathbb{N}$ : $H(X_0) \rightarrow H(X_1) \rightarrow \dots$.

I was looking for decomposition theorems and stability results for the signature made from this objects.