# Research activity

## Status :

I am a PhD sstudent at INRIA-Saclay, in France located near Paris. My research focus on Persistence Modules.

## Research field :

I study persistent homology.

I am interested by linear representations of certains quiver, usualy obtained from partialy ordered set.

For example, one can consider the category $$(\mathbb{R}, \leq)$$ composedof 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.

One can also consider sub-sets of $$\mathbb{R}$$ like $$\mathbb{N}$$ or $$\mathbb{Z}$$.

This functor 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.

We speack of 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'm looking for decomposition theorems and stability results for the signature made from this objects.