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.

Papers :