# Research activity

## Current research interests

- Neural networks (CNN, RNN, GAN, VAE, etc..)
- Signal processing
- Genetic algorithm
- Probabilistic programming

## Brief introduction to Persistent homology

During my thesis, I worked on **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 usual 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 have a natural structure of modules over a “polynomial” ring,
function of the index set. In the case of $\mathbb{N}$, the ring is $k[x]$.
The name of **persistence module** come from this structure of module over a ring.

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 name this construction as *multipersistence*.
If the order of the index set alternate, like it is the case with the quiver given by $0 \leq 1 \geq 2 \leq 3 \geq \dots$, the construction is called *zigzag persistence*.

This functor,
with values in the category of vector spaces,
can be obtain by computing the homology of a filtered topological space.
Hence the name **persistent homology**.
The simplest construction consist of taking a point cloud $P \subset \mathbb{R}^m$,
and then computing the sub-levelset of the (euclidean) 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
$\mathbb{N}$ : $H(X_0) \rightarrow H(X_1) \rightarrow \dots$.

During my PhD I was looking for decomposition theorems and stability results for the signature
made from this multipersistence modules.

## Dissertation

I wrote my PhD at the INRIA-Saclay, Palaiseau(France) located near Paris under
the direction of Steve Oudot as a member
of the Datashape (former geometrica) team.
My research focused on **Persistence Modules** and its generalization where
the index set is $\mathbb{R}^2$.

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