Research interests

I am mainly interested in developing topological and geometrical methods that can extract structure from data and help understand the behavior of learning systems. Therefore, my research sits at the intersection of topology, data analysis and deep learning.

Projects

Large language models and the geometry of embeddings

Abstract LLM pipeline diagram

Large language models (LLMs) map words, sentences, and concepts into high-dimensional vector spaces called embedding spaces. These spaces are far from arbitrary: meaningful semantic and syntactic relationships tend to self-organize into geometric structures.

I study these structures using tools from computational topology, notably persistent homology. The goal is to better understand what a LLM learns, how reliably it represents data, and how its internal geometry evolves through training. By obtaining a clearer picture of these geometric structures, we can improve the explainability and robustness of LLMs.

Abstract LLM pipeline diagram
Gradient field of a multidimensional discrete Morse function

Multiparameter discrete Morse theory

Gradient field of a multidimensional discrete Morse function

Classical Morse theory studies how the topology of a space changes as one sweeps through the sublevel sets of a real-valued function, tracking when connected components, holes or voids either form or disappear. Discrete Morse theory carries this idea to a combinatorial setting, allowing us to analyze the shape of complex datasets in a computationally efficient way.

In my PhD thesis, I develop the theoretical and algorithmic foundations needed to extend this framework to a multidimensional setting, where a space is filtered simultaneously by several functions rather than one. This generalized theory is strongly linked to multiparameter persistent homology, which provides a way to characterize a dataset with respect to multiple descriptors.

Multicomplex fractals

3D slice of a multicomplex fractal

The complex plane can be extended to higher dimensions using multicomplex numbers, opening the door to fractals in dimensions 4, 8, 16, and beyond. This raises the question: what do these higher-dimensional fractals look like and how can they be described?

In my master's thesis, I explore the fascinating world of multicomplex numbers and their associated fractals. Notably, I classify the principal 3D slices of the generalized Mandelbrot sets using tools from dynamical systems, algebra and geometry.

3D slice of a multicomplex fractal