Research

My interests

My early work focused on the interaction between (quantum) groups and K-theory in the framework of the Baum–Connes conjecture. From there, my interests expanded naturally into new directions, guided both by mathematical curiosity and by the richness of the structures involved.

This exploration has led me to engage with areas at the interface of pure and applied mathematics, such as Quantum Information Theory and Non-local Games.

My current research programme spans both fundamental mathematics and its interdisciplinary applications.

Fundamental research

My fundamental research lies at the intersection of several areas of mathematics, with a central focus on operator algebras and their deep connections to topology, quantum groups, and noncommutative geometry.

  • Research areas:
    • Algebraic Topology
    • Noncommutative Geometry
    • Operator Algebras
    • Quantum Groups
    • Representation theory
  • Keywords:
    • C\(^*\)-algebras
    • C\(^*\)-dynamics
    • (Higher) representation theory
    • Hopf algebras
    • Homological algebra
    • K-theory
    • Triangulated and tensor categories
    • Twisting

Applied research

In recent years, I have become involved in applied mathematics projects incorporating tools from machine learning and applied topology.

  • Research areas:
    • Causal analysis
    • Computer Vision
    • Graphical models
    • Topological Data Analysis (TDA)
  • Key words:
    • Bayesian networks
    • (Graphical) Neural Networks
    • Persistent homology

This work revolves around three main applications:

  1. Codicology: Investigating the structural properties of medieval legal manuscripts.
  2. Genomics: Studying genetic regulation processes.
  3. Stochastic topology: Investigating the probability distribution of persistence diagrams.

Doctoral project

The Baum-Connes conjecture for Quantum Groups. Stability properties and K-theory computations.

The main goal of my thesis (under the direction of P. Fima) has been to compute the K-theory of the C\(^*\)-algebras associated to compact quantum groups in some (interesting and) concrete examples. The strategy to reach such computations involves the study of the quantum counterpart of the Baum-Connes conjecture.

I defended my doctoral thesis in September 2018. Here you can find the manuscript.

Master’s Dissertations

  • Bachelor’s Thesis: Existencia de funciones meromorfas en superficies de Riemann abiertas.

    I studied the Behnke-Stein’s theorem in dimension 1, that is, the generalization of Runge’s theorem about the approximation of holomorphic functions in open Riemann surfaces.

  • M1: Analyse Complexe à plusieurs variables. Le théorème de Hartogs.

    I continued to study in depth complex analysis/geometry. My M1 dissertation is about Hartogs’ theorem of holomorphic extension in complex dimension n>1.

  • M2: Groupes agissant sur un arbre et K-moyennabilité. Une introduction à la K-théorie.

    I studied Kasparov’s KK-theory in order to prove the K-amenability of a discrete group acting on a tree with amenable stabilizers (after P. Julg and A. Valette).

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