Mathieu Vallée

Ph.D.

Current position

I am currently a postdoctoral researcher at Université libre de Bruxelles (ULB) in the Algorithm team, supervised by Jean Cardinal.

Ph.D. thesis

Title: Cones, fans, and wedges: combinatorial and geometrical perspectives
Date defended: 9 September 2025
Institution: LIPN (Laboratoire d’Informatique de Paris Nord), Université Sorbonne Paris Nord, France
Advisors: Roland Grappe and Bruno Vallette
Topics: combinatorial optimization, polyhedral geometry, toric geometry
Link to the manuscript: PDF

Teaching

I teach in the Computer Science Department at Université libre de Bruxelles.

Research overview

Combinatorial optimization

Box-integrality and integer decomposition properties.
Box-integrality is a property of some integer polyhedron to remain integer after a truncation with any integer box. Integer decomposition properties is the study of integer points in convex polyhedra and, more precisely, of how to find "minimal" generators of these integer points. I study integer decomposition properties of polyhedra satisfying box-integrality. For this subject, I use plenty of discrete geometry such as triangulation of polyhedron, box-total dual integrality, and a class of matrices that generalizes totally unimodular matrices: totally equimodular matrices.

Toric topology

Toric manifolds with small Picard number.
This research focuses on toric manifolds, which correspond to complete nonsingular fans in the n-real space. In particular, we use the wedge operation on simplicial complexes to study toric manifolds with small Picard number, that is, whose associated fans have only slightly more rays than their dimension.

Operad theory

Set theoretic operads.
The purpose of operads is to encode universally all the operations acting on any category of algebras. I study operads on sets and how they behave with respect to different set theoretic operations with a specific focus on operads on simplicial complexes. In fact, they appear in the theory of polyhedral products, which for instance helps building topological models for toric manifolds.

arXiv feed (submitted)

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Publications

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Conferences

GPU Algorithm for Enumerating PL Spheres of Picard Number 4: Application to Toric Topology

2024

Co-authors: Suyoung Choi, Hyeontae Jang

Leibniz International Proceedings in Informatics (LIPIcs)

40th International Symposium on Computational Geometry (SoCG 2024), Vol. 293, pp. 41:1-41:15

Teaching

ULB

Computer science department

Combinatorial optimization, Master 1 (12h).
Randomized algorithms, Master 1 (12h).

IUT de Villetaneuse

BUT Informatique.

Chargé de Cours/TD de première année de formation initiale: Introduction Base de Données (47h).
Responsable du cours de passerelle: Développement efficace (17h).

Photography

I have been an amateur photographer starting from 2020 when I went to Korea for an exchange year. Since then I took this hobby seriously and always carry a camera with me.

Some links