Mathieu Vallée

Ph.D.

I am now a postdoc at Université libre de Bruxelles in the Algorithm team, supervised by Jean Cardinal.

I obtained my Ph.D. in Computer Science and Mathematics (September 2025) at the LIPN (Laboratoire d'Informatique de Paris Nord), at the University of Sorbonne Paris Nord in France, supervised by Roland Grappe and Bruno Vallette. My Ph.D. subject is "Cones, fans, and wedges: combinatorial and geometrical perpectives". It gathers combinatorial optimization, polyhedral geometry, and toric geometry. Link for the manuscript.

I teach at the IUT Villetaneuse.

Research overview

Combinatorial optimization

Box-integrality and the integer decomposition property

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.

Preprints

Complete non-singular toric varieties with Picard number 4: joint work with Suyoung Choi and Hyeontae Jang (2025)
arXiv:2504.18134

Totally equimodular matrices: decomposition and triangulation: joint work with Patrick Chervet and Roland Grappe (2025)
arXiv:2504.05930

Publications

Toric wedge induction and toric lifting property for piecewise linear spheres with few vertices: joint work with Suyoung Choi and Hyeontae Jang (2025)
Journal of the London Mathematical Society
Vol. 112, no. 1, pages e70248

The characterization of (n−1)-spheres with n+4 vertices having maximal Buchstaber number: joint work with Suyoung Choi and Hyeontae Jang (2024)
Journal für die reine und angewandte Mathematik (Crelles Journal)
Vol. 2024, no. 811, 2024, pages 267-292

An algorithmic strategy for finding characteristic maps over wedged simplicial complexes: joint work with Suyoung Choi (2022)
Pacific Journal of Mathematics
Vol. 320-1, pages 13–43

Cohomological rigidity of the connected sum of three real projective spaces: joint work with Suyoung Choi (2022)
Proceedings of the Steklov Institute of Mathematics
Vol. 317, pages 178–188

Conferences

GPU Algorithm for Enumerating PL Spheres of Picard Number 4: Application to Toric Topology: joint work with Suyoung Choi and Hyeontae Jang (2024)
40th International Symposium on Computational Geometry (SoCG 2024).
Leibniz International Proceedings in Informatics (LIPIcs),
Volume 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.