Nicolas Guès
About
Welcome to my personal page. Since September 2024, I have been pursuing a PhD in mathematics at LAGA under the supervision of Geoffroy Horel. My research area is algebraic topology. I am particularly interested in the interactions between differential topology and homotopy theory. I enjoy thinking about configuration spaces of manifolds and embedding spaces between them. I am also working on generalizing certain results concerning the rational homotopy of embedding spaces to tame homotopy theory, a refinement of rational homotopy theory developed by Dwyer.
I am also working on generalizing results about rational homotopy of embedding spaces to tame homotopy theory, a refinement of rational homotopy theory developed by Dwyer.
Research
(Pre)publications
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Linear stable ranges for homotopy groups of configuration spaces
Accepted in Journal of Topology · arXiv · 2025
Abstract
We prove explicit linear stable ranges for the $\mathsf{FI}$-modules $\mathrm{Hom}(\pi_p \mathrm{Conf} M, \mathbb Z)$ and $\mathrm{Ext}(\pi_p \mathrm{Conf} M, \mathbb Z)$ with $\mathrm{Conf} M$ being the configuration co$\mathsf{FI}$-space of a $d$-dimensional manifold with $ d \geq 3$. The proof of this result uses a homotopy-theoretic approach to representation stability for $\mathsf{FI}$-modules. This allows us to derive representation stability results from homotopy-theoretical statements, in particular the generalized Blakers-Massey theorem. We also generalize to $\mathsf{FI}_G$-modules and orbit configuration spaces.
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Periodicity in the homology of moduli spaces of disconnected submanifolds
arXiv · 2025
Abstract
We show that the moduli space of $n$ suitably embedded copies of a closed smooth manifold $P$ inside a closed smooth manifold $M$ satisfies cohomological periodicity over $\mathbb F_p$ when $n$ grows, with an explicit linear bound on the period and the periodicity range. This generalizes a known result about configuration spaces. We also show integral stability of the cohomology when $M$ is open, generalizing a result of Palmer and improving the slope when inverting $2$. The main input in the proof is Goodwillie and Klein's multiple disjunction lemma for embedding spaces. As a corollary we get stability and periodicity results for some classes of symmetric diffeomorphism groups of manifolds.
Other mathematical writings
- Master's thesis — "A model in groupoids for the splicing operad", written under the supervision of Paolo Salvatore.
- Research introduction note — A brief introduction to my research area for a non-specialist audience, focusing on the homotopy type of spaces of embeddings. Written in french.
Teaching
2025–2026
2024–2025
- S1 — Exercise sessions, Introduction to mathematical structures, L1 Computer Science, Paris 13
- S2 — Exercise sessions, M1 Error-correcting codes
- S2 — Exercise sessions, M2 Homotopy II
Talks
- Topology seminar, Laboratoire Paul Painlevé, Université de Lille — March 27, 2026
- Topology seminar, Stockholm University, Sweden — March 17, 2026
- Online seminar, Purdue University, Indiana, USA — February 18, 2025
- Topology seminar, Universitat de Barcelona — December 2024
- LMNO algebra and geometry seminar, Université de Caen — October 14, 2025
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Homotopical Methods in Representation Stability
- Participant in the mini working group at the 2024 meeting of the GDR in algebraic topology, Toulouse
- Participant in the working group 2023–2024 of the topology team at LAGA
Misc.
- I co-organize with Lucas Lagarde and Loth Damagui Chabi the PhD students' seminar at LAGA.
- Animations of long knots created to illustrate my Master's internship.
- A popularization talk on algebraic topology given at Lycée Thiers in September 2023.