Julien Boulanger

Since September 2024 I am a postdoc researcher at the Centro de Modelamiento matematico in Santiago de Chile, with Rodolfo Gutiérrez-Romo. Prior to that I did my PhD at the Université Grenoble alpes under the supervision of Erwan Lanneau and Daniel Massart, on some geometric problems around translation surfaces.

I am interested in dynamics and geometry of surfaces. I specifically study translation surfaces and their Veech groups, but I am also interested in interval exchange transformations, continued fractions, combinatorics on words and Anosov dynamics.

News

I’m looking for a postdoc position starting from September 2026, don’t hesitate to contact me if you are interested in my research! Here is my CV, and a research statement.

October 2025 - New preprint available on arXiv: “The hurwitz problem for abelian differentials”. With Rodolfo Gutiérrez-Romo and Erwan Lanneau. The paper aims to shed light on the problem of determining the maximal number of translations an abelian differential can have in a given genus. It is known from the work of Schlage-Puchta and Weitze-Schmidthüsen that this number is not more than 4(g-4) and that this bound is achieved if and only if g-1 is a multiple of 2 or 3. The purpose of this paper is to study the other genera g, and the possible achievable bounds in this case.

September 2025 - New preprint available on arXiv: “Algebraic interaction strength for translation surfaces with several singularities”. The paper studies how pairs of curves intersect on families of translation surfaces built from (semi-)regular polygons, and extends the results of my previous work (see (5) and (7) below) to the context where the surface has more than one singularity. In this type of study (but also in the study of the systole, for example), it is often difficult to deal with several singularities as geodesics can change direction at singularities. The main contribution of this paper is to deal with this type of surfaces and study the interaction strength of two families of translation surfaces with two (resp. any number) of singularities: the regular 4m+2-gons and the Bouw-Möller surfaces S_{m,n} with 1 < gcd(m,n) < n.

See slides (in spanish) or a very close english version for my preprint about connection points on double regular polygons, which has been accepted to the Journal of Modern dynamics!

Preprints

(9) The Hurwitz problem for abelian differentials. With Rodolfo Gutiérrez-Romo and Erwan Lanneau. October 2025.
Arxiv

(8) Algebraic interaction strength for translation surfaces with several singularities. September 2025.
Arxiv

(7) Algebraic intersections on Bouw-Möller surfaces, and more general convex polygons, with Irene Pasquinelli, 2024.
Arxiv

Publications

(6) Connection points on double regular polygons, Accepted for publication in the Journal of Modern Dynamics (2025).
Arxiv

We study connection points on the double regular n-gon translation surface, for n≥7 odd and its staircase model. For n≠9, we provide a large family of points with coordinates in the trace field that are not connection points. This family includes the central points, and for n=7 we conjecture that all the remaining points are connection points. Further, in the case where n≥7 is a prime number, we provide a constructive proof by exhibiting an explicit separatrix passing through a central point that does not extend to a saddle connection.

(5) Algebraic intersection, lengths and Veech surfaces, To appear in the Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2025).
Arxiv or Journal

(4) Lower bound for KVol on the minimal stratum of translation surfaces. Geometria Dedicata 218, 88 (2024).
Arxiv or Journal

(3) Algebraic intersection in regular polygons, with Erwan Lanneau and Daniel Massart. Annales Henri Lebesgue . Volume 7 (2024), pp. 787-821.
Arxiv or Journal

(2) There are no primitive Teichmüller curves in Prym(2,2), with Sam Freedman. Comptes rendus Mathematiques. Volume 362 (2024), pp. 167-170.
Arxiv or Journal

(1) Central points of the double heptagon translation surface are not connection points. Bulletin de la SMF, Vol.150(2) (2022), pp. 459-472.
Arxiv or Journal

Teaching

I gave several courses during my PhD and my post-doctorate, in french, english and spanish. This semester I am teaching Ecuaciones Diferenciales Ordinarias at the Universidad de Chile.

Academic responsibilities

I am now organizing the SIPo (Seminario de Inverstigadores Postdoctorales) of the Center for Mathematical Modeling. Check out the webpage.

I used to organize the Séminaire Compréhensible , which is the PhD seminar of the Institut Fourier. As part of it, I also organized two two-days conferences (2022 and 2023), the PhD days of the Institut Fourier.

PhD defense

I started my PhD on September 2021 under the supervision of Erwan Lanneau and Daniel Massart and defended in December 2023 at the Institut Fourier, Université Grenoble Alpes. The slides are available here as well as the manuscript. You can also find here a small explication of my work which was meant for non-matematicians attending my PhD defense.

A fundamental domain for the SL(2,R)-orbit of the golden L in the moduli space of translation surfaces.