# Mathieu Dutour

# About me

I am currently a postdoctoral fellow at the University of Alberta, in Edmonton, working alongside Manish Patnaik on interactions between loop groups and Arakelov geometry.

Before that, I was a Ph.D. student at Sorbonne Université, in Paris, where I worked under the supervision of Gerard Freixas i Montplet on Arakelov geometry for singular metrics, and more precisely on obtaining a Deligne-Riemann-Roch isometry for flat unitary holomorphic vector bundles over modular curves.

My main mathematical interest is Arakelov geometry, a relatively recent discipline which involves many other interesting areas of research, such as complex and algebraic geometry, number theory, spectral theory, functional analysis, to name but a few.

More recently, I re-discovered the theory of Lie and Kac-Moody algebras, which I had partly learned years ago, when I started studying loop groups. The interaction between those and Arakelov geometry, although surprising at first, ends up being perfectly natural, in view of recent works on pro-Hermitian vector bundles and theta invariants.

Here, you can find my most recent research project in English, and a French version of it.

Email address: dutour@ualberta.ca

Questions and comments are welcome! Also, please let me know if there are any dead links.

## Research articles and preprints

M. Dutour, Pseudo-Laplacian on a cuspidal end with a flat unitary line bundle: Alvarez-Wentworth boundary conditions. Arxiv: 2211.04040.

70 pages.M. Dutour, M. Patnaik, Infinite-rank Euclidean Lattices and Loop Groups, Arxiv: 2203.08976. To appear in proceedings of M.V. Subbarao Centennial volume.

36 pages.M. Dutour, Pseudo-Laplacian on a cuspidal end with a flat unitary line bundle: Dirichlet boundary conditions. Arxiv: 2106.05412.

95 pages.

### Theta-finite pro-Hermitian vector bundles from loop groups elements

This talk was held on November 28th, 2022 for the Number Theory and Combinatorics Seminar at the University of Lethbridge, in Alberta, Canada.

Sound level drops at around 7:00 due to some technical issue, but the talk can still be heard.

Note: Every available video recording of my talks can be accessed in the section "Research presentations" below.

### Quillen metrics on modular curves

This talk was held on November 6th, 2021, during the Alberta Number Theory Days at Banff International Research Station, in Alberta, Canada.

Note: Every available video recording of my talks can be accessed in the section "Research presentations" below.

## Research presentations

Séminaire Groupes Réductifs et Formes Automorphes (Reductive groups and automorphic forms seminar), IMJ-PRG, Université Paris-Cité, March 6th, 2023. Slides. The slides are in French. Website of the seminar.

Title: Fibrés vectoriels pro-hermitiens et groupes de lacets. (Pro-Hermitian vector bundles and loop groups)Number Theory and Combinatorics Seminar, University of Lethbridge, November 28th, 2022. Slides. Video of the talk. Alternative link to the video.

Website of the seminar.

Title: Theta-finite pro-Hermitian vector bundles from loop groups elements.Intercity Seminar on Arakelov Geometry 2022 (September 12th - 16th), La Cristalera (near Madrid, Spain), September 15th, 2022. Slides. (Slide 2/22 has been slightly updated after the talk). Website of the conference.

Title: Loop groups and Pro-Hermitian vector bundles.GAP Seminar (Geometry, Algebra, Physics), University of Alberta, December 1st, 2021. Slides.

Website of the seminar.

Title: A Deligne-Riemann-Roch isometry for flat unitary vector bundles over modular curves.Alberta Number Theory days, Banff research station, November 6th, 2021. Slides. Video. Alternative link to the video. Other alternative link.

Website of the conference.

Title: Quillen metrics on modular curves.Thesis defense, Sorbonne Université, September 22nd, 2020. Slides (in French). Thesis (in English).

Title: A Deligne-Riemann-Roch isometry for flat unitary vector bundles on modular curves.

Jury: Jean-Benoît Bost, José Ignacio Burgos Gil, Gerard Freixas i Montplet, Colin Guillarmou, Kai Köhler, Xiaonan Ma, Frederic Naud, Richard Wentworth.Inaugural conference for the Franco-Korean LIA, Bordeaux, November 25th, 2019.

Website of the conference.

Title: Quillen metrics on modular curves.Ph.D. students seminar, University of Amiens, March 20th, 2019.

Title: A Deligne-Riemann-Roch isometry on modular curves.Ph.D. students seminar, Paris-Diderot University, February 20th, 2019.

Title: A Deligne-Riemann-Roch isometry on modular curves.

## CV

2021-2023, Postdoctoral fellow at the University of Alberta (Edmonton, Alberta, Canada), department of mathematical and statistical sciences.

Supervisor: Manish Patnaik

Recruited in 2020, though officially started in 2021.2016-2020, Ph.D. student at Sorbonne Université (Paris, France), IMJ-PRG (Institut de mathématiques de Jussieu - Paris Rive Gauche).

Supervisor: Gerard Freixas i Montplet

Topic: A Deligne-Riemann-Roch isometry for flat unitary vector bundles on modular curves.

Defended on September 22nd, 2020 in front of a jury composed of:Jean-Benoît Bost (president of the jury)

José Ignacio Burgos Gil

Gerard Freixas i Montplet (adviser)

Colin Guillarmou (thesis referee)

Kai Köhler (thesis referee)

Xiaonan Ma

Frederic Naud

Richard Wentworth

2015-2016, Research master degree, Sorbonne Université (Paris, France).

2014-2015, Préparation au concours de l'agrégation externe de mathématiques, Professional master degree, ENS Lyon (Lyon, France). Ranked 30th.

The agrégation externe is a competitive exam which allows you to become a teacher in middle school, high school, preparatory classes, or even university in some cases (PRAG positions) in France. At the ENS Lyon, students were encouraged to take a year between the 1st and 2nd year of master to prepare for this exam. This preparation is a great opportunity to consolidate everything that has been learned up to that point in mathematics, and includes almost exclusively material of university level.2013-2014, 1st year of Master, ENS Lyon (Lyon, France).

2012-2013, 3rd year of Licence, ENS Lyon (Lyon, France).

2011-2012, Preparatory class MP*, Lycée du Parc, Lyon, France.

2010-2011, Preparatory class MP, Lycée du Parc, Lyon, France.

2009-2010, Preparatory class MPSI, Lycée Victor Hugo, Besançon, France.

2006-2009, Highschool, Lycée du Parc, Lyon, France.

The base template used for the pdf versions of the CV is available here.

## Organization of seminars and other events

2021, Spring term, co-organization of the seminar "Geometry, Number Theory, and Representation Theory" at the University of Alberta (Edmonton, Alberta, Canada), along with Manish Patnaik and Valentin Buciumas. Seminar page.

## Teaching

2021, Spring term, University of Alberta, Math 125: "Linear Algebra 1". In English.

2019-2020, 2nd semester, Sorbonne Université, TA for LU2MA211: "Lebesgue integrals on R^n". 2nd year students. In French.

2019-2020, 1st semester, Sorbonne Université, TA for LU3MA270: "Algebra". 3rd year students. In French.

2019-2020, 1st semester, Sorbonne Université, TA for LU2MA220: "Arithmetics and algebra". 2nd year students. In French.

2018-2019, 1st semester, Sorbonne Université, TA for 3M270: "Algebra". 3rd year students. In French.

Course notes (written by Alberto Minguez).

Exercise sessions (only the solutions and the written tests were written by me):

Sheet 1 - Solutions

Sheet 2 - Solutions

Sheet 3 - Solutions

Sheet 4 - Solutions

Written Test 1 - Solutions

Written Test 2 - Solutions

In addition, there was a midterm and a final exam.2017-2018, 2nd semester, Sorbonne Université, TA for 3M270: "Algebra". 3rd year students. In French.

Additional document on short exact sequences (written by me).

Exercise sessions (only the solutions and the written tests were written by me):

Sheet 1 - Solutions

Sheet 2 - Solutions

Sheet 3 - Solutions

Sheet 4 - Solutions

Sheet 5 - Solutions

Sheet 6 - Solutions

Sheet 7 - Solutions

Sheet 8 - Solutions

Sheet 9 - Solutions

Sheet 10 - Solutions

Written test 1 - Solutions

Written test 2 - Solutions

In addition, there was a midterm and a final exam.2017-2018, 1st semester, Sorbonne Université, TA for 2M220: "Arithmetics and algebra". 2nd year students. In French.

Course notes (written by Alain Kraus).

Additional document on subfields of a finite field (written by me).

Exercise sessions (only the solutions and the homework assignment were written by me):

Sheet 1 - Solutions

Sheet 2 - Solutions

Sheet 3 - Solutions

Sheet 4 - Solutions

Sheet 5 - Solutions

Sheet 6 - Solutions

Sheet 7 - Solutions

Homework assignment - Solutions

In addition, there was a midterm and a final exam.2016-2017, 2nd semester, Sorbonne Université, TA for 2M256: "Vector analysis and multiple integrals". 2nd year students. In French.

Exercise sessions (only the solutions were written by me):

Sheet 1 - Solutions

Sheet 2 - Solutions

Sheet 3 - Solutions

Sheet 4 - Solutions2016-2017, 1st semester, Sorbonne Université, TA for 1M001: "Analysis and algebra for science". 1st year students. In French.

2013-2014, Lycée du Parc, Lyon, TA in preparatory classes MP*1 and MP*2.

Preparatory classes MPSI/MP/MP* in France constitute an alternative to the first two years of university to study mathematics and physics. They are typically extremely demanding, and students go through two oral exams every week in groups of two or three: one in mathematics, the other one alternatively in physics or in English (or whichever language the student primarily learns). These oral exams are given by a "sort of TA".

## Research internships

2016, Sorbonne Université, Research internship for the 2nd year of Master.

Supervisor: Gerard Freixas i Montplet

Topic: Hilbert modular surfaces

Dissertation (French).2014, Université d'Orsay, Research internship for the 1st year of Master.

Supervisor: Daniel Perrin

Topic: Algebraic liaison of finite sets in the projective space, theorems of Davis-Geramita-Orecchia and Cayley-Bacharach, stratification of finite sets.

Dissertation (French).2013, Université Paris-Diderot, Research internship for the 3rd year of Licence.

Supervisor: Rached Mneimné

Topic: Affine geometry, conics, Clifford algebras.