Margaret Bilu

I am a CNRS researcher (chargée de recherche) at the Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique. I completed my PhD in 2017 under the supervision of Antoine Chambert-Loir.

Note:

I am on maternity leave until September 2025, and therefore currently do not accept requests for referee reports, committees, internships, etc.

Contact

Centre de Mathématiques Laurent Schwartz
Ecole Polytechnique
91128 Palaiseau Cedex
FRANCE

firstname.lastname@polytechnique.edu

Curriculum vitae

Research

My research interests lie in number theory and algebraic geometry. I am particularly interested in developing and proving motivic analogues of number-theoretic tools and results in the Grothendieck ring of varieties. In my thesis, I developed a notion of motivic Euler product which has since found several applications.

Organization

I helped organize an online seminar on rational points and related topics: Zoom on Rational Points

Past and future talks and travel (real or virtual)

• Rencontre d'ouverture de l'ANR Cyclades, Strasbourg, January 13-15th, 2025
• Séminaire Des Mathématiques, DMA, ENS, December 18th, 2024
• Séminaire TNGA, Caen, October 25th, 2024
• Séminaire AGATA, Montpellier, October 17th, 2024
• Moduli Spaces and Arithmetic, Nagoya, Japan, September 9-13 2024
• Séminaire Mathematic Park, Institut Henri Poincaré, April 27th, 2024
  
• 8th Mini-Symposium of the Roman Number Theory Association, Rome, April 17-19th
• Workshop Enumerative Geometry and Arithmetic, Lorentz Center, Leiden, March 11-15, 2024
• Logic Seminar, University of Oxford, January 25th, 2024
• WAARP, University of Bristol, September 11-15, 2023
  
• SAGA, Laboratoire de Mathématiques d'Orsay, April 25th, 2023
  
• Séminaire Géométrie et Théorie des Modèles, IHP, April 21st, 2023
• Séminaire Algèbre et Géometries, Institut Fourier, Grenoble, April 3rd, 2023
• Séminaire de Géométrie Arithmétique et Motivique, LAGA, Sorbonne Paris Nord, March 17th, 2023
  
• Rencontre ANR Fracasso, Bordeaux, March 9-10, 2023
  
• Non-archimedean methods in arithmetic and geometry, Les Diablerets, Switzerland, February 12-17, 2023
• Séminaire d'Arithmétique, UMPA, Lyon, January 26th, 2023
  
• Séminaire de théorie des nombres, Bordeaux, December 2nd, 2022
• Women in Numbers Europe (Project group leader), Utrecht, August 29th-September 2nd 2022
• Rational Points on Higher-Dimensional Varieties, ICMS, Edinburgh, April 25-29, 2022
  
• Séminaire de théorie des nombres de l'IMJ-PRG, March 21st, 2022
  
• Algebra, Geometry and Physics seminar (MPIM Bonn/HU Berlin), July 20th, 2021
  
• BIRS workshop "Geometry via arithmetic", July 11-16, 2021
• Browning group working seminar, IST Austria, July 2nd, 2021
• Around Frobenius distributions and related topics II, June 28-29, 2021
  
• Séminaire de théorie des nombres, Université de Bordeaux, May 21st, 2021
  
• Séminaire de théorie des nombres, Institut Fourier, April 13th, 2021
• Browning group working seminar, IST Austria, January 27th, 2021
  
• Séminaire de géométrie, Université Savoie Mont Blanc, Chambéry, January 21st, 2021
  
• Oberseminar Zahlentheorie und Arithmetische Geometrie, Hannover, November 27th, 2020
  
• Duke Algebraic Geometry Seminar, November 20th, 2020
• Browning group working seminar, IST Austria, October 7th, 14th, 21st, 2020
  
• Arithmetic Statistics and Local-Global Principles, Erwin Schrödinger Institute, Vienna, Austria, July 20-24, 2020. (cancelled)
• Stanford Algebraic Geometry Seminar online, June 12th, 2020
  
• Fifth International Workshop on Zeta Functions in Algebra and Geometry, Nice, France, May 25-28th, 2020 (postponed)
  
• Simons Symposium on the Geometry of Arithmetic Statistics, Schloss Elmau, Krün, Germany, May 3-9, 2020 (postponed)
  
• Georgia Algebraic Geometry Symposium, Emory University, Atlanta, April 17-19, 2020 (postponed)
  
• Number Theory Seminar, Duke University, March 18th, 2020 (cancelled)
  
• Quebec-Vermont Number Theory Seminar, Mc Gill University, Montreal, March 12th, 2020 (cancelled)
  
• PU/IAS Number Theory Seminar, Princeton University, January 23, 2020
  
• Number Theory seminar, Boston University, November 4th, 2019
• Geometry/Topology seminar, University of Chicago, October 21st, 2019
  
• Algebraic Geometry seminar, University of Michigan, October 16th, 2019
• Algebraic Geometry seminar, University of Stony Brook, September 18th, 2019
  
• Conference "Rational points on Fano and similar varieties", Institut Henri Poincaré, Paris, May 20-24, 2019 Video
  
• Algebraic Geometry seminar, Princeton University, April 1st, 2019
  
• Algebra Seminar, Brown University, March 11, 2019
  
• Number Theory seminar, MIT, November 20, 2018
  
• CANT 2018, Graduate Center at the City University of New York, May 23, 2018
  
• Algebra seminar, University of Pennsylvania, October 23, 2017
  
• Atkin Memorial Workshop on Height zeta functions, University of Illinois at Chicago, April 28-30, 2017
• Séminaire de théorie des nombres de l'IMJ-PRG, Paris, March 20th 2017
• Algebraic Geometry Seminar, Courant Institute of Mathematical Sciences, New York University, October 25, 2016
• Special day on motivic integration and non-archimedean geometry, Université Lille 1, October 10, 2016
• Intercity Number Theory Seminar, Leuven, Belgium, June 10, 2016
• Number Theory Seminar, ETH Zürich, March 4, 2016
• Séminaire Géométrie et Singularités, IRMAR, Université Rennes 1, February 10, 2016
  

Publications and preprints

• A motivic circle method (joint with T. Browning). Link to arXiv
  
• A quadratic enrichment of the logarithmic derivative of the zeta function (joint with W. Ho, P. Srinivasan, I. Vogt and K. Wickelgren). Trans. Amer. Math. Soc. Ser. B 11 (2024), 1183-1225 Link to arXiv
  
• Zeta statistics and Hadamard functions (joint with R. Das and S. Howe), Adv. Math. 407 (2022), 108556. Link to arXiv.
  
• Motivic Euler products in motivic statistics (joint with S. Howe),  Algebra Number Theory 15 (2021), no. 9, 2195–2259.Link to arXiv.
• Motivic Euler products and Motivic Height Zeta Functions, Mem. Amer. Math. Soc. 282 (2023), no. 1396. Link to arxiv. This is an updated version of my PhD Thesis.
  
• Rational points on analytic varieties (joint with A. Beshenov, Yu. Bilu and P. Rath), EMS Surv. Math. Sci. 2 (2015), p. 109-130. Link to arXiv. These are notes I wrote up following a mini-course by Yuri Bilu. It starts with some elementary results by Jarník and Dörge, continues with the theorem of Bombieri and Pila about rational points with given denominator on analytic curves, and then presents the powerful theorem of Pila and Wilkie about rational points of bounded height on sets definable in an o-minimal structure. Finally, applications to the Manin-Mumford and André-Oort conjectures are given.

Theses and essays

• My PhD thesis.Here are notes from talks I have given on the subject. 
• My Introduction à un domaine de recherche: this is a short thesis written towards the end of one's studies at the ENS. Its aim is to present, in a way accessible to non-specialists, the research area one wants to specialise in and the questions one intends to address during one's PhD.  It summarises in particular the contents of my Master's thesis.
  
• My Master's thesis was about Hrushovski and Kazhdan's motivic Poisson summation formula. It is a motivic analogue of the classical Poisson summation formula for functions over the adele group of a global field, which is proved in the first part of the text, following Tate's thesis.
  
• MyPart III essay written during my year at Cambridge: this is an account of the theory of complex multiplication of abelian varieties.
  
• My Mémoire de maîtrise: this is a thesis written in my first year at the ENS, together with Guillaume Dubach. The u-invariant of a field is the maximal dimension of an anisotropic quadratic form over this field. A conjecture by Kaplansky states that the u-invariant of a field should always be a power of 2. This text presents the construction, due to Merkurjev, of a field with u-invariant 2n for every positive integer n, which thereby disproves Kaplansky's conjecture. This beautiful argument uses division algebras and Clifford algebras. Our text also contains the proof, by Pfister, of another conjecture by Kaplansky, stating that the level of a field, that is, the minimal integer n such that -1 is a sum of n squares, is always a power of 2. 

Teaching

Outreach/Mathématiques périscolaires

Les activités suivantes peuvent intéresser tout collégien ou lycéen aimant les mathématiques.

• L'association Animath et la Préparation Olympique Française de Mathématiques (POFM) fournissent une préparation à diverses compétitions internationales de mathématiques, appelées olympiades. Il s'agit de s'entraîner à résoudre en temps limité des problèmes assez courts provenant de plusieurs domaines des mathématiques. Plus d'informations ici.
• Le  Tournoi Français des Jeunes Mathématiciens est un tournoi où il s'agit de réfléchir par équipes pendant quelques mois à des problèmes comprenant des questions de difficulté variable, dont certaines ouvertes, puis de présenter ce qu'on a trouvé sous forme écrite et orale.
• Le club Parimaths est un club de mathématiques ayant lieu à l'ENS tous les samedis en période scolaire. Sur le site on peut consulter les feuilles distribuées lors de certaines séances.

Voici une sélection de cours que j'ai donnés à l'occasion de stages olympiques ainsi que de séances du club Parimaths:

• Séries génératrices
• Graphes
• Théorie extrémale des graphes
• Polynômes et nombres entiers
• Fractions continues
• Polygones équidécomposables

Attention: pour la plupart, ces textes n'ont servi qu'une fois. Ils peuvent donc encore contenir un certain nombre de coquilles.