ED Mathématiques et Informatique
Geodesics on Convex Flat Cone Spheres
by Kai FU (IMB - Institut de Mathématiques de Bordeaux)
The defense will take place at 15h00 - Amphithéâtre LaBRI Université de Bordeaux, 351 cours de la Libération, Bâtiment A30 33405 TALENCE
in front of the jury composed of
- Elise GOUJARD - Maîtresse de conférences - Université de Bordeaux, Institut de Mathématiques de Bordeaux (IMB) - Directeur de these
- Vincent DELECROIX - Chargé de recherche - Université de Bordeaux, Laboratoire Bordelais de Recherche en Informatique - CoDirecteur de these
- Samantha FAIRCHILD - Assistant professor - Technical University of Eindhoven - Examinateur
- Duc-Manh NGUYEN - Professeur - Université de Tours, Département de mathématiques - Examinateur
- Peter SMILLIE - Assistant professor - Max Planck Institute for Mathematics in the Sciences in Leipzig - Examinateur
- Dylan THURSTON - Professor - Indiana University, Bloomington - Examinateur
- Carlos MATHEUS SILVA SANTOS - Directeur de recherche - École Polytechnique - Rapporteur
- Howard MASUR - Professeur émérite - Department of Mathematics, University of Chicago - Rapporteur
A flat cone sphere is a Riemann sphere equipped with a flat metric with finitely many conical singularities. When all cone angles are strictly less than 2 pi, the flat cone sphere is said to be convex. Such structures arise in the study of polygonal billiards and in geometric structures on moduli spaces of Riemann spheres with labeled points. In this thesis, we study the behavior of geodesics on flat cone spheres, with a focus on saddle connections and regular closed geodesics. While much is known when all cone angles are rational multiples of pi, the irrational case remains largely unexplored. Our goal is to develop new tools to investigate this irrational setting and to extend existing techniques beyond the rational case. In the rational case, the geometry of flat cone spheres becomes tightly connected to the rich theory of translation surfaces. One of the central results in this area is the Siegel–Veech formula. The classical Siegel mean value theorem computes the average number of lattice points in bounded subsets of Euclidean Spaces. Motivated by this result, Veech introduced a counterpart for translation surfaces, known as the Siegel–Veech formula, which describes the average number of saddle connections of bounded length over moduli spaces of translation surfaces. We extend the Siegel–Veech framework to the setting of convex flat cone spheres. We define a generalized Siegel–Veech transform and prove that it is essentially bounded on the moduli space. This leads to the definition of a Siegel–Veech measure on the positive real line, obtained by integrating Siegel–Veech transforms over the moduli space. This measure can be viewed as a generalization of the classical Siegel–Veech formula. We show that it is absolutely continuous and piecewise real analytic. Finally, we compute the asymptotic behavior of the measure on small intervals (0,a) as a tends to 0, thereby obtaining an analogue of Siegel–Veech constants for convex flat cone spheres.
Arithmetics and algorithmics of algebraic curves and applications to error-correcting codes and cryptography
by Jean GASNIER (IMB - Institut de Mathématiques de Bordeaux)
The defense will take place at 11h00 - Salle 1 351 Cours de la Libération, Batiment A33, 33400, Talence
in front of the jury composed of
- Jean-Marc COUVEIGNES - Professeur des universités - Université de Bordeaux - Directeur de these
- Pierrick GAUDRY - Directeur de recherche - CNRS - Rapporteur
- Alain COUVREUR - Directeur de recherche - INRIA - Rapporteur
- David KOHEL - Professeur des universités - Aix-Marseille Université - Examinateur
- Elisa LORENZO-GARCIA - Maîtresse assistante - Université de Neuchâtel - Examinateur
The elementary arithmetics and algorithmics of algebraic curves is at the heart of major contributions to coding theory and cryptology. This PhD thesis draws on more advanced concepts, from class field theory, equivariant Riemann-Roch theory and the arithmetic geometry of jacobian varieties, to establish a general framework adapted to these constructions and improve their efficiency. In particular, we study the properties of linear codes endowed with a module structure over the algebra of a finite group G. We study more specifically the codes endowed with a structure of free submodule of a free module, and their duality. Specifically, we show that these codes can be described by parity check matrices whose coefficients belong to the algebra of the group G. When G is commutative, the fast Fourier transform provides nice algorithmic properties to these error-correcting codes. We also show how to build these codes, using unramified abelian coverings of smooth projective curves, and we give the first examples of excellent codes encodable in quasi-linear time and decodable in quasi-quadratic time. Another application involves the generation of families of pairing-friendly elliptic curves, used in some cryptographic protocols. The complex multiplication theory allows to reduce the underlying geometric problem to a problem of cyclotomic arithmetics. We deduce from the study of this problem an unified method of generation of families of pairing-friendly elliptic curves.
ED Sciences Physiques et de l'Ingénieur
Lead halide Perovskite nanostructures as single photon sources
by Mathias STAUNSTRUP (Laboratoire Photonique, Numérique & Nanosciences)
The defense will take place at 14h00 - Amphithéâtre Andre Ducasse 1 Rue François Mitterrand, 33400 Talence
in front of the jury composed of
- Jean-Sébastien LAURET - Professeur - École normale supérieure Paris-Saclay - Examinateur
- Stéphanie BUIL - Maîtresse de conférences - Université de Versailles Saint-Quentin-en-Yvelines - Rapporteur
- Laurent LEGRAND - Maître de conférences - Sorbonne Université - Rapporteur
- Lionel CANIONI - Professeur - Université de Bordeaux - Examinateur
The generation of pure and indistinguishable photon states is essential for quantum photonic devices, which are expected to enable secure communication and enhanced computational capabilities. Lead halide Perovskite nanocrystals have become an attractive platform for photovoltaics and light emitting devices and more recent research has focused on lead halide perovskite nanostructures as single-photon emitters. The first part of this thesis investigates the photophysical properties of inorganic colloidal perovskite nanorods of cesium lead halide using magneto-photoluminescence and time-resolved spectroscopy. The results reveal a strong dependence of the fine structure on nanorod size and aspect ratio, with emission from a short-lived sublevel concentrating the oscillator strength into a highly polarized, narrow zero-phonon line. Single-photon emission is demonstrated by g^{(2)} correlation function measurements, showing strong antibunching. Fourier spectroscopy analysis of decoherence time and its temperature dependence shows photon coherence up to the limit set by the lifetime of the excited state. Photon indistinguishability in two-photon Hong-Ou-Mandel interference experiments is demonstrated with a visibility up to 60%. The second part explores a new platform of buried organic lead halide perovskite quantum dots of FAPbI3-xBrx embedded in a 3D perovskite thin film of FAPbI3 via flash annealing. This integration mitigates issues due to surface interactions usually found in colloidal nanostructures, achieving performance on par with colloidal emitters. Using magneto-photoluminescence spectroscopy, a bright triplet exciton structure is revealed with a red shifted dark singlet and trion state. The emission spectra show lines of good stability and narrow sub ~130μeV linewidths, and single photon antibunching. This system holds great promise as a low-cost, on-chip single-photon source for quantum devices.