PhD positions

The proposed thesis lies at the interface between the theory of Liouville integrable systems and the semiclassical analysis of geometric quantization. The goal is to describe the joint spectrum of certain pairs of operators acting on spaces of holomorphic sections of large tensor powers of some complex line bundle and quantizing an integrable system with two degrees of freedom whose momentum map possesses a singular value of focus-focus type with multiple singular points on the corresponding level. This description would help studying inverse questions for the spectral theory of such operators. More precisely, on a four-dimensional quantizable compact Kähler manifold, we consider two commuting Berezin-Toeplitz operators whose joint principal symbol is the momentum map for an integrable system, and assume that there exists a singular value of this momentum map of focus-focus type for which the corresponding level is connected and contains several singular points. The goal is to describe the joint spectrum of the operators in the semiclassical limit, locally near this singular value. The case of semiclassical pseudodifferential operators, with a single singular point on the critical level, has been investigated in the early 2000s; the case of several singular points has never been explored, and neither has the single critical point case in the setting of Berezin-Toeplitz operators, which would consitute an interesting first step. The multiple singular points case should involve the semi-local symplectic invariants recently obtained by Pelayo and Tang. To illustrate the results, it will be possible and interesting to rely on the numerous examples investigated in the last few years and for some of which the aforementioned invariants have been computed. As a natural follow-up, the results could be applied to the inverse spectral problem for Berezin-Toeplitz operators; a natural question is to understand whether the knowledge of the semiclassical joint spectrum of two such operators (which commute), whose joint principal symbol is the momentum map for a semitoric integrable system, determines this integrable system up to isomorphism. A positive answer has been given recently by Le Floch and Vũ Ngọc in the case where every focus-focus singular level has a single singular point, but the lack of description of the joint spectrum near focus-focus values led to working with a double limit (first on the semiclassical parameter, then when a regular value goes to a given focus-focus value). A first application for the results of the thesis would be to obtain a more straightforward proof, without using this double limit. A second application would be to determine if a similar inverse result could be obtained in the case where the focus-focus singular levels may contain several singular points.

Advanced differential geometry (fibers, symplectic and
Kählerian geometry, etc.) 

Foundations of semiclassical analysis

The construction of a volumetric mesh for a given geometric domain is a complex problem that has been addressed for many years. The generation of purely hexahedral meshes for domains of any shape is still an open problem. Such meshes would be very useful in numerical simulations such as fluid dynamics. As part of the work proposed in this thesis, we aim to develop an efficient and automatic algorithm that, starting from a domain defined by a surface mesh or a point cloud, uses the variational approach [4-HKTB24] to obtain a skeleton, which is then used as a scaffold [2-VKB23] to construct a hexahe-dral volume mesh. 

Numerous problems must be solved in order to obtain a complete and integrated solution.

I. A rigorous mathematical demonstration of the robustness of the algorithm could prove useful in ensuring the long-term viability of our method.

II. The remeshing of the internal topology of the skeleton composed of segments (1D) and triangles (2D), obtained by the variational method, will need to be implemented for its coupling with mesh gen-eration. In addition, the management of special cases that we have identified in order to maintain com-patibility with our scaffolding structure needs to be studied rigorously.

III. Particular attention must be paid to preserving the topological properties of meshes, which is necessary if we wish to retain specialised optimisations for simulation. In this context, methods for subdividing and adapting mesh sampling will need to be explored.

IV. Characterisation of the geometric domains that can be represented by skeletons (1D-2D) and then meshed by our algorithm is also required in order to control the domain of validity of the methodology. 

V. Finally, validating the results by applying simulation codes to the meshes produced by experts would enable practical validation of the work and might lead to the discovery of new problems to be solved. 

[4-HKTB24] Q. Huang, P. Kraemer, S. Thery, D. Bechmann, Dynamic Skeletonization via Variational Medial Axis Sampling, Full paper at ACM SIGGRAPH ASIA 2024, Tokyo, Japan, décem-bre 2024.
[2-VKB23] P. Viville, P. Kraemer, D. Bechmann, Meso-Skeleton Guided Hexahedral Mesh Design, Full paper at Pacific Graphics 2023, Computer Graphics Forum, Volume 42, Number 7.

The candidate holds a master’s degree in computer science with expertise in computer graphics, specifically geometric modeling. 

He or she possesses the skills necessary to address scientific problems and develop 3D applications (C++ programming and graphics). 

Mathematical skills in geometry would also be a major asset for this position.