Stage de Master en mathématique

Company Description ENSTA is a leading French public institution of higher education and research under the supervision of the Ministry of the Armed Forces and a founding member of the Institut Polytechnique de Paris. Formed in 2025 by the merger of ENSTA Paris and ENSTA Bretagne, the institute operates two campuses in Paris-Saclay (Palaiseau) and Brest, hosting over 2,000 students, 200 teacher-researchers, 300 PhD candidates, and 9 research laboratories. ENSTA offers a wide range of programs including engineering degrees, master’s programs, PhDs, and specialized master’s degrees, with a strong focus on cutting-edge research and innovation. Its work addresses strategic fields such as transport and mobility, sustainable energy, defense and security, marine engineering, digital technologies, environmental sciences, space, and healthcare engineering. Since 1741, ENSTA has been recognized for scientific and technical excellence and a culture of bold innovation. Scientific background Diffraction problems involving obstacles can be solved using representations of the field in terms of auxiliary singular sources. Recent methods, such as the one proposed by Costa (arXiv:2601.19020), exploit analytic continuation and conformal mappings to determine the locations of these sources. As the geometry of an obstacle evolves, the behavior of these singularities raises fundamental mathematical questions: some move, others appear to emerge or disappear, while their contributions to a representation of the field may undergo substantial redistribution. These phenomena suggest an underlying geometric and analytic structure whose understanding goes beyond numerical methods for diffraction. Internship objectives The internship aims to investigate the evolution of complex singularities associated with a family of deformable domains, with an emphasis on rigorous mathematical analysis. Possible lines of investigation include:
• Geometry of conformal mappings: meromorphic continuation of families of conformal maps, trajectories of critical points, and associated branched structures.
• Bifurcations of singularities: characterization of collisions, unfoldings, changes of multiplicity, and possible emergence or disappearance of singularities, particularly through discriminants and Puiseux expansions.
• Schwarz functions and analytic continuation: relationships between boundary deformation, the evolution of geometric singularities, and singularities of solutions to partial differential equations, particularly the Helmholtz equation.
• Singular representations: investigation of the redistribution of weights among point sources and the possibility of representing certain singular structures by continuous distributions. The work will begin with explicitly parameterized families of domains (disk, ellipse, nonconvex geometries), allowing analytic results to be obtained and precise conjectures to be formulated. The goal is to identify structural properties and, wherever possible, establish theoretical results. Targeted numerical experiments may support the proofs but will not be the main focus of the internship. Candidate profile A second-year Master’s student (M2) in mathematics, with a solid background in complex analysis and an interest in geometry and analysis. Knowledge of partial differential equations, Riemann surfaces, algebraic geometry, or bifurcation theory would be an advantage, but expertise in all of these areas is not required. Indicative duration: 4–6 months. Supervision and

location:
ENSTA Brest campus. Starting reference S. Costa, arXiv:2601.19020 (2026), https://arxiv.org/abs/2601.19020. Contact: prof. Arnaud Coatanhay (arnaud.coatanhay@ensta.fr) and prof. Yves-Marie Scolan (yves-marie.scolan@ensta.fr)