Bohr-Sommerfeld conditions in the multiple focus-focus case

Supervision

Yohann Le Floch (IRMA, Strasbourg)

Laboratory and team

IRMA, Strasbourg - Team "ANA"

Subject description

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.

Related mathematical skills

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

Foundations of semiclassical analysis

Other public funding

Candidates recruited as PhDs will benefit from IRMIA++ funding and will have to follow the Graduate Program "Mathematics and Applications: Research and Interactions" (https://irmiapp.unistra.fr/training/presentation).

IRMIA++ is one of the 15 Interdisciplinary Thematic Institute (ITI) of the University of Strasbourg. It brings together a research cluster and a master-doctorate training program, relying on 12 research teams and 9 master tracks.

It encompasses all mathematicians at Université de Strasbourg, with partners in computer science and physics. ITI IRMIA++ builds on the internationally renowned research in mathematics in Strasbourg, and its well-established links with the socio-economic environment. It promotes interdisciplinary academic collaborations and industrial partnerships.

A core part of the IRMIA++ mission is to realize high-level training through integrated master-PhD tracks over 5 years, with common actions fostering an interdisciplinary culture, such as joint projects, new courses and workshops around mathematics and its interactions.

Selection will rely on the professional project of the candidate, his/her interest for interdisciplinarity and academic results.