Harmonic flows of SU(n) structures (HARFLO)

Call for expression of interest description

The Marie S. Curie Postdoctoral Fellowship (MSCA-PF) programme is a highly prestigious renowned EU-funded scheme. It offers talented scientists a unique chance to set up 2-year research and training projects with the support of a supervising team. Besides providing an attractive grant, it represents a major opportunity to boost the career of promising researchers. 

The LMBA located in Brest, France is thus looking for excellent postdoctoral researchers with an international profile to write a persuasive proposal to apply for a Marie S. Curie Postdoctoral Fellowship grant in 2024 (deadline of the EU call set on 11 September 2024). The topic and research team presented below have been identified in this regard.

Main Research Field

Mathematics (MAT)

Research sub-field(s)

Differential geometry, Analysis on manifolds, Mathematical Physics

Keywords

Harmonic maps, Geometric structures, Special geometries

Research project description

In the theory of connections and their holonomy groups, G-structures provide a general framework to study geometric structures.

This is a central subject in differential geometry, essentially because of Berger’s classification of Riemannian holonomy groups, yielding a large list of special geometries: Kähler, hyper-Kähler, quaternion-Kähler, Calabi-Yau, G 2 and Spin(7)-manifolds, each related to a Lie subgroup G ⊂ SO(n).

However, the existence of torsion free G-structures, i.e. manifolds with holonomy group contained in G, is a hard problem, since it involves solving a challenging non-linear system of PDE’s. To ease the difficulty, an easier approach would be to start work with the relaxed softer condition of non-integral geometries, i.e. G-structures admitting torsion, such as nearly Kähler or nearly G 2 -manifolds.

Meanwhile, the theory of harmonic maps offers a powerful tool to determinate the best member of a given homotopy class of maps between Riemannian manifolds, based on a variational principle. The main tool is the heat flow method.

This theory and its results can be applied to maps with a specific geometrical meaning, in particular sections of fiber bundles.

SU(n)-Structures are exceptionally important, as they correspond to the geometric reduction wherein Yau’s solution to the Calabi conjecture takes place. The significance of this result goes far beyond the realm of geometry

and its relevance to theoretical physics can be found in the model of string theory, understanding the complement of the Minkowski space-time into a 10-dimensional manifold using a Calabi-Yau 3-fold (i.e. a torsion free SU(3)-structure).

Moreover, non-integrable SU(n)-structures have also been considered from the physical and mathematical points of view, with the study of holonomy groups, namely, the 6-dimensional nearly Kähler manifolds and G 2 -manifolds.

The existence of harmonic SU(n)-structures then becomes a natural question of great consequence, underscored by its connections to other structures.

We propose to use geometric flow methods to investigate harmonic  and biharmonic SU(n)-structures.

Motivated by the recent results of W. He on the existence of minimisers of the bienergy functional on four-dimensional Hermitian manifolds, we propose to extend the programme of harmonic maps to harmonic and biharmonic SU(n) structures.

As bienergy measures the failure of harmonicity, a bienergy minimizing structure can be seen as being as close to harmonic as possible.

This introduces a new generalization of Kähler and Calabi-Yau structures.

As proved by W. He, energy-minimizers exist and therefore likely to be reached by the gradient flow. It is a far more difficult challenge, as the biharmonic flow will be a non-linear system of fourth-order PDE’s.

References :

1) S. Dwivedi, P. Gianniotis and S. Karigiannis, A Gradient Flow of Isometric G 2 -structures, J. Geom. Anal. (2019) 1-79.

2) W. He, Biharmonic almost complex structure, arXiv:2006.05958, (2020).

Supervisor(s)

The Postdoctoral Fellow will be supervised by Eric Loubeau.

Eric Loubeau was recruited at the University of Brest in 2000, after taking a Ph.D. in Pure Mathematics at the University of Leeds (U.K.) in 1996, under the supervision of J.C. Wood, and a post-doctoral position at the K.T.H. in Stockholm.

E. Loubeau has written around 45 research articles on harmonic maps, biharmonic maps, harmonic morphisms and harmonic sections.

He organized international conferences, notably in 1997, 2001, 2017 and 2019 and supervised several post-doctoral students. He is currently the supervisor of a PhD thesis.

Recently, he developed a collaboration network between France and Brazil on Special Geometries and Gauge Theories, funded by a CAPES-COFECUB cooperation project.

His current research interests are in the flow of geometric structures, Nearly Kähler manifolds and biharmonic submanifolds.

References :

1) E. Loubeau, H. Sá Earp, Harmonic flows of geometric structures, in Annals of Global Analysis and Geometry, 2023, 64

2) S. Dwivedi, E. Loubeau, H.N. Sa Earp, Harmonic flow of Spin(7)-structures, to appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (2022).

3) E. Loubeau, A. Moreno, H. Sa Earp & J. Saavedra, Harmonic Sp(2)-invariant G2-structures on the 7-sphere. Journal of Geometric Analysis 32 (2022).

Department/

Research                   

The Laboratoire de Mathématiques de Bretagne Atlantique is a UBO-CNRS joint research unit (UMR 6205) counting some fifty-seven members.

The research group in Differential Geometry has established in Brest a widely recognised expertise in geometric flows, harmonic maps and morphisms, as witness the eleven international conferences organised since 1997 and the summer school “Geometric analysis” in 2010.

Its nine permanent members cover a wide range of subjects revolving around geometric analysis: heat flows for (p)-harmonic maps and curvature problems, harmonic vector fields, spin geometry, biharmonic maps, foliations, Einstein manifolds, twistor theory and special geometries.

Recently, four postdoctoral researchers have been guests of the Differential Geometry group, including a Marie Curie IEF fellow.

The Laboratory has developed its expertise in international networks. with more than ten years of experience in the management of European projects, with four Marie Curie Training Networks, an IEF grant in 2008, an IRSES exchange programme between Europe and Brazil and a Brazil-France CAPES-COFECUB cooperation programme.

Overall, since 2013, nearly two-hundred research articles have been written by members of the Laboratory, in collaboration with mathematicians from twenty different countries.

www.math.univ-brest.fr

Location

LMBA at the Faculty of Science, Brest

Suggestion for interdisciplinary / intersectoral secondments and placements

A six-month secondment in Brazil or in England will be included in the project.

Skills Requirements 

Knowledge of basic Differential Geometry and Analysis, experience in analytical flows will be a bonus.

Eligibility criteria

for applicants

Academic qualification: By 11 September 2024, applicants must bein possession of a doctoral degree, defined as a successfully defended doctoral thesis, even if the doctoral degree has yet to be awarded.

Research experience: Applicants must have a maximum of 8 years full-time equivalent experience in research, measured from the date applicants were in possession of a doctoral degree. Years of experience outside research and career breaks (e.g. due to parental leave), will not be taken into account.

Nationality & Mobility rules:Applicants can be of any nationality but must not have resided more than 12 months in France in the 36 months immediately prior to the MSCA-PF call deadline on 11 September 2024.

Application process

We encourage all motivated and eligible postdoctoral researchers to send their expressions of interest through the EU Survey application form (link here), before 5^th of May 2024. Your application shall include:

• a CV specifying: (i) the exact dates for each position and its location (country) and (ii) a list of publications;

• a cover letter including a research outline (up to 2 pages) identifying the research synergies with the project supervisor(s) and proposed research topics described above.

Estimated timetable