Description du projet
Une étude susceptible de faire progresser la compréhension de la géométrie des variétés de caractères
Les variétés de caractères sont des espaces géométriques omniprésents en mathématiques et en physique qui saisissent d’importants invariants topologiques et quantiques. Elles paramètrent les équations différentielles ordinaires complexes singulières et leurs généralisations. Toutefois, les variétés de caractères sont généralement des espaces singuliers compliqués, de sorte que les informations précieuses qu’elles encodent sont difficilement accessibles. Financé par le programme Actions Marie Skłodowska-Curie, le projet AbQuantumSpec envisage de développer une nouvelle méthode de description des variétés de caractères appelée abélianisation. Cette nouvelle méthode devrait permettre la construction de systèmes de coordonnées spéciaux sur les variétés de caractères (appelées coordonnées spectrales) qui pourraient rendre compte de leurs importantes structures géométriques.
Objectif
This cross-disciplinary project lies at the interface of geometry, mathematical physics, perturbation theory, and integrable systems, combining techniques from algebraic topology, cluster algebras, ordinary differential equations, and asymptotic analysis. The main goal is to advance our understanding of the geometry of character varieties and their quantisation. This project -- carried out by Nikita Nikolaev under the supervision of Marta Mazzocco at the University of Birmingham -- is expected to result in a fundamental innovation in geometry and have important implications for quantum field theory. It will open a vast new scientific arena and will serve to establish Nikolaev amongst research leaders in this highly active research area. Character varieties are geometric spaces which are ubiquitous in mathematics and physics, where they capture important topological and quantum invariants. These spaces parameterise singular complex ordinary differential equations (such as the Airy and Bessel equations, or even time-independent Schrödinger equations), as well as their generalisations: meromorphic connections on vector bundles over a Riemann surface. However, character varieties are usually complicated singular spaces, so the valuable information they encode is not easy to access. This project will develop a new method to describe character varieties called abelianisation. Ideas behind abelianisation stem from the WKB method in quantum mechanics and have recently resurfaced in the context of quantum field theory and string theory. Abelianisation will allow to construct special coordinate systems on character varieties (called spectral coordinates) which naturally capture crucially important geometric structure of character varieties (most prominently the symplectic and cluster structures). In other words, spectral coordinates will be a geometric gadget to decrypt mathematical and physical information encrypted in character varieties.
Champ scientifique
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equations
- natural sciencesphysical sciencesquantum physicsquantum field theory
- natural sciencesmathematicspure mathematicstopologyalgebraic topology
- natural sciencesphysical sciencestheoretical physicsstring theory
- natural sciencesmathematicspure mathematicsgeometry
Programme(s)
Régime de financement
MSCA-IF-EF-ST - Standard EFCoordinateur
B15 2TT Birmingham
Royaume-Uni