Descrizione del progetto
Uno studio potrebbe migliorare la comprensione della geometria delle varietà di caratteri
Le varietà di caratteri sono spazi geometrici onnipresenti in matematica e fisica che permettono di acquisire importanti invarianti quantistici e topologici. Sebbene parametrizzino complesse equazioni differenziali ordinarie singolari, nonché le loro generalizzazioni, le varietà di caratteri sono solitamente spazi singolari complicati, per cui le preziose informazioni da esse codificate non sono di facile accesso. Finanziato dal programma di azioni Marie Skłodowska-Curie, il progetto AbQuantumSpec intende sviluppare un nuovo metodo per descrivere le varietà di caratteri, chiamato abelianizzazione. Questo nuovo metodo dovrebbe consentire la costruzione di speciali sistemi di coordinate su varietà di caratteri (chiamati coordinate spettrali) che potrebbero permettere l’acquisizione delle loro importanti strutture geometriche.
Obiettivo
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.
Campo scientifico
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equations
- natural sciencesphysical sciencesquantum physicsquantum field theory
- natural sciencesmathematicspure mathematicstopologyalgebraic topology
- natural sciencesphysical sciencestheoretical physicsstring theory
- natural sciencesmathematicspure mathematicsgeometry
Programma(i)
Argomento(i)
Meccanismo di finanziamento
MSCA-IF-EF-ST - Standard EFCoordinatore
B15 2TT Birmingham
Regno Unito