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Differential Geometry

Person in charge of the Unit : Oui

The research activities of this unit are devoted to various aspects of differential geometry : symplectic geometry, Kähler geometry, deformation quantization,  symplectic and contact topology. 


Definite connections

An SO(3)-connection on a 4-manifold is called definite if its curvature is non-zero on every tangent 2-plane. Given such a connection the corresponding 2-sphere bundle carries a natural symplectic structure. Definite connections carry a sign, + corresponds to Fano manifolds, - to Calabi-Yaus. The study of definite connections involves both the construction of examples, most notably via hyperbolic geometry, as well as attempts to understand which 4-manifolds admit definite connections. There is a geometric flow which attempts to deform a given definite connection into one which solves a certain PDE. Understanding singularity formation in this flow is an important step forward in understanding which 4-manifolds support definite connections. 

Complex geometry of flat bundles over hyperbolic 3-manifolds

The frame bundle X of a hyperbolic 3-manifold M is naturally a complex manifold with trivial canonical bundle. From this it is possible to give a holomorphic interpretation of flat vector bundles over M. In one direction, one can interpret a flat SL(2,C)-bundles over M a defining an alternative complex structure on X. Alternatively, one can fix the holomorphic structure on X, then the pull-back of flat complex vector bundles on M are naturally holomorphic vector bundles on X. This procedure embeds the Chern-Simons theory of M in the holomorphic Chern-Simons theory of X and one may hope to use techniques from one side to solve problems in the other. For example, given a flat bundle over M with a Hermitian metric, the pull-back metric in the holomorphic bundle is Hermitian-Einstein if and only if the original metric is harmonic. From here we intend to investigate if the theory of harmonic maps can shed light on the behaviour of Hermitian-Einstein connections over X.

Kahler Geometry

Extremal Kähler metrics, when they exist, are ''canonical'' representatives of their Kähler class. Their existence is conjecturaly equiavlent to the stability of the underlying polarised variety. Via quanitsation, there is a strong connection between extremal metrics and balanced projective embeddings. In addition to these aspects we also consider the production of extremal metrics via geometric analysis. One tool for this is the Calabi flow which attempts to deform a given Kähler metric to an extremal one. The quantization of this flow is balancing flow, a certain flow on the space of projective embeddings. We are intereseted in better understanding Calabi flow via the projective geometry of balancing flow.  

Complex, symplectic and contact geometry, quantisation and interactions (ARC)

This research project involves three inter-linked subjects: quantisation, symplectic geometry and Kähler geometry. The research will follow two main directions: the use of ideas from moment-map geometry and the use of geometric flows.