King's College London
Geometric structures often capture certain properties of the physical world, they
might represent solutions to equation like Maxwell of Einstein equations, or might
play the role of a background for certain theory in the way symplectic structure
underlines Hamiltonian mechanics. Probably because of this initial connection to
physics these structures often have a rich internal life and obey non-trivial laws.
One of important slogans in this area is the principle "local to global". Namely,
geometric structure is something that we can observe locally in the space and we
would like to know if the mere existence of a geometric structure on our space
tells us something about the global shape of the space. A very simple application
of this slogan would be a mathematical deduction of the fact that the Earth is a
sphere from the fact that it is (approximatively) locally convex. Here convexity
plays the role of a geometric structure.
There are two possible extremes for a geometric structure. The structure can be
rigid enough so that we can completely predict the shape of the space, in this
case one can say that a certain Law of Nature is observed, an example would be
the positive solution to Poincare conjecture in dimension three. The structure on
the contrary can be too flabby and bare no information about the space, one
would expect to encounter such phenomena especially for higher-dimensional
spaces. For example it is expected that any manifold a dimension five and higher
admits infinite amount of metrics satisfying Einstein equation.
An important goal in the area is to understand which geometric structures lead to
laws restricting the shape of the space and which do not.