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Dmitri Panov

Dr Dmitri Panov

Research Fellow


King's College London

Research summary

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.

Grants awarded

Geometric structures on manifolds

Scheme: University Research Fellowship

Dates: Oct 2015 - Sep 2018

Value: £272,364.05

Geometric structures on complex and symplectic manifolds

Scheme: University Research Fellowship

Dates: Oct 2010 - Sep 2015

Value: £492,397.76