Complex geometry in equilibrium

Research Project, 2012 – 2015

The main aim of this project is to introduce a new approach to certain geometric problems which play a central role in current mathematics. More precisely, the project will study canonical metrics on complex manifolds, notably Kähler-Einstein metrics. Such metrics appear as special solutions to Einstein´s highly non-linear field equations. The main research problems are (A) To show that a certain explicit and algebraically defined random point process recently introduced by the applicant, can be used to statistically produce Kähler-Einstein metrics (B) To apply new analytic tools (closely related to problem A) to Donaldson´s new program for constructing Kähler-Einstein metrics with conical singularities. The proposed approaches combine complex geometry and analysis with probability and statistical mechanics in a new way. The project is motivated by exciting recent developments in differential and algebraic geometry (such as Donaldson´s new program for the Tian-Yau-Zelditch conjecture, Perelman´s work on the Ricci flow, etc. Although the project is mainly motivated by research problems in current pure mathematics, it is connected to some previous approaches to study turbulence in fluids. It may, in the long term, result in new numerical methods to obtain and approximate solutions to other highly non-linear partial differential equations.

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