Objective In broad terms the aim of this proposal is to introduce a new probabilistic approach to the study of Kähler-Einstein (K-E) metrics on complex manifolds. A precise procedure, based on a blend of Statistical Mechanics, Pluripotential theory and Kähler Geometry. will be used to show that• when a K-E metric exists on a complex manifold X it can be obtained from the “largeN limit” of certain canonical random point processes on X with N particles.The canonical point processes are directly defined in terms of algebro-geometric data and the thrust of this approach is thus that it gives a new link between algebraic geometry on one and hand and complex differential (Kähler) geometry on the other. A major motivation for this project comes from the fundamental Yau-Tian-Donaldson conjecture in Kähler geometry, which aims at characterizing the obstructions to the existence of a K-E metric on a Fano manifold in terms of a suitable notion of algebro-geometric “stability”, notably K-Stability. In this project a new “probabilistic/statistical mechanical” version of stability will be introduced referred to as Gibbs stability, which also has an interesting purely algebro-geometric definition in the spirit of the Minimal Model Program in current algebraic geometry and another specific aim of this project is to prove or at least make substantial progress towards proving,• There is a (unique) K-E metric on a Fano manifold X precisely when X is asymptotically Gibbs stableThe canonical random point processes will be defined as certain “beta-deformations” of determinantal point processes and share certain properties with the ones appearing in Random Matrix Theory and in the study of quantum chaos and zeroes of random polynomials (and random holomorphic sections) But a crucial new feature here is that the processes are independent of any back-ground data, such as a potential or a metric. Fields of science natural sciencesphysical sciencesclassical mechanicsstatistical mechanicsnatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsalgebraalgebraic geometry Programme(s) FP7-IDEAS-ERC - Specific programme: "Ideas" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013) Topic(s) ERC-SG-PE1 - ERC Starting Grant - Mathematical foundations Call for proposal ERC-2012-StG_20111012 See other projects for this call Funding Scheme ERC-SG - ERC Starting Grant Coordinator CHALMERS TEKNISKA HOGSKOLA AB Address - 412 96 Goteborg Sweden See on map Region Södra Sverige Västsverige Västra Götalands län Activity type Higher or Secondary Education Establishments Principal investigator Robert Berman (Dr.) Administrative Contact Helena Rafstedt (Ms.) Links Contact the organisation Opens in new window Website Opens in new window EU contribution No data Beneficiaries (1) Sort alphabetically Sort by EU Contribution Expand all Collapse all CHALMERS TEKNISKA HOGSKOLA AB Sweden EU contribution € 1 200 000,00 Address - 412 96 Goteborg See on map Region Södra Sverige Västsverige Västra Götalands län Activity type Higher or Secondary Education Establishments Principal investigator Robert Berman (Dr.) Administrative Contact Helena Rafstedt (Ms.) Links Contact the organisation Opens in new window Website Opens in new window Other funding No data
