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Real and complex Monge-Ampère equations, statistical mechanics and canonical metrics
Doctoral thesis, 2018

Statistical Mechanics

Point Processes

Hessian manifolds

Kähler geometry

Optimal Transport

Canonical metrics

Complex Monge-Ampère equations

Real Monge-Ampère equations

Kähler-Einstein metrics

## Author

### Jakob Hultgren

Chalmers, Mathematical Sciences, Algebra and geometry

This thesis contains four papers contributing to this field. The first paper explores a recently discovered connection to probability and statistical mechanics in which the objects of interest can be described as clouds of large numbers of interacting particles. The second paper proves that solutions to an equation related to the Einstein Field Equations minimise a certain energy type quantity, which can be formulated in terms of the theory of Optimal Transport. The motivation for this work is twofold. On the one hand, it paves the way for an interpretation in terms of statistical mechanics for these equations. On the other hand, it is a preparation to adress a question about certain types of degenerating geometric objects, asked independently by several mathematicians in the early 2000’s, motivated by the notion of mirror symmetry in string theory. In the third paper, the discovery of a new type of geometric objects is presented, which puts Einstein Field Equations into a more general setting. This paper also provides a connection to the algebraic point of view in geometry, similar to a link which has been studied by many mathematicians during the last 20 years that relates the Einstein Field Equations to the algebraic point of view in geometry. In the fourth paper we show that in many situations this connection can be expressed in surprisingly concrete terms using the geometry of polytopes.

### Subject Categories

Algebra and Logic

Geometry

Probability Theory and Statistics

Mathematical Analysis

### Roots

Basic sciences

### ISBN

978-91-7597-719-5

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 4400

### Publisher

Chalmers University of Technology

Lecture hall Euler, Mathematical Sciences, Skeppsgränd 3

Opponent: Associate Professor Song Sun, University of California, Berkeley, US