Real and complex Monge-Ampère equations, statistical mechanics and canonical metrics
Doctoral thesis, 2018
Recent decades has seen a strong trend in complex geometry to study canonical metrics and the way they relate to geometric analysis, algebraic geometry and probability theory. This thesis consists of four papers each contributing to this field. The first paper sets up a probabilistic framework for real Monge-Ampère equations on tori. We show that solutions to a large class of real Monge-Ampère equations arise as the many particle limit of certain permanental point processes. The framework can be seen as a real, compact analog of the probabilistic framework for Kähler-Einstein metrics on Kähler manifolds. The second paper introduces a variational approach in terms of optimal transport to real Monge-Ampère equations on compact Hessian manifolds. This is applied to prove existence and uniqueness results for various types of canonical Hessian metrics. The results can, on one hand, be seen as a first step towards a probabilistic approach to canonical metrics on Hessian manifolds and, on the other hand, as a remark on the Gross-Wilson and Kontsevich-Soibelmann conjectures in Mirror symmetry. The third paper introduces a new type of canonical metrics on Kähler manifolds, called coupled Kähler-Einstein metrics, that generalises Kähler-Einstein metrics. Existence and uniqueness theorems are given as well as a proof of one direction of a generalised Yau-Tian-Donaldson conjecture, establishing a connection between this new notion of canonical metrics and stability in algebraic geometry. The fourth paper gives a necessary and sufficient condition for existence of coupled Kähler-Einstein metrics on toric manifolds in terms of a collection of associated polytopes, proving this generalised Yau-Tian-Donaldson conjecture in the toric setting.
Statistical Mechanics
Point Processes
Hessian manifolds
Kähler geometry
Optimal Transport
Canonical metrics
Complex Monge-Ampère equations
Real Monge-Ampère equations
Kähler-Einstein metrics
Lecture hall Euler, Mathematical Sciences, Skeppsgränd 3
Opponent: Associate Professor Song Sun, University of California, Berkeley, US
Author
Jakob Hultgren
Chalmers, Mathematical Sciences, Algebra and geometry
Complex geometry is a central research field in current mathematics that also forms part of the backbone of many subjects in theoretical physics, for example quantum mechanics and string theory. During the last 50 years, use of the Einstein Field Equations in complex geometry has become more and more popular. The main reason for this is that in complex geometry the Einstein Field Equations constitute an intersection point between several different areas in mathematics, Geometry, Algebra and Probability, providing surprising links between concepts that were not previously known to be related.
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.
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
Lecture hall Euler, Mathematical Sciences, Skeppsgränd 3
Opponent: Associate Professor Song Sun, University of California, Berkeley, US