Geometry advanced class: Einstein manifolds

Overview

In Riemannian geometry, Einstein manifolds of dimension 4 and higher are of fundamental importance and key interest, since they satisfy the natural and deceptively simple-looking constraint that their Ricci curvature is constant. However, outside the realm of Kähler geometry or special holonomy, many of the main questions remain unanswered. In this advanced class, we will survey some of the exciting recent progress in this field, including the construction of new examples with both positive and negative scalar curvature, and attempts to study Einstein manifolds via Ricci flow.

Schedule

The class will be in Classrom C5 in the Mathematical Institute at 9:30-11:00 on Thursdays in each week of Trinity Term 2025.
The plan is to have a one hour talk with a further 30 minutes allowed for discussion.
The organizers are Andrew Dancer and Jason Lotay.

The planned timetable is as follows.

Week 1 1 May 2025	Introduction and overview Speaker: Jason Lotay Abstract: This talk will introduce Einstein manifolds, describe the fundamental open questions and give an overview of the advanced class. [Jason's handwritten notes]
Week 2 8 May 2025	Homogeneous Einstein manifolds Speaker: Andrew Dancer Abstract: This talk will survey the state-of-the-art in the study of homogeneous Einstein metrics, focussing on existence questions in the compact setting. [Andrew's handwritten notes]
Week 3 15 May 2025	Cohomogeneity one Einstein manifolds Speaker: Andrew Dancer Abstract: The cohomoeneity one technique has proven to effective in various geometric problems, including the study of Einstein metrics. This talk will explain the method then describe some of its various applications and success, including in 4 dimensions and in the construction of exotic Einstein metrics on spheres. [Andrew's handwritten notes]
Week 4 22 May 2025	New cohomogeneity one Einstein manifolds and numerical methods Speaker: Qiu Shi Wang This talk will discuss the recent work of Buttsworth-Hodgkinson concerning Einstein metrics on the 12-sphere which combines cohomogeneity one techniques with numerical approximation. Potentially, follow-up work by the speaker will also be described. [Qiu Shi's handwritten notes]
Week 5 29 May 2025	Ricci flow in dimension 4 and orbifold singularities (Part 1) Speaker: John Hughes These talks will give an overview of work by Deruelle-Ozuch relating Ricci flow in dimension 4 to orbifold singularities and Ricci-flat ALE metrics. [John's handwritten notes]
Week 6 5 June 2025	Ricci flow in dimension 4 and orbifold singularities (Part 2) Speaker: John Hughes These talks will give an overview of work by Deruelle-Ozuch relating Ricci flow in dimension 4 to orbifold singularities and Ricci-flat ALE metrics.
Week 7 12 June 2025	Construction of non-locally symmetric Einstein manifolds with negative curvature (Part 1) Speaker: Thibault Langlais These talks will describe the work of Hamenstädt-Jäckel constructing infinitely many new Einstein n-manifolds for each n at least 4 which have negative scalar curvature. [Thibault's notes]
Week 8 19 June 2025	Construction of non-locally symmetric Einstein manifolds with negative curvature (Part 2) Speaker: Thibault Langlais These talks will describe the work of Hamenstädt-Jäckel constructing infinitely many new Einstein n-manifolds for each n at least 4 which have negative scalar curvature. [Thibault's notes]