MAT 682 
Advanced Topics in
Differential Geometry 
Fall 2023 

TΘ 11:30-12:50
Physics P-129

Prof. Claude LeBrun.
Office: Math Tower 3-108.
Office hours: TΘ 2:00--3:30 pm.

Einstein Manifolds

An Einstein metric is, by definition, a Riemannian metric of constant Ricci curvature. This course will attempt to survey the current state of our knowledge concerning the existence and uniqueness of Einstein metrics on smooth, compact manifolds. The first part of the course will begin by quickly reviewing the situation in dimensions 2 and 3, and will then go on to provide a detailed introduction to the world of Einstein manifolds in dimensions ≥ 5. The remainder of the course will then offer a comprehensive introduction to the 4-dimensional case, where the relationship between curvature and differential topology depends on features that that are quite unlike the phenomena seen in any other dimension.

The lectures will presuppose a basic knowledge of Riemannian geometry, roughly at the level of MAT 568. A familiarity with Kähler metrics, say at the level of MAT 545, might also be helpful.

Grades will be based upon class participation.

The Professor can be reached via e-mail by . This is probably the best method for scheduling appointments outside of regular office hours.

Illustration: Smoothing of a Kummer surface with 16 real nodes. Graphic produced using the surf software package.

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