MAT 644 

Topics in
Differential Geometry 

Fall 2024 

TΘ 2:00-3:20 pm,
Math Tower 4-130

Prof. Claude LeBrun.

Office hours:
in Math Tower 3-108, TΘ 12:30-1:45 pm;
or, by appointment, via Zoom.

Gravitational Instantons

A gravitational instanton is, by definition, a complete, non-flat, non-compact, Ricci-flat Riemannian 4-manifold. As the terminology might seem to indicate, the study of these objects originated in physics, where they first arose in connection with theories of quantum gravity. However, they later turned out to also play a fundamental role in pure mathematics; in particular, they play an essential role in understanding the moduli theory of compact 4-dimensional Einstein manifolds, where they arise as models of curvature concentration, and thereby help govern the formation of singularities.

We will begin by exploring hyper-Kähler gravitational instantons, constructing many explicit examples and carefully discussing their twistor theory. We will also explore recent classification results governing the hyper-Kähler case. The last part of the course will then focus on the general-holonomy case, where much less is known, but where interesting theorems can nonetheless be proved when reasonable additional hypotheses are imposed.

The lectures will presuppose a basic knowledge of core topics in Riemannian geometry, roughly at the level of MAT 568. Some familiarity with Kähler metrics, say at the level of MAT 545, would also be helpful. While enrollment in the course is limited to Mathematics PhD students who have already passed the department's comprehensive exam, well-behaved auditors will of course also be welcome to attend the lectures. Grades will be based on class participation.

You can e-mail Prof. LeBrun by clicking .
This is the best way to make an appointment to speak with the professor via Zoom.

Illustration: 2-dimensional slice of the Riemannian Schwarzschild gravitational instanton.

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