MAT 644 

Topics in
Differential Geometry 

Fall 2020 

TΘ 1:15-2:35 pm
via Zoom

Prof. Claude LeBrun.
Office hours, via Zoom:
TΘ 2:45-4:15 pm.

The Curvature of Four-Manifolds

This course will offer a one-semester introduction to the Riemannian geometry of smooth compact 4-manifolds. Our main objective will be to study the interplay between differential topology and Riemannian geometry in dimension four. For example, we will explore the question of when a given 4-manifold can be geometrized by endowing it with an Einstein metric, or with some more general "canonical" Riemannian metric. We will learn that there are differential-topological obstructions that actually imply the non-existence of such metrics on many 4-manifolds. On the other hand, we will also become acquainted with a complementary toolbox of techniques and results that can be used to prove the existence of such metrics on many other 4-manifolds.

The lectures will presuppose a basic, core knowledge of 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.

Grades will be based upon class participation.

You can e-mail Prof. LeBrun by clicking .
This is the best way to arrange to speak with the professor outside of regular office hours.

Illustration: Real slice of a smooth quartic K3. Graphic produced using the surf software package.

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