# Gustav Nilsson: Ricci-Flat 4-Manifolds with Toric Symmetry

**Time: **
Tue 2020-10-13 13.00 - 14.00

**Location: **
Zoom Meeting ID: 666 9376 0005

**Participating: **
Gustav Nilsson

**Supervisor: **
Mattias Dahl

### Abstract

In this thesis, we study non-compact, complete Ricci-flat 4-manifolds with toric symmetry and asymptotically locally Euclidean (ALE) or asymptotically locally flat (ALF) geometry. The toric symmetry is studied in terms of the rod structure formalism, and topological invariants of the manifold are expressed in terms of the rod structure. Versions of the Hitchin–Thorpe inequality for non-compact manifolds are then applied, yielding conditions on the rod structure as a consequence of Ricci-flatness. The conditions are considered in the case of a rod structure with three turning points, and are seen to substantially narrow down the set of possible such rod structures.