Geometry working seminar

Ren Zhu & Ruxandra Moraru, Pure Mathematics Deptartment, University of Waterloo

Please note room.

Ren Zhu will speak on “An introduction to sheaves”

Frequently one tries to understand geometric objects such as topological spaces or manifolds by studying algebras of functions or vector bundles on it. However, it may be the case that we can only get information locally and not globally. For example, a function might only be holomorphic in an open set but cannot be lifted to a globally holomorphic function. Sheaves give us a way of encoding such local information on geometric objects, allowing for a systematic study of them. In this talk I will define sheaves as well as discussing some of their properties. I will also talk about ways of making new sheaves, including the sheafification of a presheaf.

Ruxandra Moraru will speak on “Metric connections II: the Levi-Civita connection (continued)”

This is the second in a series of lectures on metric connections. In the first talk, I introduced the notions of smooth vector bundle and metric connection, and presented some of their properties. In this talk, I will specialize to affine connections on the tangent bundle, in particular giving a geometric interpretation of the curvature and the torsion of an affine connection in terms of parallel transport. I will finally discuss the Levi-Civita connection on a Riemannian manifold, which is the unique torsion-free connection that is compatible with the Riemannian metric.