Summer School on Surgery and the Classification of Manifolds

The summer school is aimed at graduate students and postdocs, although all are welcome. It is being sponsored by the Pacific Institute of Mathematical Sciences (PIMS) Collaborative Research Group (CRG) in Applied, Algebraic, and Geometric Topology and is supported by the National Science Foundation. Many of the summer school participants will be invited to the 2-day BIRS workshop on Surgery and Geometry which will be held immediately after the summer school.

Scientific Plan: The goal is for participants to learn about the classification of manifolds, up to homeomorphism and diffeomorphism within a fixed homotopy type, using the technique of surgery. This technique uses the close relations between algebraic and geometric topology, passing between homology and homotopy groups via the Hurewicz theorem and between handles in a manifold and cells in a CW complex through Morse theory. The classification techniques are most effective for manifolds of dimension four or greater, although there are be applications to and interactions with low-dimensional topology.

This mathematics interacts with algebraic topology, geometric topology, differential topology, geometric group theory, and geometric functional analysis. The capstone speaker will be Guoliang Yu, who will discuss connections of surgery theory with index theory, with the Dirac operator and the existence of metrics of positive scalar curvature, and with the Novikov and Baum-Connes conjectures.

Format: There will be four hours of lectures given by the organizers each day. In addition, there will be minilectures given by participants on topics suggested by the organizers in advance. There will also be exercise sessions and question sessions, promoting a good deal of interactions between the participants and the organizers.

Prerequisites: We hope to draw participants from multiple backgrounds, including geometric topology, algebraic topology, geometric group theory, geometry, and geometric functional analysis. But in order to effectively participate in the summer school, participants should come to the summer school with a working knowledge of smooth manifolds, vector bundles, transversality, the cohomology ring, the Hurewicz Theorem, and the statement of Poincaré duality. Familiarity with bordism theory and generalized (co)homology theory, classifying spaces for vector bundles, characteristic classes and Morse Theory would also be helpful.

Downloads:

Participants 

Lectures and Minilectures

Exercises 

Schedule