Differential Topology

The aim of the course is to introduce fundamental concepts and examples in differential topology. Key concepts that will be discussed include differentiable structures and smooth manifolds, tangent bundles, embeddings, submersions and regular/critical points. Important examples of spaces are surfaces, spheres, and projective spaces. Other key concepts are homotopy, transversality, intersection numbers and cobordism. Applications presented in the course may range from Brouwer's fixed point theorem to vector fields on spheres. These methods and ideas have been influential to and are used in many other parts of mathematics, but also in physics and other areas of application.

1. Knowledge. The student has knowledge of fundamental concepts and methods in differential and algebraic topology, and of examples of manifolds.

2. Skills. The student is able to apply his or her knowledge of differential and algebraic topology to formulate and solve problems of a geometrical nature in mathematics.

The learning methods and activities depend on the course teacher, but will in general consist of lectures and exercises.