Course content

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.

Learning outcome

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.

Learning methods and activities

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

Recommended previous knowledge

The course is based on TMA4100/05/15/ Calculus 1/2/3 or equivalent. For students who have their background from mathematical sciences: The course is based on MA1101 Basic Calculus I, MA1102 Basic Calculus II, MA1103 Vector Calculus, MA1201 Linear Algebra and Geometry and MA1202 Linear Algebra with Applications or equivalent. The course can be taken together with TMA4190 Introduction to Topology (or as a follow-up to TMA4190).

Course materials

Will be announced at the start of the course.