Admission requirements
Differentiable manifolds 1 (required)
Topology (required)
Linear Algebra 1,2 (required)
Analysis 2,3 (required)
Algebra 1 (required)
Algebra 2 (recommended)
Complex Analysis (recommended)
Description
In this course, we introduce abstract smooth manifolds. This notion generalises that of a smooth surface in three dimensional space, which serves as a motivating example. We discuss the main properties of and structures on manifolds, such as vector bundles, vector fields and derivations, flows, differential forms, orientation and integrals. We then prove Stokes’ theorem. We introduce Riemannian metrics and prove the divergence theorem. This paves the way for a discussion of DeRham cohomology, its fundamental properties and its relation to topology.
Course objectives
Obtain a thorough understanding of the geometry and topology of manifolds. Develop the ability to carry out explicit calculations of differential and integral calculus on manifolds in a concrete setting. Apply these techiques in examples from geometry and dynamical systems.
Mode of instruction
weekly lectures and question hour (optional)
Assessment Method
Written exam and homework. During the course, there will be six homework assignments to be handed in at predetermined dates. Out of these, the five best count towards the final grade. For the final grade, the homework counts for 25% and the written exam for 75%, unless the grade for the written exam is higher than the aforementioned weighted average. In the latter case, the final grade is equal to the the grade for the written exam.
Literature
John. M. Lee, “Introduction to smooth manifolds”
Website
Enrollment in Usis/Brightspace is compulsory
Contact
Bram Mesland: b[dot]mesland[at]math[dot]leidenuniv[dot]nl