- The student is familiar with the notion of smooth manifold and smooth map
- The student understands the notions of tangent space, vector fields, cotangent space and differential forms
- The student understands flows of vector fields
- The student can manage integration theory on smooth manifolds, and knows the general Stokes' Theorem
A smooth n-dimensional manifold M is a topological space which locally looks like Rn. For example, a sphere in R3 is a 2-dimensional manifold, since locally it can be smoothly parameterized using two coordinates (e.g. the latitude and the longitude). This course consists of the study of this fundamental concept of mathematics and the basic constructions associated with it. Multivariable calculus on Rn extends to any smooth n-dimensional manifold M, and it can be used to understand global properties of M, which leads to a subtle interplay between analysis and topology.
The course should be considered as a natural and necessary preparation for a large number of master courses: algebraic geometry, differential geometry, dynamical systems, global analysis, Lie groups, Riemann surfaces, symplectic geometry.
|This course will be taught in English.There will be 2 X 45 minutes of lecture per week, and 2 X 45 minutes of exercise class per week, organized as follows: on each of the two days of teaching, there will be 45 minutes of lecture followed by 45 minutes of exercise class.|
|Written exam and homework. Possibly also a midterm exam.|
|Analysis 1, 2 and Topology|
• L.W.Tu, An introduction to manifolds, Second edition. Universitext. Springer, New York, 2011.
• Other books that might be useful:
• J.M. Lee, Introduction to smooth manifolds. Second edition. Graduate Texts in Mathematics, 218. Springer, New York, 2013.
• Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, 94, Springer-Verlag, New York-Berlin, 1983.
• 32 hours lecture
• 32 hours problem session
• 104 hours individual study period
|L.W.Tu, An introduction to manifolds, Second edition. Universitext. Springer, New York, 2011|
|• J.M. Lee, Introduction to smooth manifolds. Second edition. Graduate Texts in Mathematics, 218. Springer, New York, 2013|
|Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, 94, Springer-Verlag, New York-Berlin, 1983|
|Gelegenheden||Blok KW2, Blok KW3|