Material covered until now: Chapter 1, Sections 1, 2 and 3 of Chapter 2 and the start of section 1 of Chapter 4.
Wednesday, December 21, 2011 (week 51)
This lecture we will learn Stokes theorem and take a new look at the de Rham cohomology (Chapter 4, Sections 3 and 4)
Wednesday, December 14, 2011 (week 50)
This lecture we will learn integration on manifolds (Chapter 4, Section 2).
Wednesday, December 7, 2011 (week 49)
This lecture we will finish whatever is left of from tensors on manifolds. If we have time, we will start dealing with orientation on manifolds and manifolds with boundary (section 1 from Chapter 4)
Wednesday, November 30, 2011 (week 48)
This lecture we will continue our study of tensors. We will look again at the wedge product of forms and interior product by vectors and the effects of linear transformations of V on forms. Then we will transfer all this knowledge to manifolds where we will also introduce the exterior derivative and Lie derivative. Hopefully we will define de Rham cohomology today.
Wednesday, November 23, 2011 (week 47)
This lecture we started our study of tensors. We defined the tensor product of two vector spaces and the tensor algebra of a vector space. We also introduced the exterior algebra of a vector space.
Wednesday, November 16, 2011 (week 46)
This lecture we proved Frobenius theorem.
Wednesday, November 9, 2011 (week 45)
Exam 1, from 13.30 to 16.30 in Educatorium Thetazaal.
Material present in the First exam: Chapter 1, sections 1, 2, 3, 4, 5, 6 and 7 (all of chapter 1 except Frobenius theorem).
Wednesday, November 2, 2011 (week 44)
This lecture we finished the inverse function theorem.
Wednesday, October 26, 2011 (week 43)
This lecture we will cover submanifolds and the inverse and implicit function theorems.
Wednesday, October 19, 2011 (week 42)
This week we learnt about flows of vector fields.
Wednesday, October 12, 2011 (week 41)
This week we recalled the definition of vector bundle and section. Saw that sections of the trivial bundle M x R correspond to functions on M , which we also called “zero forms” and denoted the space of zero forms by Omega^0(M). Then saw that sections of TM correspond to our intuitive notion of vector field and named the sections of T*M 1-forms, denoted by Omega^1(M). Then we went on to define an algebra over the reals, a real Lie algebra, Lie brackets and state that the set of vector fields with Lie bracket is a Lie algebra. We finished introducing the differential, an R-linear map d: Omega^0 ---> Omega^1.
Wednesday, October 5, 2011 (week 40)
This week Andre talked about the tangent and cotanget bundles, and proved in detail that the tangent space of M at p corresponds to the derivations of germs of functions at p. He introduced the notion of a vector bundle and section.
Wednesday, September 28, 2011 (week 39)
This week we will cover the definition of tangent vectors and 1-forms, as well as introduce the first instance of the exterior derivative. Homework: finish homework from previous lectures plus the following extra homework.
Wednesday, September 21, 2011 (week 38)
In this lecture we will cover functions between manifolds and the consequences of second countability. Homework: Chapter 1, exercises 3, 4, 5 and 6 and the following extra exercise:
Show that Diff(M), the set of diffeomorphisms of a manifold M,
Diff(M) = { f:M ---> M: f is a diffeomorphism}
is a group.
Wednesday, September 14, 2011 (week 37)
We covered notation that will be used consistently thoughout the course and then went on to define locally Euclidean spaces, C^k structures on such spaces and finally manifolds. For us a smooth manifold is a second countable, locally Euclidean space with a C^∞ structure. Variations on the definition lead to the concepts of topological, C^k, analytic and complex manifold. We finished the lecture with examples of manifolds: R^n, vector spaces, n x n matrices and the general linear group. Homework: Chapter 1, exercises 1 and 2.
