Differential Geometry Seminar



Autumn 2016

Crash course on Line bundles, Connections and Hamiltonian actions

Starting Monday October 31, Ieke Moerdijk and Kirill Mackenzie will offer a short course, partly based on Kostant's famous 1970 notes on geometric prequantisation, and involving notions such as line bundles, symplectic manifolds and Hamiltonian actions, all central in differential geometry and beyond. More specifically, we will address at least the following questions in detail:

  • When is a closed real-valued 2-form the curvature of a connection in a circle bundle (equivalently, a complex line bundle) ?
  • In what sense are coadjoint orbits the universal examples of Hamiltonian symplectic manifolds ?
The lectures are aimed at postgraduates having a nodding acquaintance (or better) with manifolds and Lie groups. All are welcome.

The course will start on Monday 31 October and will last about five weeks, with two lectures per week. The planned times are Mondays at 2pm and Thursdays at 10am.

The venue is Hicks J 11 unless otherwise stated.

Lecture I, Monday 31st (Ieke)

Line bundles and cocycles: we define complex line bundles over manifolds and show that these can be described in terms of 1-cocycles of nowhere zero complex functions.

Lecture II, Thursday 3nd (Ieke)

Cech cohomology: we introduce Cech cohomology with coefficients in a presheaf and prove some of its basic properties (long exact sequence, acyclicity of fine sheaves, double complex lemma). We deduce that line bundles are characterized by degree 2 cohomology classes with coefficients in (the constant presheaf given by) the integers.

Lecture III, Monday 7th (Ieke)

Connections: after completing the proof of the double complex lemma which we didn't quite finish in the previous lecture, we will discuss some differential geometry of line bundles: connections, connection 1-forms, and curvature.

Lecture IV, Thursday 10th (Ieke)

The integrality theorem: We will use (the proof of) the double complex to show that the de Rham cohomology class given by the curvature 2-form corresponds to the integral cocycle cohomology class of lecture 2. This will show that the de Rham cohomology class is integral.

Lecture V, Monday 14th (Ieke)

In the fourth lecture, we did not get quite as far as described above. In the fifth lecture, everything that we did previously will come together, to allow for a comparison between the de Rham 2-form given by the curvature, and the integral 2-form given by the cocycle description of a line bundle. The integrality theorem will just drop out. We will conclude with a few remarks about to what extent the de Rham class determines the line bundle.

Lecture VI, Thursday 17th (Kirill)

I will give a one-off lecture on a nonabelian extension of the principal question treated in Ieke's lectures (When is a closed real-valued 2-form the curvature of a connection in a complex line bundle?)

This will be designed to be accessible to people who have not come to Ieke's lectures, but who are familiar with the basics of manifolds and Lie groups.

Roughly speaking the question is: when is a 2-form on a manifold, which takes values in a `bundle of Lie algebras', the curvature of a connection in a principal bundle? The answer is recognizably similar to the answer in the case of complex line bundles, but the techniques are substantially different.

I will cover principal bundles and their connections at the start.

(If you happen to be familiar with the classification of extensions of discrete groups, or Lie algebras, by cohomology, the differences are roughly the same as the differences between when the kernel is abelian and when it is not.)

Lecture VII, Monday 21st (Kirill)

I'll conclude the account, begun on Thursday, of the non-abelian extension of the Weil Lemma. I will be able to go a bit slower than I did on Thursday.

Lecture VIII, Friday 25th at 11am in F 20 (Kirill)
Note unusual day time and location

I will begin the lectures on coadjoint orbits and Hamiltonian actions. I will start with concrete examples, at a leisurely pace. No knowledge of the preceding lectures is needed.

Lecture IX, Monday 28th at 4pm in LT 10 (Kirill)
Note unusual time and location

Lecture X, Thursday 1st (Kirill)

I will give an exposition of the two main results stated on Monday; that the coadjoint orbits of a connected Lie group are the symplectic leaves of the Lie algebra dual with its Poisson structure, and that the Marsden-Weinstein reduced spaces of a Hamiltonian action are the symplectic leaves of the quotient manifold (assumed to exist) with its Poisson structure.

The proofs won't be complete in every detail, but should give an idea of what is involved.

I intend to be faster than last Friday but slower than Monday.

Seminars to summer 2016