The lecture will focus mainly on generalisations of finite-dimensional manifolds and their treatment as objects of (pre)sheaves.
This will roughly divided into two major parts: The treatment of infinite-dimensional objects, such as manifolds of mappings,
and the treatment of categorified objects (objects with internal symmetries), such as Lie groupoids. In both cases the
perspective from sheaf theory provides important insights and results, for instance in understanding the "correct"
notion of morphism between Lie groupoids, which leads to the description of Lie groupoids as smooth stacks.
Both structures (infinite-dimensional and categorified ones) are important in various parts of modern mathematics, for instance
in algebraic geometry and in mathematical physics. I will point out these connections throughout the lecture and can provide
literature for further reading on demand.
Here is a very preliminary version of the lecture notes that I will provide. Beware: This is a non-proofread working copy of my own notes, so take them with a grain of salt. In particular, I am grateful for any kind of feedback on them.
The exercise sheets will be posted above roughly one week before they are discussed (there will be no exercise class in the
first week), and will be discussed in the class on Wednesday. The students are supposed to work on the exercises before
the class. This means that each student is expected to have worked on at least one exercise and should be able to
present a solution or to nail down the problems that she or he has with solving it.
Note that I will not provide solution sheets this semester.