Monday 1st of May 2023, 15:15-16:15 in HG03.085
Philip Schlösser
Complex ODEs or: How I Learned to Stop Worrying and Love the Multifunction.
Tired of having to argue if \(\sqrt{xy} = \sqrt{x} \sqrt{y}\)? Annoyed by branch cuts?
Did you ever stop to wonder if there might be more to life complex analysis than simply connected domains?
What if I told you that not only have your worries been heard, but that they can also be tied to (linear)
complex ODEs, (vector bundle) connections, and locally constant sheaves?
Join me on my quest to explore different facets of monodromy and even get insight into when and how the
Frobenius-Ansatz works.
More seriously: When I first learned about ODEs (on the real line), we showed the existence and uniqueness
of local solutions, and then showed that every local solution can be extended to a maximal ('global')
solution, so that all information about the solutions is contained in the maximal solutions. On the
complex plane, a similar local existence and uniqueness result holds as well. However, when attempting
to get a global solution by gluing local solutions, we quickly run into topological obstructions in the
form of monodromy. It is therefore useful to look for solutions that are multifunctions. We will explore
how (holomorphic) multifunctions of finite determination are related to (linear) complex ODEs, and under
which conditions we can locally write such a multifunction in terms of generalised power series and
logarithms (Frobenius-Ansatz).