I am a third year PhD student at the Mathematical Institute of the university of Bonn under the supervision of Carl-Friedrich Bödigheimer, in the area of algebraic topology. I am funded by the Max Planck Institute for Mathematics in Bonn and am a member of the International Max Planck research school (IMPRS) on moduli spaces.
My PhD project deals with various coloured topological operads, which allow arguments of ‘higher multiplicity’. Particular examples are clustered versions of May’s little cubes or of Tillmann’s surface operad. On the one hand, their free algebras are certain clustered configuration spaces which have been studied recently by Tran and Palmer. On the other hand, they act on moduli spaces of Riemann surfaces with multiple boundary curves. This gives rise to a system of operations on the homology of moduli spaces, helping us to understand unstable classes, and also exhibits ‘parametrised’ variants of the classical moduli spaces as infinite loop spaces in the spirit of Madsen and Weiss.
I am working together with Andrea Bianchi from the university of Copenhagen (former PhD student in Bonn) and Jens Reinhold from the university of Münster.
In the upcoming summer term 2021, I will be tutor for the lecture Algebraic Topology 2 given by Carl-Friedrich Bödigheimer. In the summer term 2020, I co-organised a graduate seminar on Operads in Algebra and Topology together with Andrea Bianchi. Moreover, I have been tutor for the following topology lectures:
winter 2020/21 | Algebraic Topology 1 | Carl-Friedrich Bödigheimer |
summer 2020 | Algebraic Topology 2 | Christoph Winges |
winter 2019/20 | Algebraic Topology 1 | Wolfgang Lück |
summer 2019 | Topology 2 | Daniel Kasprowski |
winter 2018/19 | Topology 1 | Wolfgang Lück |
summer 2018 | Einführung in die Geometrie und Topologie | Wolfgang Lück |
We study the \(E_2\)-algebra \(\Lambda\mathfrak{M}_{*,1}=\coprod_{g\geqslant 0}\Lambda\mathfrak{M}_{g,1}\) consisting of free loop spaces of moduli spaces of Riemann surfaces with one parametrised boundary component, and compute the homotopy type of the group completion \(\Omega B\Lambda\mathfrak{M}_{*,1}\): it is the product of \(\Omega^\infty\mathbf{MTSO}(2)\) with a certain free \(\Omega^\infty\)-space depending on the family of all boundary-irreducible mapping classes in all mapping class groups \(\Gamma_{g,n}\) with \(g\geqslant 0\) and \(n\geqslant 1\).
It is a classical result that configuration spaces of labelled particles in \(\mathbb{R}^d\) are free algebras over the little \(d\)-cubes operad \(\mathscr{C}_d\), and their \(d\)-fold bar construction is equivalent to the \(d\)-fold suspension of the labelling space. The aim of this paper is to study a variation of these spaces, namely the configuration space of labelled clusters of points in \(\mathbb{R}^d\). This configuration space is again an \(E_d\)-algebra, but in general not a free one. We give geometric models for their iterated bar construction in two different ways: one uses an additional verticality constraint, and the other one uses a description of these clustered configuration spaces as cellular \(E_1\)-algebras. In the last section, we show a stable splitting result and present some applications.
We introduce ordered and unordered configuration spaces of ‘clusters’ of points in an Euclidean space \(\mathbb{R}^d\), where points in each cluster have to satisfy a ‘verticality’ condition, depending on a decomposition \(d=p+q\). We compute the homology in the ordered case and prove homological stability in the unordered case.
This is my master thesis which was finished in the summer term 2018. Its main result is a Poincaré–Lefschetz correspondene between the Mumford–Miller–Morita classes in the cohomology of moduli spaces and certain simplicial subcomplexes of Bödigheimer’s model of parallel slit domains.
These are the notes of a talk I gave in the winter term 2020/21 in our IMPRS seminar at the MPI. They tell a classical story about H-spaces, loop spaces, and May’s recognition principle.
In summer 2020, the first ‘corona term’, I gave an introductory talk on model categories in our IMPRS seminar. It was the first in a series of four talks and essentially covers the first chapter of Hovey’s book on the topic.