I am a postdoctoral researcher at the Karlsruhe Institute of Technology in the working group of Manuel Krannich, specialising in the area of algebraic topology. Before that, I was a doctoral and a postdoctoral researcher at the University of Bonn, in the working group of Carl-Friedrich Bödigheimer.
My research interests lie in the area of algebraic topology: more specifically, I am interested in (coloured) topological operads and their algebras, configuration spaces and their homological stability, as well as diffeomorphism groups and mapping class groups, moduli spaces of surfaces and their (unstable) homology. More recently, I have been studying low-dimensional equivariant and parametrised cobordism categories and their homotopy type.
In this article we give a survey of homology computations for moduli spaces \(\smash{\mathfrak{M}_{\smash{g,1}}^m}\) of Riemann surfaces with genus \(g\geqslant 0\), one boundary curve, and \(m\geqslant 0\) punctures. While rationally and stably this question has a satisfying answer by the Madsen–Weiss theorem, the unstable homology remains notoriously complicated. We discuss calculations with integral, mod-2, and rational coefficients. Furthermore, we determine, in most cases, explicit generators using homology operations
We study the
It is a classical result that configuration spaces of labelled particles
in \(\mathbb{R}^d\) are free E_{d}-algebras
and that their
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.
While it is a classical result that the collection of moduli spaces of surfaces with a single boundary curve is an \(E_2\)-algebra (more precisely: it admits an action of the little 2-cubes operad \(\mathscr{C}_2\)), we need a coloured version of \(\mathscr{C}_2\) which understands a cluster of squares as a single input with a certain multiplicity, if we want to establish an action on the collection of moduli spaces of surfaces with multiple boundary curves in a similar way. Moreover, Bödigheimer introduced a finite multisimplicial model for moduli spaces, which is useful for explicit homological calculations. In order to construct an operadic action on this specific model, we have to additionally require a certain coupling behaviour among squares belonging to the same input. This gives rise to a family of suboperads, called vertical operads. We analyse these operads from several perspectives: on the one hand, their operation spaces and free algebras are modelled by clustered and vertical configuration spaces, whose homology, homological stability, and iterated bar constructions we investigate in the first chapters. On the other hand, we study the homotopy theory and the homology of their algebras and use the arising operations to describe the unstable homology of moduli spaces. Finally, it turns out that the developed methods are also useful to solve a problem of a seemingly different flavour: for a fixed space \(A\), the collection of moduli spaces of surfaces parametrised over \(A\) is itself an \(E_2\)-algebra, and its group completion is an infinite loop space. We identify the underlying spectrum in the spirit of Madsen and Weiss.
This is my master’s 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.
This is the first half of my public talk of my PhD defence (‘Kolloquium’). It illustrates how stability phenomena and operadic techniques can help us to understand the geometry of configuration spaces and moduli spaces, and ends with the central questions my PhD thesis starts with.
These are the notes of a talk I gave in the summer term 2021 in our
These are the notes of a talk I gave in the winter term 2020/21 in our
In summer 2020, the first ‘corona term’, I gave an introductory talk on
model categories in our
Karlsruhe Institute of Technology
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