I am a postdoctoral researcher at the Mathematical Institute of the University of Bonn in the working group of Carl-Friedrich Bödigheimer, in the area of algebraic topology. I am funded by the Hausdorff Center for Mathematics (HCM).

• My ORCID record,
• Professional homepage at the University of Bonn,
• Curriculum Vitæ,
• Bonn Topology Group,
• Max Planck Institute for Mathematics.

Starting with October 2022, I will be a postdoctoral researcher in the working group of Manuel Krannich at the Karlsruhe Institute of Technology (KIT).

Research interests

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.

Publications

• Computations in the unstable homology of moduli spaces of Riemann surfaces

  16 Sep 2022, arXiv:2209.08148, with Carl-Friedrich Bödigheimer and Felix Boes

  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

• Parametrised moduli spaces of surfaces as infinite loop spaces

  12 May 2021, arXiv:2105.05772, with Andrea Bianchi and Jens Reinhold
  Forum of Mathematics, Sigma 10 (2022), DOI
  Talk via Zoom at Purdue University: Video (YouTube), Notes (PDF 1.8 MiB)

  We study the \(E_2\)-algebra \(\Lambda\mathfrak{M}\) 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 \(\Omega B\Lambda\mathfrak{M}\): 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 \(n\geqslant 1\).

• Configuration spaces of clusters as E_d-algebras

  6 Apr 2021, arXiv:2104.02729
  Talk at the University of Copenhagen: Video (YouTube)

  It is a classical result that configuration spaces of labelled particles in \(\mathbb{R}^d\) are free algebras over the little \(d\)-cubes operad, 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.

• Vertical configuration spaces and their homology

  22 Mar 2021, arXiv:2103.12137, with Andrea Bianchi
  Quarterly Journal of Mathematics (2022), DOI
  Talk via Zoom at the University of Bucharest: Notes (PDF 1.7 MiB)

  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.

Theses

• Coloured topological operads and moduli spaces of surfaces with multiple boundary curves

  PhD thesis, 7 Jul 2022, Download (PDF 1.9 MiB)
  Published in the Institutional Repository of the University of Bonn, HDL

  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.

• Moduli spaces of Riemann surfaces and symmetric products: A combinatorial description of the Mumford–Miller–Morita classes

  Master’s thesis, 25 Sep 2018, Download (PDF 911 kiB)

  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.

Teaching

In the summer term 2022, I am the teaching assistant for the lectures Einführung in die Geometrie und Topologie by Koen van den Dungen and Topology 2 by Daniel Kasprowski.

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:

summer 2021	Algebraic Topology 2	Carl-Friedrich Bödigheimer
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	Geometrie und Topologie	Wolfgang Lück

Selected survey talks

• Coloured topological operads and moduli spaces of surfaces with multiple boundary curves

  12 May 2022, Slides (PDF 200 kiB)

  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.

• The homotopy type of the cobordism category

  19 Jul 2021, Notes (PDF 1.2 MiB)

  These are the notes of a talk I gave in the summer term 2021 in our IMPRS seminar at the MPI. It summarises the celebrated results of Galatius, Madsen, Tillmann, and Weiss on cobordism categories.

• Homotopy coherent multiplications and loop spaces

  5 Nov 2020, Notes (PDF 1.1 MiB)

  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.

• An introduction to model categories

  10 Jun 2020, Notes (PDF 1.3 MiB)

  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.