Ich bin ein Wissenschaftlicher Mitarbeiter („Postdoktorand“) am Karlsruher Institut für Technologie im Bereich der Algebraischen Topologie, in der Arbeitsgruppe von Manuel Krannich. Zuvor war ich Doktorand und Postdoktorand an der Universität Bonn, in der Arbeitsgruppe von Carl-Friedrich Bödigheimer.

### Forschungsinteressen

Meine Forschungsinteressen liegen im Bereich der Algebraischen Topologie: ich beschäftige mich mit (gefärbten) topologischen Operaden und ihren Algebren, Konfigurationsräumen und dem Phänomen der homologischen Stabilität sowie Diffeomorphismen- und Abbildungsklassengruppen, Modulräumen von Flächen und deren (instabiler) Homologie. In letzter Zeit untersuche ich außerdem verschiedene niedrigdimensionale äquivariante und parametrisierte Kobordismuskategorien und ihren Homotopietyp.

### Publikationen

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

16. 09. 2022, arXiv:2209.08148, mit Carl-Friedrich Bödigheimer und 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. 05. 2021, arXiv:2105.05772, mit Andrea Bianchi und Jens Reinhold
Forum of Mathematics, Sigma 10 (2022), DOI
Vortrag über Zoom an der Purdue University: Video (YouTube), Notizen (PDF 1.8 MiB)

We study the E2-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 Ed-algebras

06. 04. 2021, arXiv:2104.02729
Vortrag an der Universität Kopenhagen: Video (YouTube)

It is a classical result that configuration spaces of labelled particles in $$\mathbb{R}^d$$ are free Ed-algebras and that their d-fold bar construction is equivalent to the $$d$$-fold suspension of the labelling space. In this paper, we study a variation of these spaces, namely configuration spaces of labelled clusters of particles. These configuration spaces are again Ed-algebras, and we give geometric models for their iterated bar construction in two different ways: one establishes a description of these configuration spaces of clusters as cellular E1-algebras, and the other one uses an additional verticality constraint. In the last section, we apply these results in order to calculate the stable homology of certain vertical configuration spaces.

• #### Vertical configuration spaces and their homology

22. 03. 2021, arXiv:2103.12137, mit Andrea Bianchi
Quarterly Journal of Mathematics (2022), DOI
Vortrag über Zoom an der Universität Bukarest: Notizen (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.

### Abschlussarbeiten

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

Veröffentlicht auf dem Publikationsserver der Universität 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

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.

### Lehre

Im Sommersemester 2022 bin ich Assistent für die Vorlesungen Einführung in die Geometrie und Topologie, die von Koen van den Dungen gehalten wird, sowie für die Topology 2 von Daniel Kasprowski.

Im Sommersemester 2020 habe ich gemeinsam mit Andrea Bianchi ein Masterseminar über Operads in Algebra and Topology organisiert. Darüber hinaus war ich Tutor für folgende Topologievorlesungen:

 SoSe 2021 Algebraic Topology 2 Carl-Friedrich Bödigheimer WiSe 2020/21 Algebraic Topology 1 Carl-Friedrich Bödigheimer SoSe 2020 Algebraic Topology 2 Christoph Winges WiSe 2019/20 Algebraic Topology 1 Wolfgang Lück SoSe 2019 Topology 2 Daniel Kasprowski WiSe 2018/19 Topology 1 Wolfgang Lück SoSe 2018 Geometrie und Topologie Wolfgang Lück

### Ausgewählte Übersichtsvorträge

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

12. 05. 2022, Folien (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. 07. 2021, Notizen (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

05. 11. 2020, Notizen (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. 06. 2020, Notizen (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.

Kontakt

Karlsruher Institut für Technologie
Büro 1.019
Englerstraße 2
76131 Karlsruhe

Öffentlicher PGP-Schlüssel (0xD2890F65)

Kommende Veranstaltungen
• Weihnachtskonzerte des Chors des Collegium musicum Bonn
13. 12. 2022 + 14. 12. 2022