Ich bin ein Wissenschaftlicher Mitarbeiter („Postdoktorand“) am Mathematischen Institut der Universität Bonn im Bereich der Algebraischen Topologie, in der Arbeitsgruppe von Carl-Friedrich Bödigheimer. Ich werde finanziert vom Hausdorff Center für Mathematics (HCM).

Ab Oktober 2022 werde ich Wissenschaftlicher Mitarbeiter in der Arbeitsgruppe von Manuel Krannich am Karlsruher Institut für Technologie (KIT) sein.

### 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.

Meine Koautoren sind Andrea Bianchi von der Universität Kopenhagen und Jens Reinhold von der Universität Münster.

### Publikationen

• #### 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 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 Ed-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 E1-algebras. In the last section, we show a stable splitting result and present some applications.

• #### 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

Mathematisches Institut
Büro 4.020
Endenicher Allee 60
53115 Bonn

Öffentlicher PGP-Schlüssel (0xD2890F65)

Kommende Veranstaltungen
• Young Topologists Meeting 2022
Universität Kopenhagen
18. 07. 2022 – 22. 07. 2022
• Seminar „Topologie“
Oberwolfach
24. 07. 2022 – 30. 07. 2022