Florian Kranhold
Address 
Karlsruhe Institute of Technology Englerstraße 2 76131 Karlsruhe 

Room  Kollegiengebäude, 1.019 
PGP Public Key (0xD2890F65) 
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 Mathematical Institute of the University of Bonn, in the working group of CarlFriedrich Bödigheimer.
My research interests lie in the area of algebraic topology: more specifically, I am interested in (coloured) topological operads andtheir 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 lowdimensional equivariant and parametrised cobordism categories and their homotopy type.
Here is my CV and my ORCID record, and you can watch a Zoom talk I gave in the Purdue Topology Seminar in September 2021, as well as a Talk at a blackboard I gave at a Workshop in Copenhagen in November 2021.
Research

Computations in the unstable homology of moduli spaces of Riemann surfaces
16 Sep 2022, arXiv:2209.08148, with CarlFriedrich 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, mod2, and rational coefficients. Furthermore, we determine, in most cases, explicit generators using homology operations

Parametrised moduli spaces of surfaces as infinite loop spaces
Forum of Mathematics, Sigma 10 (2022), DOI
12 May 2021, arXiv:2105.05772, Recorded talk, with Andrea Bianchi and Jens Reinhold
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 boundaryirreducible 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, Recorded talk
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. 
Vertical configuration spaces and their homology
Quarterly Journal of Mathematics (2022), DOI
22 Mar 2021, arXiv:2103.12137, with Andrea Bianchi
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
7 Jul 2022, PDF (1.9 MiB), HDL
This is my PhD thesis which was finished in January 2022 under the supervision of CarlFriedrich Bödigheimer.

Moduli spaces of Riemann surfaces and symmetric products: A combinatorial description of the Mumford–Miller–Morita classes
25 Sep 2018, PDF (911 kiB)
This is my master thesis which was finished in the summer term 2018 under the supervision of CarlFriedrich Bödigheimer. Its main result is a Poincaré–Lefschetz correspondence between the Mumford–Miller–Morita classes in the cohomology of moduli spaces and certain subcomplexes of Bödigheimer’s simplicial model of parallel slit domains.
Teaching
In the summer term 2022, I was a 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 coorganised a graduate seminar on Operads in Algebra and Topology together with Andrea Bianchi. Moreover, I have been tutor for the following topology lectures:
SoSe 2021  Algebraic Topology 2  CarlFriedrich Bödigheimer 
WiSe 2020/21  Algebraic Topology 1  CarlFriedrich 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  Einführung in die Geometrie und Topologie  Wolfgang Lück 
News
Dr. Regula Krapf receives university teaching award
Prof. Catharina Stroppel joined the North RhineWestphalia Academy for Sciences and Arts
Prof. Daniel Huybrechts receives the Compositio Prize for the periode 20172019
Prof. Catharina Stroppel receives Gottfried Wilhelm Leibniz Prize 2023
Rajula Srivastava receives Association for Women in Mathematics Dissertation Prize
Prof. Ana Caraiani wins a New Horizons in Mathematics Prizes 2023
Pius XI Medal awarded to Professor Peter Scholze
Prof. Valentin Blomer und Prof. Georg Oberdieck erhalten ERC grants
Grants for Mathematics students from Ukraine
Prof. Jessica Fintzen is awarded a Whitehead Prize of the London Mathematical Society
Prof. Peter Scholze elected as Foreign Member of the Royal Society
Bonner Mathematik belegt bei Shanghai Ranking den 1. Platz in Deutschland und weltweit den 13. Platz
Prof. Daniel Huybrechts erhält gemeinsam mit Debarre, Macri und Voisin ERC Synergy Grant