About Me

I'm a second year master student at Uni Bonn. My interest lie in higher category theory and algebraic topology. Currently, I am writing my Master Thesis supervised by Markus Hausmann.

My Master Thesis:

On equivariant formal group laws and complex bordism over profinite abelian groups

To be submitted in August '25.

Classical Story

To every complex oriented cohomology theory one assigns a formal group law. Formal group laws are a great invariant for oriented cohomology theories: In fact, Landweber's theorem tells us that in particular cases its a complete invariant. Moreover, the category of spectra itself can be stratified using a notion of height of a formal group law. The connection between topology (spectra) and formal group laws on the algebraic side is extremely firm: While, the complex bordism spectrum $MU$ is the initial example of a complex oriented cohomology theory, it's a theorem of Quillen that the formal group law associated to $MU$ is the initial example of a formal group law.

Thesis Abstract

In my master thesis we fix a compact, first countable and abelian group $A$. I will show that the equivariant (homotopical) complex bordism spectrum $MU_A$ is the initial example of a complex oriented cohomology theory. Moreover, I use the presentation of $MU$ as a global spectrum, in the sense of Schwede, to generalize Quillen's theorem to the $A$-equivariant setting. These results are known for compact abelian Lie groups due to [Cole, Kriz, Greenlees, 2000] and [Hausmann, 2022]. The krux of my thesis is to generalize these results from compact abelian Lie groups to countable inverse limits of compact abelian Lie groups. I will explore examples of $A$-equivariant complex oriented spectra like $A$-equivariant topological $K$-theory $KU_A$ and Borel theories, as well as there associated $A$-equivariant formal group law.

Our work will allow for the following insight: It was shown by [Hausmann, Meier, 2023] that for a compact abelian Lie group the universal support theory on finite $A$-spectra is given by $MU_A$-homology. We generalize this to inverse limits of compact abelian Lie groups. The Balmer Spectrum of $A$-spectra, for profinite groups $A$, has been computed by [Balchin, Barnes, Barthel, 2024]. We get an explicit isomorphism to the prime spectrum of the moduli-stack of $A$-equivariant formal groups.