Research

I am a PhD student of Magdalena Kędziorek studying equivariant homotopy theory. I am mostly interested in multiplicative structures in global homotopy theory. Global homotopy theory studies objects which have compatible actions by all compact Lie groups. In my research I have studied operads in global homotopy theory, especially the global N∞-operads that represent intermediate levels of commutativity between ultracommutativity and naive equivariant commutativity (introduced in [2]).

Papers and preprints:

[3] Global transfer systems of abelian compact Lie groups [Accepted for publication in the Proceedings of the American Mathematical Society] https://doi.org/10.1090/proc/16310

Abstract: 

Global transfer systems are equivalent to global N∞-operads, which parametrize different levels of commutativity in globally equivariant homotopy theory, where objects have compatible actions by all compact Lie groups. In this paper we explicitly describe and completely classify global transfer systems for the family of all abelian compact Lie groups.

[2] Global N∞-operads (2023) Bulletin of the London Mathematical Society https://doi.org/10.1112/blms.12790

Abstract: 

We define N∞-operads in the globally equivariant setting and completely classify them. These global N∞-operads model intermediate levels of equivariant commutativity in the global world, i. e. in the setting where objects have compatible actions by all compact Lie groups. We classify global N∞-operads by giving an equivalence between the homotopy category of global N∞-operads and the partially ordered set of global transfer systems, which are much simpler, algebraic objects. We also explore the relation between global N∞-operads and N∞-operads for a single group, recently introduced by Blumberg and Hill. One interesting consequence of our results is the fact that not all equivariant N∞-operads can appear as restrictions of global N∞-operads.

[1] Operads in Unstable Global Homotopy Theory [Accepted for publication in Algebraic & Geometric Topology]

Abstract: 

In this paper we study operads in unstable global homotopy theory, which is the homotopy theory of spaces with compatible actions by all compact Lie groups. We show that the theory of these operads works remarkably well, as for example it is possible to give a model structure for the category of algebras over any such operad. We define global E∞-operads, a good generalization of E∞-operads to the global setting, and we give a rectification result for algebras over them.

Master Thesis:

I did my master thesis under the supervision of Markus Hausmann. Most of the contents of my master thesis appear in my first paper [1], which has additional results and examples.