Since May 2024 I am a PhD student in algebraic topology at the Münster Topology Group under supervision of Thomas Nikolaus. My main interests lie in (stable) homotopy theory, algebraic K-theory and higher category theory / higher algebra.
Previously, I completed my master’s at the University of Bonn under supervision of Stefan Schwede. Even earlier, I obtained Bachelor degrees in Mathematics and Computer Science at RWTH-Aachen University under supervision of Erich Grädel.
Publications & Preprints
Norms in Equivariant Homotopy Theory, joint with Tobias Lenz and Sil Linskens, preprint available on arXiv
We show that the \(\infty\)-category of normed algebras in genuine \(G\)-spectra, as introduced by Bachmann–Hoyois, is modelled by strictly commutative algebras in \(G\)-symmetric spectra for any finite group \(G\). We moreover provide an analogous description of Schwede’s ultra-commutative global ring spectra in higher categorical terms.Using these new descriptions, we exhibit the \(\infty\)-category of ultra-commutative global ring spectra as a partially lax limit of the \(\infty\)-categories of genuine \(G\)-spectra for varying \(G\), in analogy with the non-multiplicative comparison of Nardin, Pol, and the second author.
Along the way, we establish various new results in parametrized higher algebra, which we hope to be of independent interest.
Global Picard Spectra and Borel Parametrized Algebra, preprint available on arXiv or here as pdf
We answer a question of Schwede on the existence of global Picard spectra associated to his ultra-commutative global ring spectra; given an ultra-commutative global ring spectrum \(R\), we show there exists a global spectrum \(\textup{pic}_\textup{eq}(R)\) assembling the Picard spectra of all underlying \(G\)-equivariant spectra \(\textup{res}_G R\) of \(R\) into one object, in that for all finite groups \(G\), the genuine fixed points are given by $$ \textup{pic}_\textup{eq}(R)^G \simeq \textup{pic}(\textup{Mod}_{\textup{res}_G R}(\textup{Sp}_G)). $$ Along the way, we develop a generalization of Borel-equivariant objects in the setting of parametrized higher algebra. We use this to assemble the symmetric monoidal categories of \(G\)-spectra for all finite groups \(G\) together with all restrictions and norms into a single 'normed global category’, and build a comparison functor which allows us to import ultra-commutative \(G\)-equivariant or global ring spectra into the setting of parametrized higher algebra. This is project is a revised version of my master’s thesis.Logics of Dependence and Independence: The Local Variants, joint with Erich Grädel, Journal of Logic and Computation (2021), arXiv
We extend the idea of localizing logics of dependence and independence in a systematic way, taking into account local variants of standard atomic dependency properties: besides dependence and independence, also inclusion, exclusion, and anonymity. We study decidability issues of the local logics, establish characterisation theorems via appropriate notions of bisimulation and present the complexity of model checking problems for these logics. This is a revised version of my bachelor’s thesis.
Other writing
Lecture Notes: Sheaves on Manifolds, joint with Achim Krause and Thomas Nikolaus. Last Update: 07.11.2024 pdf
I helped write these lecture notes for the course “Sheaves on Manifolds” held at the University of Münster in WS23/24 and SS24.
Despite the title, the lecture was mostly concerned with the recent theory of compactly assembled categories and continuous K-Theory,
as initiated by Sasha Efimov.
Parametrized Higher Algebra and Global Picard Spectra pdf
My master’s thesis. A revised version is available as a preprint, see above.
Decidability and Bisimulation for Logics of Functional Dependence pdf
My bachelor thesis. A condensed and improved version has been published in the Journal of Logic and Computation, see above.
I gave a talk on the main results at the Amsterdam Logic and Interactive Rationality Seminar,
a recording of which can be found here.
CV and contact
My CV as of 10.08.24: Curriculum Vitae
You can reach me under: philpuetzstueck at gmail dot com