### About me

I am a mathematician, currently a Benjamin Peirce Fellow at Harvard. I graduated from my PhD in August 2019 at Northwestern University, under the supervision of Paul Goerss.

My CV is available here. A couple of talks I gave: a short one introducing basic ideas of chromatic homotopy theory and a longer one about its relationship with motives.

I will be at the IAS for the 2022/2023 academic year.

### Notes

Finite height chromatic homotopy theory (Harvard Math 252Y, Spring 2021): PDF

### Papers

Dirac Geometry I: Commutative Algebra. - Joint with Lars Hesselholt. (To appear in Peking Mathematical Journal)

Morava K-theory and Filtrations by Powers. Joint with Tobias Barthel. (To appear in Journal of the Institute of Mathematics of Jussieu)

The Intrinsic Normal Cone for Artin Stacks. Joint with Dhyan Aranha. (To appear in Annales l'Institut Fourier)

Moduli of spaces with prescribed homotopy groups. (Journal of Pure and Applied Algebra, 2023)

Synthetic spectra and the cellular motivic category. (Inventiones mathematicae, 232, 553–681, 2023)

Chromatic Picard groups at large primes. (Proceedings of the American Mathematical Society, 150, 2022, 4981-4988)

Adams-type maps are not stable under composition. Joint with Robert Burklund and Ishan Levy. (Proceedings of the American Mathematical Society, Ser. B Vol. 9, 2022, 373-376)

On dualizable objects in monoidal bicategories. (Theory and Applications of Categories, Vol. 38, No. 9, 2022)

Abstract Goerss-Hopkins theory. Joint with Paul Vankoughnett. (Advances in Mathematics, Volume 395, 2022, 108098, ISSN 0001-8708)
Chromatic homotopy is algebraic when p > n

^{2}+n+1. (Advances in Mathematics, Volume 391, 2021, 107958, ISSN 0001-8708)

### Preprints

Quivers and the Adams spectral sequence (arXiv:2305.08231) - In this joint work with Robert Burklund, we describe a novel way of identifying the second page of the Adams spectral sequence in terms of homological algebra of quiver representations. Our method applies much more broadly than the standard techniques based on descent-flatness, bearing on a varied array of ring spectra. In the particular case of p-local integral homology, we are able to give a decomposition of the second page, describing it completely in terms of the classical Adams spectral sequence. In the appendix, we develop functoriality of deformations of ∞-categories.

Perfect even modules and the even filtration (arXiv:2304.04685) - We introduce a variant of the even filtration which is naturally defined on associative ring spectra and their modules. We show that our variant satisfies flat descent and so agrees with the Hahn-Raksit-Wilson filtration on ring spectra of arithmetic interest, showing that various "motivic" filtrations are in fact invariants of the associative structure alone. We prove that our filtration can be calculated via appropriate resolutions in modules and apply it to the study of even cohomology of connective rings, proving vanishing above the Milnor line, base-change formulas, and explicitly calculating cohomology in low weights.

Dirac geometry II: Coherent cohomology (arXiv:2303.13444) - In this joint work with Lars Hesselholt, we continue our study of the geometry of graded-commutative rings, embedding Dirac schemes from the prequel into the larger ∞-category of Dirac stacks, and developing their coherent cohomology. As applications of the general theory to stable homotopy theory, we use Quillen’s theorem on complex cobordism and Milnor’s theorem on the dual Steenrod algebra to identify the Dirac stacks corresponding to MU and mod p homology in terms of their functors of points.

Adams spectral sequences and Franke's algebraicity conjecture (arXiv:2110.03669) - In this joint work with Irakli Patchkoria, to any well-behaved homology theory we associate a derived ∞-category which encodes its Adams spectral sequence. As applications, we prove a conjecture of Franke on algebraicity of certain homotopy categories and establish homotopy-coherent monoidality of the Adams filtration.

On dualizable objects in monoidal bicategories, framed surfaces and the Cobordism Hypothesis (arXiv:1411.6691) - I prove coherence theorems related to dualizability in symmetric monoidal bicategories, classify two-dimensional framed topological field theories and give a new proof of the Cobordism Hypothesis in dimension two. This paper was written as my Master's thesis at Bonn University and was supervised by Christopher Schommer-Pries. The first part of the paper was already published, see above.

### Non-mathematical interests

I love spending time with my dogs, though unfortunately, they live in Poland. Here's a picture of one of them, Ida.

日本語を勉強して、少し話せる。

© 2022 Piotr Pstrągowski