### 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.

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

### Teaching

Spring 2022: Math 155R Combinatorics.

I also co-organize the eCHT reading seminar on synthetic spectra with William Balderrama.

### Notes

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

### Papers

Chromatic Picard groups at large primes. (To appear in Proceedings of the American Mathematical Society)

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

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

Dirac Geometry I: Commutative Algebra (arXiv:2207.09256) - In this joint work with Lars Hesselholt, we develop the commutative algebra of graded-commutative rings (such as those given by the homotopy groups of a commutative ring spectrum). We name the resulting geometry Dirac geometry, since the grading exhibits the hallmarks of spin, distinguishing between symmetric and anti-symmetric behavior.

Morava K-theory and Filtrations by Powers (arXiv:2111.06379) - In this joint work with Tobias Barthel, we prove the convergence of the Adams spectral sequence based on Morava K-theory and relate it to the filtration by powers of the maximal ideal in the Lubin-Tate ring through a Miller square. We use the filtration by powers to construct a spectral sequence relating the homology of the K-local sphere to derived functors of completion and express the latter as cohomology of the Morava stabilizer group. As an application, we compute the zeroth limit at all primes and heights.

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.

Synthetic spectra and the cellular motivic category (arXiv:1803.01804) - To any Adams-type homology theory one can associate a notion of a synthetic spectrum, this is a spherical sheaf on the site of finite E-projective spectra. I show that ∞-category of synthetic spectra based on E is in a precise sense a deformation of Hovey's stable homotopy theory of comodules whose generic fibre is given by the ∞-category of spectra. In the case of MU, I show that the even variant of this construction coincides with the cellular motivic category after p-completion.

Moduli of Π-algebras (arXiv:1705.05761) - I describe a homotopy-theoretic approach to the moduli of Π-algebras of Blanc-Dwyer-Goerss using the ∞-category of product-preserving presheaves on finite wedges of positive-dimensional spheres, reproving their results in this setting.

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。

© 2020 Piotr Pstrągowski