### About me

I am a graduate student in Mathematics at Northwestern University, supervised by Paul Goerss.

### Preprints

Abstract Goerss-Hopkins theory (arXiv:1904.08881) - In this joint work with Paul VanKoughnett, we present an abstract version of Goerss-Hopkins theory in the setting of a prestable ∞-category equipped with a suitable periodicity operator. In the case of the ∞-category of synthetic spectra, this yields obstructions to realizing a comodule algebra as a homology of a commutative ring spectrum, recovering the results of Goerss and Hopkins.

Chromatic Picard groups at large primes (arXiv:1811.05415) - As a consequence of the algebraicity of chromatic homotopy at large primes, I show that the Hopkins' Picard group of the K(n)-local category coincides with the algebraic one when 2p−2 > n

^{2}+n.Chromatic homotopy is algebraic when p > n

^{2}+n+1 (arXiv:1810.12250) - I show that if E is a p-local Landweber exact homology theory of height n and p > n^{2}+n+1, then there exists an equivalence between homotopy categories of E-local spectra and differential E_{∗}E-comodules, generalizing Bousfield's and Franke's results to heights n > 1.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.

### 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。

© 2018 Piotr Pstrągowski