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.
Teaching
Fall 2020 - Math 155R Combinatorics
Preprints
Intrinsic normal cone for Artin stacks (arXiv:1909.07478) - In this joint work with Dhyan Aranha, we extend the construction of the intrinsic normal cone due to Behrend and Fantechi to any locally of finite type morphism of higher Artin stacks. As an application, we associate to any morphism of Artin stacks equipped with a choice of a global perfect obstruction theory a relative virtual fundamental class in the Chow group of Kresch.
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 > n2+n. This paper is a part of my PhD thesis at Northwestern.
Chromatic homotopy is algebraic when p > n2+n+1 (arXiv:1810.12250) - I show that if E is a p-local Landweber exact homology theory of height n and p > n2+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. This paper is a part of my PhD thesis at Northwestern.
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。
© 2020 Piotr Pstrągowski