arXiv subjects

Peter Marek

Publications and source records attributed to Peter Marek.

3 recordsLinked to original sources

arXiv2024

Stable comodule deformations and the synthetic Adams-Novikov spectral sequence

We study the Adams-Novikov spectral sequence in $\mathbb{F}_p$-synthetic spectra, computing the synthetic analogs of $\mathrm{BP}$ and its cooperations to identify the synthetic Adams-Novikov $\mathrm{E}_2$-page, computed in a range with a synthetic algebraic Novikov spectral sequence. We then identify deformations associated to the Cartan-Eilenberg and algebraic Novikov spectral sequences in terms of stable comodule categories, categorifying an algebraic Novikov spectral sequence result of Gheorghe-Wang-Xu. We then apply Isaksen-Wang-Xu methods in $\mathbb{F}_2$-synthetic spectra to deduce differentials in the $p=2$ synthetic Adams-Novikov for the sphere, producing almost entirely algebraic computations through the 45-stem.

math.AT↗

arXiv2022

$\mathrm{H}\mathbb{F}_2$-synthetic homotopy groups of topological modular forms

To any Adams-type spectrum $E$, Pstr\k{a}gowski produced a symmetric monoidal stable $\infty$-category $Syn_E$ whose objects are, in a sense, ''formal Adams spectral sequences''. $Syn_E$ comes equipped with a lax symmetric monoidal functor $\nu_E:Sp\to Syn_E$ from classical spectra, which embeds $Sp$ fully and faithfully in $Syn_E$, and is a category with a natural notion of bigraded homotopy groups. The bigraded homotopy groups $\pi_{*,*}\nu_EX$ systematically record information about the homotopy groups $\pi_*X$ and the $E$-Adams spectral sequence of $X$. In this paper, we compute the $\nu_{\mathrm{H}\mathbb{F}_2}\mathbb{F}_2$-Adams spectral sequence of $\nu_{\mathrm{H}\mathbb{F}_2}tmf_2^{\wedge}$, synthetic versions of hidden $2$-, $\eta$-, $\nu$-, and $\overline{\kappa}$-extensions, and use this to deduce information about the homotopy ring structure of $\pi_{*,*}\nu_{\mathrm{H}\mathbb{F}_2}tmf_2^{\wedge}$.

math.AT↗

arXiv2018

Improving angular resolution of telescopes through probabilistic single-photon amplification?

The use of probabilistic amplification for astronomical imaging is discussed. Probabilistic single photon amplification has been theoretically proven and practically demonstrated in quantum optical laboratories. In astronomy it should allow to increase the angular resolution beyond the diffraction limit at the expense of throughput: not every amplification event is successful -- unsuccessful events contain a large fraction of noise and need to be discarded. This article indicates the fundamental limit in the trade-off between gain in angular resolution and loss in throughput. The practical implementation of probabilistic amplification for astronomical imaging remains an open issue.

astro-ph.IM↗