Ryan Quinn

About me

I am currently a fourth-year PhD student at Utrecht University, supervised by Lennart Meier. In the spring, I will start a postdoc position at the Max Planck Institute for Mathematics in Bonn.

Before this, I was a master’s student in maths at the University of Amsterdam, and an undergrad in maths and theoretical physics at Maynooth University.

I am funded by the National University of Ireland Travelling PhD Studentship.

email: r.quinn@uu.nl

Research

My main focus is in equivariant and chromatic homotopy theory, with an emphasis on structured ring spectra. I'm particularly interested in multiplicative structures on $\mathrm{BP}^{(\!(G)\!)}\langle m\rangle$ and related computations. I have a growing interest in applying equivariant multiplicative structures to the study of equivariant algebraic $\mathrm{K}$-theory, as well as to Grothendieck-Witt and $\mathrm{L}$-theory.

Publications & Preprints

Structured Real Snaith Equivalences, joint with Qi Zhu, preprint (2026). [PDF] [arXiv]

Summary: We prove Real equivariant multiplicative versions of Snaith Theorems. As an application, we compute $\operatorname{THR}(\mathrm{MUP}_{\mathbb{R}})$ and $\operatorname{THR}(\mathrm{KU}_{\mathbb{R}})$. Along the way, we improve on our previous work by constructing more refined $\mathbb{E}_{2\rho}$-orientations, as well as $\mathbb{E}_{3\rho}$-orientations in the periodic setting.
Multiplicative equivariant Thom spectra & structured Real orientations, joint with Qi Zhu, submitted (2025). [PDF] [arXiv]

Summary: Let $\rho$ be the regular representation of $C_2$. We show that $\mathrm{BP}_{\mathbb{R}}$ admits an $\mathbb{E}_\rho$-ring structure, and that $\mathrm{BP}^{(\!(G)\!)}$ admits a $\mathrm{coind}^G_{C_2}\mathbb{E}_\rho$-ring structure. We do this by lifting Quillen's idempotent to an $\mathbb{E}_\rho$-ring map. More generally, we show that any homotopy commutative ring map $\mathrm{MU}\to E^e$ lifts to an $\mathbb{E}_\rho$-ring map $\mathrm{MU}_{\mathbb{R}}\to E$ for any strongly even $C_2$-$\mathbb{E}_\infty$-ring $E$. We do this by developing an obstruction theory for structured orientations of equivariant Thom spectra. The essential technology that goes into this is: generalizing the work of Antolín-Camarena–Barthel to the parameterized setting; and constructing a good theory of parameterized left module categories.
Spectral properties of light and charm mesons from $N_f=2+1$ anisotropic lattice QCD, joint with Jonas Glesaaen, Alexander Rothkopf, and Jonivar Skullerud. Proceedings of Science (2019). [PDF] [arXiv]

Summary: We compute temporal correlators and spectral functions for light, open charm and charmonium mesons in the pseudoscalar and vector channel for a range of temperatures below and above the deconfinement transition. The study is carried out using anisotropic lattice QCD with $2+1$ dynamical flavours, $a_s = 0.123$ fm and $a_s/a_\tau = 3.5$. The high-temperature results are benchmarked by comparing them to reconstructed correlators obtained by direct summation of the zero-temperature correlator. We use two Bayesian methods to reconstruct the spectral functions: the maximum entropy method and the more recent BR method.

Notes

Truncated Brown–Peterson spectra as generalized Thom spectra, (2025). [PDF]

Summary: We show that the truncated Brown–Peterson spectra $\mathrm{BP}\langle n\rangle$ can be constructed as generalized Thom spectra.
A convenient approach to gradings in homotopy theories, (2025). [PDF]

Summary: We give a German-approved way to talk about $\mathrm{RO}(G)$-graded homotopy groups.
Localization theorems in equivariant cohomology (2025). [PDF]

Summary: This note reviews Atiyah–Bott localization theorems in equivariant cohomology. The goal is to highlight the following three things:
- First, that proofs of Atiyah–Bott style localization theorems can be made completely formal.
- Second, give easy-to-check conditions for when an Atiyah–Bott style localization theorem holds in the integer-graded setting, and show that they always hold in the $\mathrm{RO}(G)$-graded setting.
- Finally, clarify the situation for non-abelian groups.
This is essentially a cleaned-up version of my master’s thesis, supervised by Lennart Meier.

Teaching & Supervision

Topological quantum field theories mini-course
Morse theory reading project

Misc

code for nicer underlines in $\LaTeX$, e.g. compare Sp versus Sp, or Alg versus Alg

%---(nicer underlines)-------------------------------------------------------------------START
\makeatletter
\usepackage[normalem]{ulem}
\usepackage{contour}
\usepackage{mathtools} % for \mathllap

\renewcommand{\ULdepth}{1.8pt} % how low to make the underline
\contourlength{0.8pt} % how big to make the gap in words like map

\makeatletter
% usage: there are two ways to use this
% \myuline{text} puts an underline under text
% \myuline[7em]{text} puts an underline under text but shortens it by 7em; of course 7em can be changed. This is useful for stuff like \({\myuline[0.5pt]{\mathrm{Map}}}_G\), vs \({\myuline{\mathrm{Map}}}_G\)
\DeclareRobustCommand{\myuline}[2][0pt]{%
  \ifmmode
    % draw a slightly shorter underline
    \uline{\hphantom{#2}\kern-#1}%
    % restore overall width so following text lines up
    \kern#1%
    % overlay the actual content, with proper math style
    \mathllap{\mathpalette\my@cont@{#2}}%
  \else
    \uline{\phantom{#2}\kern-#1}%
    \kern#1%
    \llap{\contour{white}{#2}}%
  \fi
}
% #1 = current math style, #2 = content
\newcommand{\my@cont@}[2]{\contour{white}{\mbox{$\m@th#1#2$}}}
\makeatother
%------------------------------------------------------------------------------------------END