Ryan Quinn

About me

I am a third year PhD student at Utrecht University supervised by Lennart Meier. Before this, I was a master 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 a particular focus in structured ring spectra. I'm particularly interested in multiplicative structures on \(\mathrm{BP}^{(\!(G)\!)}\langle m\rangle\) and related computations.

Publications & Preprints

• Multiplicative equivariant Thom spectra & structured Real orientations, joint with Qi Zhu, preprint (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 masters 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

  copy \(\LaTeX\)

  Copy

  %---(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