Peter Humphries

About

Peter Humphries is an Assistant Professor in the Department of Mathematics at the University of Virginia. His research focuses on Analytic Number Theory, Automorphic Forms, and Representation Theory. He is involved in exploring various aspects of Number Theory and Representation Theory, contributing to the advancement of these fields through his academic work.

Research topics

• Computer Science
• Physics
• Ophthalmology
• Neuroscience
• Biology
• Medicine
• Bioinformatics
• Endocrinology

Selected publications

• Findings from a Genotyping Study of over 1000 People with Inherited Retinal Disorders in Ireland

  Genes · 2020-01-16 · 57 citations

  articleOpen access

  The Irish national registry for inherited retinal degenerations (Target 5000) is a clinical and scientific program to identify individuals in Ireland with inherited retinal disorders and to attempt to ascertain the genetic cause underlying the disease pathology. Potential participants first undergo a clinical assessment, which includes clinical history and analysis with multimodal retinal imaging, electrophysiology, and visual field testing. If suitable for recruitment, a sample is taken and use…

  PublisherOA PDFDOI

• Optimal Small Scale Equidistribution of Lattice Points on the Sphere, Heegner Points, and Closed Geodesics

  Communications on Pure and Applied Mathematics · 2022-07-21 · 10 citations

  articleOpen access1st authorCorresponding

  Abstract We asymptotically estimate the variance of the number of lattice points in a thin, randomly rotated annulus lying on the surface of the sphere. This partially resolves a conjecture of Bourgain, Rudnick, and Sarnak. We also obtain estimates that are valid for all balls and annuli that are not too small. Our results have several consequences: for a conjecture of Linnik on sums of two squares and a “microsquare”, a conjecture of Bourgain and Rudnick on the number of lattice points lying in…

  PublisherOA PDFDOI

• Towards a 𝐺𝐿_{𝑛} variant of the Hoheisel phenomenon

  Transactions of the American Mathematical Society · 2021-12-20 · 6 citations

  article1st authorCorresponding

  Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π</mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a unitary cuspidal automorphic representation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper G normal upper L Subscript n"> <mml:semantics> <mml:msub>…

  PublisherDOI

• Distributing Points On The Torus Via Modular Inverses

  The Quarterly Journal of Mathematics · 2021-02-23 · 6 citations

  articleOpen access1st authorCorresponding

  Abstract We study various statistics regarding the distribution of the points $$ \left\{\left(\frac{d}{q},\frac{\overline{d}}{q}\right) \in \mathbb{T}^2 : d \in (\mathbb{Z}/q\mathbb{Z})^{\times}\right\} $$ as q tends to infinity. Due to non-trivial bounds for Kloosterman sums, it is known that these points equidistribute on the torus. We prove refinements of this result, including bounds for the discrepancy, small-scale equidistribution, bounds for the covering exponent associated with these poi…

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• $L^p$-Norm Bounds for Automorphic Forms via Spectral Reciprocity

  arXiv (Cornell University) · 2022-08-11 · 3 citations

  preprintOpen access1st authorCorresponding

  Let $g$ be a Hecke-Maass cusp form on the modular surface ${\rm SL}_2(\mathbb{Z})\backslash\mathbb{H}$, namely an $L^2$-normalised nonconstant Laplacian eigenfunction on ${\rm SL}_2(\mathbb{Z})\backslash\mathbb{H}$ that is additionally a joint eigenfunction of every Hecke operator. We prove the $L^4$-norm bound $\|g\|_4\ll_{\varepsilon}λ_g^{3/304+\varepsilon}$, where $λ_g$ denotes the Laplacian eigenvalue of $g$, which improves upon Sogge's $L^4$-norm bound $\|g\|_4\llλ_g^{1/16}$ for Laplacian e…

  PublisherOA PDFDOI

Frequent coauthors

• Paul F. Kenna

  Royal Victoria Eye and Ear Hospital

  189 shared

• N. J. O’Higgins

  University College Dublin

  156 shared

• Marian M. Humphries

  Trinity College Dublin

  144 shared

• G. Jane Farrar

  Trinity College Dublin

  129 shared

• E. W. M. McDermott

  St. Vincent's University Hospital

  120 shared

• Matthew Campbell

  University of British Columbia

  105 shared

• P. M. Mercer

  96 shared

• M. J. Duffy

  95 shared

Education

• Ph.D., Mathematics

  Princeton University

  2017

• M.Phil., Mathematics

  Australian National University

  2012

• Ph.B. (Hons) (Science), Mathematics

  Australian National University

  2010

Similar researchers at University of Virginia

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• Jennifer Morseh-38
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