About

Lillian Beatrix Pierce is a Professor of Mathematics at Duke University, specializing in Analytic Number Theory and Harmonic Analysis. Her research is supported by the National Science Foundation under grant DMS-2200470. She serves on the editorial boards of several prestigious mathematical journals, including Essential Number Theory, where she is a founding editor, the Journal of the American Mathematical Society, Duke Mathematical Journal, Discrete Analysis, and the Annals of Mathematical Studies. Previously, she has contributed to the editorial boards of Transactions of the American Mathematical Society, Memoirs of the American Mathematical Society, Journal of Geometric Analysis, and Experimental Mathematics. Professor Pierce was an invited speaker at the International Congress of Mathematicians (ICM) in 2022, where she presented a talk titled "The Hedgehog and the Fox," which is also documented in the ICM 2022 proceedings under the theme of counting problems related to class groups, primes, and number fields. During the COVID-19 pandemic, she participated in "Paths," a project of recollection by mathematicians conducted in March and April 2020.

Research topics

• Mathematics
• Discrete mathematics
• Mathematical analysis
• Pure mathematics
• Combinatorics
• Philosophy
• Quantum mechanics
• Mathematical economics
• Geometry
• Physics

Selected publications

• Counting points in thin sets: A survey

  ArXiv.org · 2026-03-24

  articleOpen access

  In the 1980's Serre asked how many points of bounded height can lie in a thin set. This has motivated significant research ever since, culminating in a series of recent breakthroughs. It is a good time to take stock of the central questions that have been resolved, and also to highlight remaining open questions. First, we survey recent progress on counting points of bounded height in the four types of thin sets, according to the projective/affine and type I/type II designations. Second, we turn to questions of uniformity. Famously, in the setting of type I thin sets, the best-known upper bound for the number of points of bounded height is independent of the maximum size, say $\|F\|$, of the coefficients of the polynomials that define the thin set; such an upper bound is called uniform. A uniform upper bound in the setting of type II thin sets is not known. For type II thin sets, we explore the dependence on $\|F\|$ via several strategies, and construct counterexamples that suggest the question of uniformity is quite subtle in the setting of type II thin sets.

  PublisherOA PDF

• Real Analysis, Harmonic Analysis, and Applications

  Oberwolfach Reports · 2026-02-16

  articleOpen access

  This workshop in real and harmonic analysis surveyed recent investigations in areas including geometric measure theory and restriction theory, multiple ergodic averages, local smoothing estimates, Schrödinger and Hörmander-type oscillatory integral operators, multilinear estimates and analysis on the Hamming cube. In particular, the workshop emphasized the recent solutions of two longstanding problems: the Kakeya phenomenon in dimension three, and almost everywhere convergence of long time averages associated to multiple commuting measure-preserving transformations. The methods presented during the workshop have yielded applications in ergodic theory, number theory, and computer science.

  PublisherOA PDFDOI

• Counting points in thin sets: A survey

  arXiv (Cornell University) · 2026-03-24

  preprintOpen access

  In the 1980's Serre asked how many points of bounded height can lie in a thin set. This has motivated significant research ever since, culminating in a series of recent breakthroughs. It is a good time to take stock of the central questions that have been resolved, and also to highlight remaining open questions. First, we survey recent progress on counting points of bounded height in the four types of thin sets, according to the projective/affine and type I/type II designations. Second, we turn to questions of uniformity. Famously, in the setting of type I thin sets, the best-known upper bound for the number of points of bounded height is independent of the maximum size, say $\|F\|$, of the coefficients of the polynomials that define the thin set; such an upper bound is called uniform. A uniform upper bound in the setting of type II thin sets is not known. For type II thin sets, we explore the dependence on $\|F\|$ via several strategies, and construct counterexamples that suggest the question of uniformity is quite subtle in the setting of type II thin sets.

  PublisherDOI

• A guide to Tauberian theorems for arithmetic applications

  arXiv (Cornell University) · 2025-04-22

  preprintOpen access1st authorCorresponding

  A Tauberian theorem deduces an asymptotic for the partial sums of a sequence of non-negative real numbers from analytic properties of an associated Dirichlet series. Tauberian theorems appear in a tremendous variety of applications, ranging from well-known classical applications in analytic number theory, to new applications in arithmetic statistics, group theory, and the intersection of number theory and algebraic geometry. The goal of this article is to provide a useful reference for practitioners who wish to apply a Tauberian theorem. We explain the hypotheses and proofs of two types of Tauberian theorems: one with and one without an explicit remainder term. We furthermore provide counterexamples that illuminate that neither theorem can reach an essentially stronger conclusion unless its hypothesis is strengthened.

  PublisherOA PDFDOI

• Counting integral points in thin sets of type II: singularities, sieves, and stratification

  ArXiv.org · 2025-05-16

  preprintOpen access

  Consider an absolutely irreducible polynomial $F(Y,X_1,\ldots,X_n) \in \mathbb{Z}[Y,X_1,\ldots,X_n]$ that is monic in $Y$ and is a polynomial in $Y^m$ for an integer $m \geq 1$. Let $N(F,B)$ count the number of $\mathbf{x} \in [-B,B]^n \cap \mathbb{Z}^n$ such that $F(y,\mathbf{x})=0$ is solvable for $y \in\mathbb{Z}$. In nomenclature of Serre, bounding $N(F,B)$ corresponds to counting integral points in an affine thin set of type II. Previously, in this generality Serre proved $N(F,B) \ll_F B^{n-1/2}(\log B)^γ$ for some $γ<1$. When $m \geq 2$, this new work proves $N(F,B) \ll_{n,F,ε} B^{n-1+1/(n+1) + ε}$ under a nondegeneracy condition that encapsulates that $F(Y,\mathbf{X})$ is truly a polynomial in $n+1$ variables, even after performing any $\text{GL}_n(\mathbb{Q})$ change of variables on $X_1,\ldots,X_n$. Under GRH, this result also holds when $m=1$. We show that generic polynomials satisfy the relevant nondegeneracy condition. Moreover, for a certain class of polynomials, we prove the stronger bound $N(F,B) \ll_{F} B^{n-1}(\log B)^{e(n)}$, comparable to a conjecture of Serre. A key strength of these results is that they require no nonsingularity property of $F(Y,\mathbf{X})$. The Katz-Laumon stratification for character sums, in a new uniform formulation appearing in a companion paper of Bonolis, Kowalski and Woo, is a key ingredient in the sieve method we develop to prove upper bounds that explicitly control any dependence on the size of the coefficients of $F$.

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• Application of a polynomial sieve: beyond separation of variables

  Algebra & Number Theory · 2024-09-18 · 2 citations

  articleOpen accessSenior author

  Let a polynomial f 2 OEX 1 ; : : : ; X n be given.The square sieve can provide an upper bound for the number of integral x 2 OE B; B n such that f .x/ is a perfect square.Recently this has been generalized substantially: first to a power sieve, counting x 2 OE B; B n for which f .x/D y r is solvable for y 2 ; then to a polynomial sieve, counting x 2 OE B; B n for which f .x/D g.y/ is solvable, for a given polynomial g.Formally, a polynomial sieve lemma can encompass the more general problem of counting x 2 OE B; B n for which F.y; x/ D 0 is solvable, for a given polynomial F .Previous applications, however, have only succeeded in the case that F.y; x/ exhibits separation of variables, that is, F.y; x/ takes the form f .x/g.y/.In the present work, we present the first application of a polynomial sieve to count x 2 OE B; B n such that F.y; x/ D 0 is solvable, in a case for which F does not exhibit separation of variables.Consequently, we obtain a new result toward a question of Serre, pertaining to counting points in thin sets.

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• On Polynomial Carleson Operators Along Quadratic Hypersurfaces

  Journal of Geometric Analysis · 2024-08-23

  articleCorresponding
  PublisherDOI

• GENERALISED QUADRATIC FORMS OVER TOTALLY REAL NUMBER FIELDS

  Journal of the Institute of Mathematics of Jussieu · 2024-04-11 · 1 citations

  articleOpen access

  Abstract We introduce a new class of generalised quadratic forms over totally real number fields, which is rich enough to capture the arithmetic of arbitrary systems of quadrics over the rational numbers. We explore this connection through a version of the Hardy–Littlewood circle method over number fields.

  PublisherOA PDFDOI

• Biliary Complications After Liver Transplantation

  Contemporary Diagnostic Radiology · 2023-11-09 · 1 citations

  article1st authorCorresponding

  Liver transplantation continues to be an optimal treatment choice for end-stage liver disease, acute liver failure, and primary liver cancer. An understanding and anticipation of potential complications of liver transplantation is integral to the medical and interventional management of these postoperative patients. Complications related to the biliary system are the most common posttransplant complications. Potential complications include biliary strictures, leakage, intraductal stone/sludge, and biliary cast syndrome. Multiple diagnostic imaging modalities are available including ultrasound, CT angiography, and hepatobiliary iminodiacetic acid scanning for evaluation of the biliary system and associated complications. The approach of interventional access to the biliary system after surgery is tailored based on surgical transplantation techniques. Endoscopic retrograde cholangiopancreatography remains the imaging modality of choice for evaluation and potential therapeutic intervention. If the biliary tree is not accessible by endoscopy due to postsurgical anatomy changes, interventional radiology may play a vital role in complication management and can provide percutaneous transhepatic cholangiography as a less invasive alternative to surgical intervention.

  PublisherDOI

• Generalizations of the Schrödinger maximal operator: building arithmetic counterexamples

  arXiv (Cornell University) · 2023-09-11

  preprintOpen accessSenior author

  Let $T_t^{P_2}f(x)$ denote the solution to the linear Schrödinger equation at time $t$, with initial value function $f$, where $P_2 (ξ) = |ξ|^2$. In 1980, Carleson asked for the minimal regularity of $f$ that is required for the pointwise a.e. convergence of $T_t^{P_2} f(x)$ to $f(x)$ as $t \rightarrow 0.$ This was recently resolved by work of Bourgain, and Du and Zhang. This paper considers more general dispersive equations, and constructs counterexamples to pointwise a.e. convergence for a new class of real polynomial symbols $P$ of arbitrary degree, motivated by a broad question: what occurs for symbols lying in a generic class? We construct the counterexamples using number-theoretic methods, in particular the Weil bound for exponential sums, and the theory of Dwork-regular forms. This is the first case in which counterexamples are constructed for indecomposable forms, moving beyond special regimes where $P$ has some diagonal structure.

  PublisherOA PDFDOI

Recent grants

• The circle method, character sums, and sieves: Applications to number theory and harmonic analysis

  NSF · $157k · 2014–2017

• Class Groups, Character Sums, and Oscillatory Integrals

  NSF · $350k · 2022–2026

• PostDoctoral Research Fellowship

  NSF · $135k · 2009–2013

• CAREER: Research and Training at the Intersection of Number Theory and Analysis

  NSF · $450k · 2017–2023

Frequent coauthors

• Russell Simpson

  16 shared

• Sean M. Haney

  Accreditation Council for Graduate Medical Education

  16 shared

• Karen L. Xie

  Medical Education Institute

  16 shared

• Kevin D. Hendzel

  University of Illinois Chicago

  16 shared

• Charles F. Pierce

  16 shared

• Po‐Lam Yung

  Australian Mathematical Sciences Institute

  15 shared

• Joris Roos

  9 shared

• Caroline L. Turnage‐Butterbaugh

  Carleton College

  8 shared

Awards & honors

• Duke Honors 38 Distinguished Professors in 2026
• Four Trinity Faculty, Three Alums Win Guggenheim Fellowships