Research Interests

Representation Theory & Arithmetic Geometry (The Langlands Program), Arithmetic Statistics, Iwasawa Theory, and Analytic Number Theory.

Papers

On Monic Abelian Trace-One Cubic Polynomials (Submitted)

Joint with Andrew O'Desky.

Read Abstract BibTeX

We compute the asymptotic number of monic trace-one integral polynomials with Galois group \( C_3 \) and bounded height. For such polynomials we compute a height function coming from toric geometry and introduce a parametrization using the quadratic cyclotomic field \( \mathbb{Q}(\sqrt{-3}) \). We also give a formula for the number of polynomials of the form \( t^3 - t^2 + at + b \in \mathbb{Z}[t] \) with Galois group \( C_3 \) for a fixed integer \( a \).

Correlations of error terms for weighted prime counting functions

Joint with Greg Martin and Reginald Simpson.

To appear in Acta Arithmetica

Read Abstract BibTeX

Standard prime-number counting functions, such as \( \psi(x) \), \( \theta(x) \), and \( \pi(x) \), have error terms with limiting logarithmic distributions once suitably normalized. The same is true of weighted versions of those sums, like \( \pi_r(x) = \sum_{p\le x} \frac1p \) and \( \pi_\ell(x) = \sum_{p\le x} \log(1-\frac1p)^{-1} \), that were studied by Mertens. These limiting distributions are all identical, but passing to the limit loses information about how these error terms are correlated with one another. In this paper, we examine these correlations, showing, for example, that persistent inequalities between certain pairs of normalized error terms are equivalent to the Riemann hypothesis (RH). Assuming both RH and LI, the linear independence of the positive imaginary parts of the zeros of \( \zeta(s) \), we calculate the logarithmic densities of the set of real numbers for which two different error terms have prescribed signs. For example, we conditionally show that \( \psi(x) - x \) and \( \sum_{n\le x} \frac{\Lambda(n)}n - (\log x - C_0) \) have the same sign on a set of logarithmic density \( \approx 0.9865 \).

AI Benchmarking

QEDBENCH: Quantifying the Alignment Gap in Automated Evaluation of University-Level Mathematical Proofs

Multiple authors.

Accepted as a regular paper at the ICML 2026.

Read Abstract BibTeX

As Large Language Models (LLMs) saturate elementary benchmarks, the research frontier has shifted from generation to the reliability of automated evaluation. We demonstrate that standard "LLM-as-a-Judge" protocols suffer from a systematic Alignment Gap when applied to upper-undergraduate to early graduate level mathematics. To quantify this, we introduce QEDBench, the first large-scale dual-rubric alignment benchmark to systematically measure alignment with human experts on university-level math proofs by contrasting course-specific rubrics against expert common knowledge criteria. By deploying a dual-evaluation matrix (7 judges x 5 solvers) against 1,000+ hours of human evaluation, we reveal that certain frontier evaluators like Claude Opus 4.5, DeepSeek-V3, Qwen 2.5 Max, and Llama 4 Maverick exhibit significant positive bias (up to +0.18, +0.20, +0.30, +0.36 mean score inflation, respectively). Furthermore, we uncover a critical reasoning gap in the discrete domain: while Gemini 3.0 Pro achieves state-of-the-art performance (0.91 average human evaluation score), other reasoning models like GPT-5 Pro and Claude Sonnet 4.5 see their performance significantly degrade in discrete domains. Specifically, their average human evaluation scores drop to 0.72 and 0.63 in Discrete Math, and to 0.74 and 0.50 in Graph Theory. In addition to these research results, we also release QEDBench as a public benchmark for evaluating and improving AI judges. Our benchmark is publicly published at https://github.com/qqliu/Yale-QEDBench.

Old Research Projects

Research Talks

Teaching & Mentoring

Other Selected Talks/Presentations

• Introduction to Modular Forms

  Student Seminar: Number Theory and Automorphic Forms
  Sep 29, 2023
• D4 and Triality

  MATH 534 (Lie Theory I) final presentation
  Dec 2022
• Algebraic Torus, Algebraic Groups course presentation
  May 4, 2022

• An Introduction to the Conjecture of Bloch and Kato

  VSRP-MATH-2021 Presentation
  Jun 15, 2021
• Nonsingular Curves
  Nov 2020

Conferences

• Arizona Winter School 2026

  Program Link
• Elliptic Curves and the Special Values of L-functions (2022, HYBRID)

  In-person participant.
• PIMS CRG Workshop on Moments of L-functions (HYBRID)
• L-functions, Circle Method, and Applications, 2022 (Hybrid, ICTS Bangalore)
• Elliptic Curves and The Special Values of L-functions, 2021 (ICTS, Bangalore)
• Dualities in Topology and Algebra, 2021 (ICTS, Bangalore)
• International Conference on Class Groups of Number Fields and Related Topics, 2021
• Modular Forms (Honouring Prof. B Ramakrishnan's 60th birthday), 2021
• A Series of Trimester Programs on Triangle Groups, Belyi Uniformization, and Modularity (2021-22)
• International Conference on Number Theory and Discrete Mathematics, 2020

Other/Old Writings

• Biases in The Distribution of Primes Modulo 4

  with Zhengheng Bao. Link to paper
• The Polya-Vinogradov Inequality

  with Reginald Simpson. Link to paper
• An Introduction to The Conjecture of Bloch & Kato (VSRP project report)

  (Available upon request)
• On a Generalization of the Ramanujan-Nagell Type Equations

  (Available upon request)
• Arbitrarily Many Primes at a Fixed Distance from Certain Composite Sequences

  (Available upon request)

Miscellaneous

• Simon Marais Mathematics Competition (2019)

  Honourable mention in the top quartile list of pairs.
• Galois Representations Reading Group

  Created with Zachary Gardner, Boston College. Devoted to discussing Galois Representations and surrounding topics.
• Math Stack Exchange Profile

  Used to post some questions/answers.

  https://math.stackexchange.com/users/466737/james-moriarty

Hobbies & Skills