HARDMath2: A Benchmark for Applied Mathematics Built by Students as Part of a Graduate Class

Large language models (LLMs) have shown remarkable progress in mathematical problem-solving, but evaluation has largely focused on problems that have exact analytical solutions or involve formal proofs, often overlooking approximation-based problems ubiquitous in applied science and engineering. To fill this gap, we build on prior work and present HARDMath2, a dataset of 211 original problems covering the core topics in an introductory graduate applied math class, including boundary-layer analysis, WKB methods, asymptotic solutions of nonlinear partial differential equations, and the asymptotics of oscillatory integrals. This dataset was designed and verified by the students and instructors of a core graduate applied mathematics course at Harvard. We build the dataset through a novel collaborative environment that challenges students to write and refine difficult problems consistent with the class syllabus, peer-validate solutions, test different models, and automatically check LLM-generated solutions against their own answers and numerical ground truths. Evaluation results show that leading frontier models still struggle with many of the problems in the dataset, highlighting a gap in the mathematical reasoning skills of current LLMs. Importantly, students identified strategies to create increasingly difficult problems by interacting with the models and exploiting common failure modes. This back-and-forth with the models not only resulted in a richer and more challenging benchmark but also led to qualitative improvements in the students' understanding of the course material, which is increasingly important as we enter an age where state-of-the-art language models can solve many challenging problems across a wide domain of fields.

Quantifying the Uniqueness of Donald Trump in Presidential Discourse

Does Donald Trump speak differently from other presidents? If so, in what ways? Are these differences confined to any single medium of communication? To investigate these questions, this paper introduces a novel metric of uniqueness based on large language models, develops a new lexicon for divisive speech, and presents a framework for comparing the lexical features of political opponents. Applying these tools to a variety of corpora of presidential speeches, we find considerable evidence that Trump's speech patterns diverge from those of all major party nominees for the presidency in recent history. Some notable findings include Trump's employment of particularly divisive and antagonistic language targeting of his political opponents and his patterns of repetition for emphasis. Furthermore, Trump is significantly more distinctive than his fellow Republicans, whose uniqueness values are comparably closer to those of the Democrats. These differences hold across a variety of measurement strategies, arise on both the campaign trail and in official presidential addresses, and do not appear to be an artifact of secular time trends.