Emily de Oliveira Santos

I work in project operations at Surge AI. My background is in pure mathematics, specifically category theory. I co-authored Humanity's Last Exam, published in Nature, and contributed problems to FrontierMath.

Press

I occasionally give interviews about my work.

Interviewed for The Sunday Times
On Surge AI and the data-labelling industry.

FrontierMath

FrontierMath is a benchmark of exceptionally challenging mathematics problems covering most major branches of modern mathematics. Problems in this dataset require PhD+ level expertise and take mathematicians several hours — or even days — to solve.

A few months after FrontierMath's initial release, Epoch AI announced an extension of the project called FrontierMath (Tier 4). Problems in this set were created through several-week research projects by mathematics professors and postdocs, representing the most extreme difficulty level in the benchmark.

I was one of the major contributors of problems across all tiers of FrontierMath, including Tier 4. I am also a co-author of the paper.

FrontierMath: A Benchmark for Evaluating Advanced Mathematical Reasoning in AI
Joint with Elliot Glazer et al. arXiv:2411.04872.

Abstract

We introduce FrontierMath, a benchmark of hundreds of original, exceptionally challenging mathematics problems crafted and vetted by expert mathematicians. The questions cover most major branches of modern mathematics — from computationally intensive problems in number theory and real analysis to abstract questions in algebraic geometry and category theory. Solving a typical problem requires multiple hours of effort from a researcher in the relevant branch of mathematics, and for the upper end questions, multiple days. FrontierMath uses new, unpublished problems and automated verification to reliably evaluate models while minimizing risk of data contamination. Current state-of-the-art AI models solve under 2% of problems, revealing a vast gap between AI capabilities and the prowess of the mathematical community. As AI systems advance toward expert-level mathematical abilities, FrontierMath offers a rigorous testbed that quantifies their progress.

Humanity's Last Exam

I contributed questions to Humanity's Last Exam, a benchmark of 2,500 expert-level questions designed to test the limits of large language models across dozens of academic subjects. In particular:

I received two Top 550 prizes and one Top 50 prize.
One of my questions appears among the six example problems included in the Nature paper (as Figure 3) and the eight on the project website.

A Benchmark of Expert-Level Academic Questions to Assess AI Capabilities (HLE)
Joint with 1,412 other authors. Nature 649, 1139–1146 (2026). DOI. arXiv:2501.14249.

Abstract

Benchmarks are important tools for tracking the rapid advancements in large language model (LLM) capabilities. However, benchmarks are not keeping pace in difficulty: LLMs now achieve over 90% accuracy on popular benchmarks like MMLU, limiting informed measurement of state-of-the-art LLM capabilities. In response, we introduce Humanity's Last Exam (HLE), a multi-modal benchmark at the frontier of human knowledge, designed to be the final closed-ended academic benchmark of its kind with broad subject coverage. HLE consists of 2,500 questions across dozens of subjects, including mathematics, humanities, and the natural sciences. HLE is developed globally by subject-matter experts and consists of multiple-choice and short-answer questions suitable for automated grading. Each question has a known solution that is unambiguous and easily verifiable, but cannot be quickly answered via internet retrieval. State-of-the-art LLMs demonstrate low accuracy and calibration on HLE, highlighting a significant gap between current LLM capabilities and the expert human frontier on closed-ended academic questions. To inform research and policymaking upon a clear understanding of model capabilities, we publicly release HLE at https://lastexam.ai.

Research

Coends of Higher Arity
Joint with Fosco Loregian. Applied Categorical Structures, 2021. DOI. arXiv:2011.13881.

Abstract

We specialise a recently introduced notion of generalised dinaturality for functors $T\colon(\mathcal{C}^{\mathsf{op}})^{p}\times\mathcal{C}^{q}\to\mathcal{D}$ to the case where the domain (resp., codomain) is constant, obtaining notions of ends (resp., coends) of higher arity, dubbed herein $(p,q)$-ends (resp., $(p,q)$-coends). While higher arity co/ends are particular instances of "totally symmetrised" (ordinary) co/ends, they serve an important technical role in the study of a number of new categorical phenomena, which may be broadly classified as two new variants of category theory.

The first of these, weighted category theory, consists of the study of weighted variants of the classical notions and construction found in ordinary category theory, besides that of a limit. This leads to a host of varied and rich notions, such as weighted Kan extensions, weighted adjunctions, and weighted ends.

The second, diagonal category theory, proceeds in a different (albeit related) direction, in which one replaces universality with respect to natural transformations with universality with respect to dinatural transformations, mimicking the passage from limits to ends. In doing so, one again encounters a number of new interesting notions, among which one similarly finds diagonal Kan extensions, diagonal adjunctions, and diagonal ends.
Weighted Category Theory
Joint with Fosco Loregian. In preparation.

Description

We study weighted variants of classical notions and constructions found in category theory, moving beyond the theory of weighted co/limits. This gives rise to weighted natural transformations, weighted ends, weighted Kan extensions, weighted adjunctions, and weighted monads.
Diagonal Category Theory
Joint with Fosco Loregian. In preparation.

Description

We investigate analogues of classical notions in category theory obtained by replacing universality with respect to natural transformations by universality with respect to dinatural transformations. This leads to a conceptual solution of the compositionality problem for dinatural transformations, as we show the reason dinatural transformations fail to compose is that they are just the degree 1 part of a more general $\mathbb{N}$-graded composition law.

Projects

The Clowder Project
An online reference for category theory, aiming to become a Stacks Project for the field.
Hundar
A free and open-source typeface for scientific and academic writing (formerly Darwin). Website.

Emily de Oliveira Santos

Press

FrontierMath

Humanity's Last Exam

Research

Projects

Contact