哈佛大学数学家劳伦·威廉士在《为什么的乐趣》播客中分享了她对「正格拉斯曼流形」（positive Grassmannian）的研究。这是一种用于对其他几何形状进行分类的组合代数结构，意外地将许多看似无关的领域联系在一起，包括交通流量模型、浅水波浪的物理行为，以及量子粒子的散射振幅。

威廉士指出，这些跨领域的关联核心在于普吕克坐标（Plücker coordinates）的排斥性质与正性。在物理学中，正格拉斯曼流形与物理学家提出的「振幅面体」（amplituhedron）概念密切相关，它能将量子场论中极其复杂的费曼图计算简化为单一几何形状的体积计算，体现了数学简洁与优雅的终极美感。

除了理论数学研究，威廉士还共同发起了「First Proof」项目，旨在通过使用数学家尚未公开的学术论文引理作为测试题目，客观评估人工智慧在解决前沿研究级数学证明方面的真实能力。该项目旨在避免AI仅通过网页检索直接获取答案，从而探索AI在未来成为人类数学研究伙伴的可能性。

Harvard mathematician Lauren Williams shared her research on the positive Grassmannian in the podcast 'The Joy of Why.' This algebraic combinatorial structure, which classifies other geometric shapes, unexpectedly connects several seemingly unrelated fields, including traffic flow models, the physical behavior of shallow-water waves, and the scattering amplitudes of quantum particles.

Williams pointed out that the core of these cross-disciplinary connections lies in the repelling nature and positivity of Plücker coordinates. In physics, the positive Grassmannian is closely related to the concept of the 'amplituhedron,' which simplifies extremely complex Feynman diagram calculations in quantum field theory into the volume calculation of a single geometric shape, embodying the ultimate beauty of mathematical simplicity and elegance.

In addition to her theoretical mathematical research, Williams co-initiated the 'First Proof' project, which aims to objectively evaluate AI's true capability in solving research-level mathematical proofs by using unpublished lemmas from mathematicians' own work. The project seeks to prevent AI systems from simply retrieving answers online, thereby exploring the potential of AI to become partners in human mathematical research in the future.