LEAPS-MPS: Deep Learning the Knot Landscape

该奖项全部由《2021年美国救援计划法案》（公法117-2）资助。结理论领域始于19世纪中期，主要受到物理学思想的启发。今天，结点通过规范理论和量子场论在物理学中扮演着重要的角色，通过蛋白质和DNA结在分子生物学中扮演着重要的角色，通过4流形的手体图和3流形的Dehn手术描述的低维拓扑发挥着重要作用。2016年，PI通过应用机器学习和人工智能技术，开创了一种研究结理论问题的新方法。虽然现在在这个方向上有一个小而不断增长的研究机构，有数学家、物理学家和计算机科学家的贡献，但大多数现有的工作都集中在监督学习技术和强化学习在解开辫子上的应用上。PI将采用生成式机器学习和强化学习的新技术来研究结的拓扑性质，并学习结的潜在分布及其不变量。PI还将通过建立新的观察和实验结果的理论基础，扩展与合作者的早期工作。作为该项目的一部分，PI将为本科生制定一个有指导的数据科学研究培训计划，在该计划中，学生将由PI和来自工业界或学术界的数据科学家指导。学生们将在PI的指导下参加一个学期的学习小组，然后完成一个密集的研讨会，在那里他们将与外部导师一起解决来自行业的数据科学问题。在这个项目的每一步，都将特别关注通过与专门向这些社区提供服务的校园组织合作，增加历史上代表性不足的群体的参与。通过参与有指导的研究，这些学生将获得经验，这将有助于他们为研究生学位和在学术界和工业界的职业生涯做准备，从而准备成为这些代表性不足的社区学生的未来榜样。PI将适应文本到图像的生成对抗网络来构建不变到结的gan，允许构建具有规定拓扑性质的结。PI还将使用变分自编码器来学习与各种拓扑性质和不变量相关的新的结的潜在分布。这些潜在的结表示将提供对结分布的更清晰的理解，并产生随机模型，允许有针对性地生成具有特定属性的结。结理论数据的任何新的潜在表示将提供给其他研究人员用于训练新的机器学习模型，提高正在开发的模型的性能。这些技术将用于指导寻找重要的开放式猜想的反例。此外，PI将使用深度强化学习算法来研究切片属和辫子带秩问题，将现有的使用强化学习的结果推广到辫子的解开。考虑到计算结的切片属的问题是物理学中低维拓扑和结构的关键开放问题的核心，这里开发的新技术将直接应用于结理论之外。这些技术的成功使用将成为未来生成机器学习和强化学习在其他数学领域应用的模板。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole under the American Rescue Plan Act of 2021 (Public Law 117-2). The field of knot theory began in the mid 1800s, motivated heavily by ideas from physics. Today, knots play an important role in physics through gauge theory and quantum field theories, molecular biology via protein and DNA knotting, and low-dimensional topology in the form of handlebody diagrams of 4-manifolds and Dehn surgery descriptions of 3-manifolds. In 2016 the PI initiated a novel approach to studying problems in knot theory by applying techniques of machine learning and artificial intelligence. While there is now a small but growing body of research in this direction, with contributions from mathematicians, physicists, and computer scientists, most of the existing work focuses on techniques of supervised learning and applications of reinforcement learning to unknotting braids. The PI will adapt new techniques from generative machine learning and reinforcement learning to study topological properties of knots and learn latent distributions of knots and their invariants. The PI will also extend earlier work with collaborators, by establishing theoretical underpinnings of new observations and experimental results. As part of this project the PI will develop a mentored data science research training program for undergraduate students, in which students will be mentored by both the PI and a data scientist from industry or academia. Students will participate in a semester-long mentored learning group with the PI, before completing an intensive workshop where they work together to solve data science problems from industry with their external mentor. At each step of this program, special attention will be given to increasing participation of historically underrepresented groups through partnerships with campus organizations that specialize in outreach to these communities. By participating in mentored research these students will gain experience that will help them prepare for graduate degrees and careers in academia and industry, thereby preparing to be future role models for students from these underrepresented communities.The PI will adapt text-to-image generative adversarial networks to construct invariant-to-knot GANs, allowing for the construction of knots with prescribed topological properties. The PI will also use variational autoencoders to learn new latent distributions of knots which are natural with respect to various topological properties and invariants. These latent representations of knots will provide a clearer understanding of knot distributions and produce random models that allow for targeted generation of knots with specified properties. Any new latent representations of knot theoretic data will be made available to other researchers for use in training new machine learning models, improving the performance of the models being developed. These techniques will be used to guide searches for counterexamples to important open conjectures. In addition, the PI will use deep reinforcement learning algorithms to study the slice genus and braid band rank problems, generalizing existing results on the use of reinforcement learning to the unknotting of braids. Given that the problem of computing the slice genus of knots is central to key open questions in low-dimensional topology and constructions in physics, new techniques developed here will have direct applications outside of knot theory. Successful use of these techniques will serve as a template for future applications of generative machine learning and reinforcement learning to other areas of mathematics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.