LEAPS-MPS: Deep Learning the Knot Landscape

该奖项是根据2021年美国救援计划法案（公法117-2）资助的。纽结理论始于19世纪中期，主要受物理学思想的影响。 今天，纽结在物理学中通过规范理论和量子场论，分子生物学通过蛋白质和DNA打结，以及低维拓扑学中的4维流形和3维流形的Dehn手术描述的身体图的形式发挥着重要作用。 2016年，PI提出了一种新的方法，通过应用机器学习和人工智能技术来研究纽结理论中的问题。 虽然现在在这个方向上有一个小但不断增长的研究机构，数学家，物理学家和计算机科学家的贡献，大多数现有的工作集中在监督学习技术和强化学习在解开辫子上的应用。 PI将采用生成机器学习和强化学习的新技术来研究节点的拓扑特性，并学习节点的潜在分布及其不变量。 PI还将通过建立新观测和实验结果的理论基础来扩展与合作者的早期工作。 作为该项目的一部分，PI将为本科生开发一个指导性的数据科学研究培训计划，学生将由PI和来自行业或学术界的数据科学家指导。 学生将与PI一起参加为期一个学期的指导学习小组，然后完成一个密集的研讨会，在那里他们与外部导师一起解决行业的数据科学问题。 在该计划的每一步，将特别注意通过与专门从事这些社区外展的校园组织合作，增加历史上代表性不足的群体的参与。 通过参与指导研究，这些学生将获得经验，帮助他们为获得研究生学位和在学术界和工业界的职业生涯做好准备，从而为来自这些代表性不足的社区的学生未来的榜样做好准备。PI将调整文本到图像生成对抗网络来构建不变到节点的GAN，允许构建具有指定拓扑属性的节点。 PI还将使用变分自编码器来学习节点的新潜在分布，这些分布对于各种拓扑属性和不变量来说是自然的。 结的这些潜在表示将提供对结分布的更清晰的理解，并产生允许具有指定属性的结的目标生成的随机模型。 结理论数据的任何新的潜在表示都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。 这些技术将被用来指导寻找重要的开放式结构的反例。 此外，PI将使用深度强化学习算法来研究切片属和编织带秩问题，将现有的关于使用强化学习的结果推广到编织物的解开。 考虑到计算节点的切片亏格的问题是低维拓扑和物理学结构中关键开放问题的核心，这里开发的新技术将在节点理论之外有直接的应用。 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NSF的法定使命，并通过使用基金会的智力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.