REU Site: Tiling Theory, Knot Theory, Optimization, Matrix Analysis, and Image Reconstruction

REU 站点：平铺理论、结理论、优化、矩阵分析和图像重建

• 批准号: 2150511
• 负责人: Casey Mann
• 金额: $ 29.18万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2150511&HistoricalAwards=false
• 关键词: REU; Site; Tiling; Theory; Knot

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). This REU Site program will host nine undergraduate students for eight weeks in summers 2022, 2023, and 2024 at the University of Washington Bothell (UWB) for research experiences. Students will provide their research preferences and work closely with a faculty mentor in groups of three. The research projects to be pursued at this REU site are chosen to be of interest to the greater mathematical research community while being accessible to students with minimal background preparation required. The research groups will be strongly encouraged to publish their results in journals and present them at professional meetings. This REU site will prepare its student participants for research careers and graduate training, contribute to increasing diversity in the mathematical sciences, and encourage and prepare its participants to pursue graduate school. The research experience will be complemented by several outings and social activities to build team cohesiveness and networking. This REU Site will focus on research problems in the areas of knot theory, tiling theory, matrix analysis, optimization, and image reconstruction. In the tiling theory group, the participants will experiment with known infinite families of tiles and use basic combinatorial methods to analyze patterns. They will also perform computer-aided explorations of potential minimal aperiodic protosets to determine aperiodicity and potential structure of associated Markov partitions. In the non-smooth optimization group, students will work to improve the numerical performance of the emerging non-smooth spectral gradient methods. This project will also help develop students' scientific computing skills using MATLAB. In the knot theory group, students will explore various open questions, for example, identifying types of knots that arise as components of hexagonal mosaic links created from saturated diagrams and understanding splittable links from specific sizes of saturated hexagonal or parallelogram diagrams. In the medical imaging and optimization group, students will study the algorithmic framework, generate simulated data, and modify existing code to train the deep-learning-based prior and incorporate it into the reconstruction algorithm. This project will offer students the opportunity to gain exposure to cutting-edge concepts in imaging science and machine learning and develop mathematical programming skills. The matrix analysis group will explore open questions on nonnegative matrices, the field of values, the geometry of polynomials, discrete geometry, and number theory.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.

该奖项是根据2021年《美国救援计划法》（公法117-2）全部或部分资助的。该REU网站计划将在2022年，2023年和2024年在华盛顿·博内尔大学（UWB）（UWB）举办九个本科生八个星期，以供研究经验。学生将提供他们的研究偏好，并与三人一组密切合作。选择在此REU站点进行的研究项目是为了使更大的数学研究社区感兴趣，同时需要最少的背景准备的学生访问。将强烈鼓励研究小组在期刊上发布其结果，并在专业会议上展示。此REU网站将为其学生参与者准备研究职业和研究生培训，为数学科学的多样性做出贡献，并鼓励和准备参与者攻读研究生院。研究经验将得到几次郊游和社交活动的补充，以建立团队的凝聚力和网络。 此REU站点将重点介绍结理论，平铺理论，矩阵分析，优化和图像重建领域的研究问题。在平铺理论组中，参与者将尝试已知的无限瓷砖家族，并使用基本的组合方法来分析模式。他们还将对潜在的最小大道原型的计算机辅助探索，以确定相关的马尔可夫分区的多个杂质和潜在结构。在非平滑优化组中，学生将努力提高新出现的非平滑光谱梯度方法的数值性能。该项目还将帮助使用MATLAB发展学生的科学计算技能。在结理论组中，学生将探索各种开放问题，例如，确定由饱和图创建的六边形镶嵌链接的组成部分，并理解从饱和的六边形或平行四边形图的特定尺寸的可分配链接。在医学成像和优化小组中，学生将研究算法框架，生成模拟数据并修改现有代码以训练基于深度学习的先验并将其纳入重建算法。该项目将为学生提供机会在成像科学和机器学习和发展数学编程技能方面接触尖端的概念。矩阵分析小组将探讨有关非负矩阵，价值领域，多项式几何，离散几何形状和数字理论的开放问题。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准来通过评估来支持的。