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

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。这个REU网站项目将在2022年、2023年和2024年夏季在华盛顿大学博塞尔分校（UWB）接待9名本科生，为期8周，进行研究体验。学生将提供他们的研究偏好，并以三人一组的方式与教师导师密切合作。在这个REU网站上进行的研究项目被选择为对更大的数学研究界感兴趣，同时对学生具有最少的背景准备要求。将强烈鼓励研究小组在期刊上发表他们的研究结果，并在专业会议上展示。这个REU网站将为其学生参与者的研究生涯和研究生培训做好准备，有助于增加数学科学的多样性，并鼓励和准备其参与者攻读研究生院。研究经验将辅以几次郊游和社会活动，以建立团队凝聚力和网络。这个REU网站将专注于结理论、平铺理论、矩阵分析、优化和图像重建等领域的研究问题。在瓷砖理论组中，参与者将对已知的无限瓷砖族进行实验，并使用基本的组合方法来分析图案。他们还将对潜在的最小非周期原集进行计算机辅助探索，以确定相关马尔可夫分区的非周期性和潜在结构。在非光滑优化组中，学生将致力于改进新兴的非光滑谱梯度方法的数值性能。这个项目也将有助于培养学生使用MATLAB的科学计算技能。在结理论小组中，学生将探索各种开放问题，例如，识别由饱和图创建的六边形马赛克链接组成的结的类型，以及从特定尺寸的饱和六边形或平行四边形图中理解可分割的链接。在医学成像和优化组，学生将学习算法框架，生成模拟数据，修改现有代码来训练基于深度学习的先验并将其纳入重构算法。该项目将为学生提供接触成像科学和机器学习前沿概念的机会，并培养数学编程技能。矩阵分析组将探讨非负矩阵、值域、多项式几何、离散几何和数论的开放性问题。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 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.