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

该奖项全部或部分根据2021年美国救援计划法案（公法117-2）资助。这个REU网站计划将在2022年，2023年和2024年夏季在华盛顿博瑟尔大学（UWB）举办为期八周的九名本科生的研究经验。学生将提供他们的研究偏好，并与三人一组的教师导师密切合作。在这个REU网站上进行的研究项目被选择为更大的数学研究社区感兴趣，同时可以访问学生所需的最低背景准备。将大力鼓励研究小组在期刊上发表其成果，并在专业会议上介绍。这个REU网站将为学生参与研究职业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。研究经验将通过几次郊游和社交活动来补充，以建立团队凝聚力和网络。 这个网站将集中在研究问题的结理论，平铺理论，矩阵分析，优化和图像重建领域。在瓷砖理论组中，参与者将使用已知的无限瓷砖家族进行实验，并使用基本的组合方法来分析图案。他们还将执行潜在的最小非周期性原型的计算机辅助探索，以确定相关的马尔可夫分割的非周期性和潜在的结构。在非光滑优化组，学生将努力提高新兴的非光滑谱梯度方法的数值性能。该项目还将帮助学生使用MATLAB开发科学计算技能。在结理论组中，学生将探索各种开放性问题，例如，识别从饱和图创建的六边形马赛克链接的组件中出现的结的类型，并从特定大小的饱和六边形或三角形图中理解可拆分的链接。在医学成像和优化组中，学生将学习算法框架，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知识并将其纳入重建算法。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。矩阵分析小组将探索非负矩阵、值域、多项式几何、离散几何和数论等方面的开放性问题。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.