Systems of Lines: Applications of Algebraic Combinatorics

这项NSF奖支持一个名为“线的系统：代数组合的应用”的数学研究研讨会，该研讨会将于2015年8月10日至8月14日在伍斯特理工学院举行。本次研讨会将汇集来自世界各地的专家，他们研究的三个主题没有明显的联系，但它们之间有着深刻的联系。组织者旨在促进数学家、计算机科学家和物理学家之间的交叉合作，以便在看似不同的数学应用中发现共同的问题、工具和技术。在代数组合专家应用工具从代数，图论和优化研究纠错码，球面码和高度规则的设计用于统计和计算机科学。量子信息理论家在量子算法、量子密码学和量子态断层扫描的工作中面临着测量和（近似）复制有限维量子系统的问题。信号处理专家采用压缩感知，以精心设计的方式通过欠采样有效地重建稀疏信号。数据科学家现在使用压缩感知的思想来降维和有效处理非常大的数据集。本次研讨会的网址是http://users.wpi.edu/~martin/MEETINGS/WPILinesWorkshop.htmlBeginning随着Candes， Tao和Donoho在2004年的突破性工作，在过去十年左右的时间里，在采样低于Nyquist-Shannon极限的稀疏信号重建方面取得了重大进展。除了核磁共振成像等信号处理应用之外，该理论还在处理大数据方面找到了新的用途。具有受限等距特性（RIP）和随机投影的矩阵现在可以有效地降维，这大大提高了我们从大规模数据集进行有用推断的能力。同时，线系统在量子信息理论中发挥着重要作用，其中互无偏基（mub）和对称信息完备的正算子值测度（sic - povm）有望为有限维量子态提供最佳测量。不幸的是，我们对如何构造这些对象知之甚少。所有这些应用都可以用代数组合学的语言来表达，在代数编码理论和图特征值的研究中，球码和格拉斯曼填充自然地出现在关联方案理论中。通过将来自压缩感知、量子信息理论和代数组合学的人们聚集在一起，本次研讨会将使每个领域的工具和结果能够应用于其他领域，并帮助确定所有三个子学科共同的突出问题。

This NSF award supports a mathematical research workshop entitled "Systems of Lines: Applications of Algebraic Combinatorics" to be held at Worcester Polytechnic Institute from August 10, 2015 to August 14, 2015. This workshop will bring together experts from around the world who study three subjects that are not obviously related but have deep connections between them. The organizers aim to foster cross-collaboration among mathematicians, computer scientists and physicists in order to identify common problems, tools and techniques among seemingly disparate applications of mathematics. Experts in algebraic combinatorics apply tools from algebra, graph theory and optimization to study error-correcting codes, spherical codes and highly regular designs for use in statistics and computer science. Quantum information theorists face problems of measuring and (approximately) replicating finite-dimensional quantum systems in their work on quantum algorithms, quantum cryptography and quantum state tomography. Signal processing experts employ compressive sensing to efficiently reconstruct a sparse signal through undersampling in a carefully designed manner. Data scientists now use ideas from compressive sensing for dimension reduction and the efficient handling of very large data sets. The URL for this workshop is http://users.wpi.edu/~martin/MEETINGS/WPILinesWorkshop.htmlBeginning with the breakthrough work of Candes, Tao and Donoho in 2004, significant advances have been made over the past decade or so in the reconstruction of sparse signals while sampling below the Nyquist-Shannon limit. Beyond signal processing applications such as MRI, this theory has found new utility in the handling of big data. Matrices with the restricted isometry property (RIP) and random projections now enable effective dimension reduction which greatly enhances our ability to make useful inferences from large-scale data sets. Meanwhile, systems of lines play an important role in quantum information theory where mutually unbiased bases (MUBs) and symmetric informationally complete positive operator-valued measures (SIC-POVMs) promise to provide optimal measurements for finite-dimensional quantum states. Unfortunately, very little is known about how to construct these objects. All of these applications can be couched in the language of algebraic combinatorics where spherical codes and Grassmannian packings naturally arise in the theory of association schemes, in algebraic coding theory and the study of graph eigenvalues. By bringing people together from compressive sensing, quantum information theory, and algebraic combinatorics, this workshop will enable the application of tools and results from each area to the other and help identify the outstanding problems common to all three subdisciplines.