CBMS Conference: Tensors and Their Uses in Approximation Theory, Quantum Information Theory, and Geometry

该奖项支持为期5天的CBMS会议“张量及其在近似理论，量子信息理论和几何中的应用”，在奥本大学，奥本，AL，2017年7月24日至28日。主题是围绕张量在近似理论，量子信息理论和其他领域的几何角度的使用。会议由J. M博士主持。兰茨贝格的得克萨斯州A M大学，在代数和几何研究张量及其应用的领先专家，谁将提供10个讲座，并产生一个专着将在CBMS系列出版，并将提供一个急需的资源，使研究人员谁使用张量开发新的思维和观点，张量理论的基础。该奖项将支持来自美国东南部地区的7名讨论领袖和25名初级研究人员组成的多元化小组。 妇女，代表性不足的少数民族，研究生，博士后助理，初级教师和教师在主要是本科院校将专门招募参加。在东南部，奥本大学是一个伟大的位置，从不同的人口招聘。会议和随后的专著将使科学和数学的不同领域之间的联系。讨论领导人来自不同的科学背景（代数几何，数值分析，应用数学和物理），因此他们的参与将有助于建立跨学科的沟通和研究。该活动将通过提高参与者的研究背景和向他们介绍与张量有关的开放问题来影响人力资源开发。日期（2017年7月24日至28日）将在2017年7月31日至8月4日在亚特兰大举行的SIAM应用代数几何会议之前举行，增加协同活动。这项活动将提升美国东南部地区的研究水平。张量在所有科学领域的重要性都在增长。在坐标系中，张量只是一个高维矩阵。正如在线性代数中，我们将矩阵视为坐标中的线性映射，在多线性代数中，即，在张量的研究中，我们将高维矩阵视为多线性映射。线性映射的观点促进了线性代数中特征值和其他与坐标无关的性质的研究，而多线性映射的观点同样能够研究张量的与坐标无关的性质。与线性代数的情况不同，多线性代数仍处于起步阶段，基本问题（例如，如何检测多线性映射的秩）是开放和微妙的。拟议会议的主要目标是向一大群人介绍理论和最近重要应用的最新技术。 更具体地说，讲座的目的是：给一个初步介绍张量，他们的几何和使用访问开始研究生。为了描述以前没有以临时形式处理过的应用程序，或者已经从完全不同的角度处理过的应用程序（例如，物理或数值分析，而不是几何角度）。介绍了张量秩和边秩研究的最新进展。为了引起人们对张量的开放性问题的关注，无论是旧的还是新的。研究生和年轻的研究人员将有能力开始解决开放性问题，使他们能够立即开始为科学发现做出贡献。http://www.auburn.edu/~lao0004/cbms.html

This award supports a 5-day CBMS conference on "Tensors and their uses in approximation theory, quantum information theory, and geometry," at Auburn University, Auburn, AL, July 24-28, 2017. The theme is centered on uses of tensors in approximation theory, quantum information theory and other areas from a geometric perspective. The conference features Dr. J.M. Landsberg of Texas A&M University, a leading expert in the algebraic and geometric study of tensors and their applications, who will deliver 10 lectures and produce a monograph to be published in the CBMS series and will provide a much-needed resource, enabling researchers who use tensors to develop the new thinking and perspectives that underlie the theory of tensors. The award will support a diverse group of 7 discussion leaders and 25 junior researchers from the southeastern region of the US. Women, underrepresented minorities, graduate students, post-doctoral associates, junior faculty, and faculty at primarily undergraduate institutions will be specifically recruited to participate. Being in the Southeast, Auburn University is a great location to recruit from a diverse population. The conference and subsequent monograph will make connections between different areas of science and mathematics. The discussion leaders come from diverse scientific backgrounds (Algebraic Geometry, Numerical Analysis, Applied Mathematics, and Physics), so their participation will help establish cross-disciplinary communication and research. The activity will impact human resource development by enhancing the research backgrounds of participants and introducing them to open problems related to tensors. The dates (July 24-28, 2017) are set to precede the SIAM meeting on Applied Algebraic Geometry to be held in Atlanta, GA, July 31-August 4, 2017, increasing synergetic activities. This activity will lift the level of research in the southeastern region of the US.Tensors are growing in importance in all areas of science. In coordinates, a tensor is just a higher dimensional matrix. Just as in linear algebra, where we view matrices as representing linear maps in coordinates, in multi-linear algebra, i.e., the study of tensors, we view a higher dimensional matrix as representing a multi-linear map. The perspective of linear maps facilitates the study of eigenvalues and other properties independent of coordinates in linear algebra, and the perspective of multi-linear maps similarly enables the study of coordinate-independent properties of tensors. Unlike the case of linear algebra, multi-linear algebra is still in its infancy and basic questions (e.g., how to detect the rank of a multi-linear map) are open and subtle. The main goal of the proposed conference is to introduce a large group of individuals to the state of the art in theory and recent important applications. More specifically, the lectures' aims are: To give an elementary introduction to tensors, their geometry and uses accessible to beginning graduate students. To describe applications that have either not been treated in expository form previously, or had been treated from a completely different perspective (e.g., of physics or numerical analysis rather than a geometric perspective). To describe recent breakthroughs in the study of tensor rank and border rank. To bring attention to open questions--both old and recent--regarding tensors. Graduate students and young researchers will become equipped to start on open problems, allowing them to begin contributing to scientific discovery right away.Conference website: http://www.auburn.edu/~lao0004/cbms.html