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”，会议将于2017年7月24日至28日在美国奥本大学举行。主题是集中在使用张量在近似理论，量子信息理论和其他领域从几何的角度。会议邀请了德州农工大学的J.M. Landsberg博士，他是张量的代数和几何研究及其应用方面的领先专家，他将发表10次讲座，并撰写专著发表在CBMS系列上，并将提供急需的资源，使使用张量的研究人员能够发展新的思维和观点，从而奠定张量理论的基础。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。女性、未被充分代表的少数民族、研究生、博士后助理、初级教师和主要本科院校的教师将被专门招募参加。奥本大学位于美国东南部，是一个招收多元化学生的好地方。会议和随后的专著将在科学和数学的不同领域之间建立联系。讨论的领导者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交流和研究。该活动将通过增强参与者的研究背景并向他们介绍与张量有关的开放问题来影响人力资源开发。会议日期（2017年7月24日至28日）定于2017年7月31日至8月4日在佐治亚州亚特兰大举行的SIAM应用代数几何会议之前举行，以增加协同活动。这项活动将提高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与日俱增。在坐标中，张量就是一个高维矩阵。就像在线性代数中，我们把矩阵看作是在坐标中表示线性映射，在多线性代数中，也就是对张量的研究中，我们把高维矩阵看作是表示多线性映射。线性映射的视角有助于研究线性代数中与坐标无关的特征值和其他性质，多线性映射的视角同样有助于研究张量的与坐标无关的性质。与线性代数不同，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。会议的主要目标是向一大群人介绍最新的理论和最新的重要应用。更具体地说，讲座的目的是：给初学研究生提供张量的基本介绍，它们的几何和用途。描述以前没有以说明性形式处理过的应用，或者从完全不同的角度（例如，物理或数值分析而不是几何角度）处理过的应用。描述张量秩和边界秩研究的最新突破。引起人们对关于张量的开放性问题的关注——无论是旧的还是最近的。研究生和年轻的研究人员将具备开始研究开放性问题的能力，使他们能够立即开始为科学发现做出贡献。会议网站：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