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

Auburn University, Auburn AL

Investigators

Abstract

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