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Mathematics Colloquium: Algebraic Geometry of Tensors
4:10 p.m. Neill Hall 5W
In many applications such as face recognitions, the collected data can be represented as elements which have more than two indices. Such high-order equivalents of vectors and matrices are called tensors. Every tensor can be written as a linear combination of so-called decomposable tensors. A tensor is said to have rank r if it can be written as a linear combination of r decomposable tensors (but not fewer). An interesting question is ``What is the least positive integer such that a generic tensor has rank less than or equal to R?". Such an integer R is called the typical tensor rank. This problem can be very naturally translated into a classical problem in Algebraic Geometry. In this talk, I will go through some basics of Algebraic Geometry. Then I will describe how objects in Algebraic Geometry can be associated to various parameter spaces of tensors of a fixed rank. At the end of the talk, I will discuss an algebro-geometric approach to find the typical tensor rank.
Hirotachi Abo is a faculty in the Department of Mathematics at University of Idaho.