This session will showcase recent tremendous activity in this area, which brings together classical algebraic geometry, computational and experimental methods, representation theory, and a wide range of applications throughout statistics, the sciences, and engineering. The basic question motivating this area is, given a tensor, what is the least number of terms in a decomposition into simple tensors? Especially important are certain particular tensors such as the matrix multiplication tensor, whose rank is somewhere between quadratic and cubic in the size of the matrix. This leads to a vast array of related questions: under what conditions is the decomposition into simple tensors essentially unique; what are the dimension and other geometric properties of the locus of tensors of a given rank; what are the generic and maximum ranks of tensors; is the rank of a sum of tensors in separate variables equal to the sum of their ranks? The same questions arise for related notions of rank, such as Waring rank, where in the last 5 years there has been a renewal of interest and some real progress: new lower and upper bounds on Waring rank, determination of Waring rank of monomials, proof that in at least some cases the rank of an ``independent'' sum is the sum of the ranks, proof of unique decomposition in many cases.
This session is an activity of the AGATES group.
|Day||Time||Speaker||Title (click for abstract)||Slides|
|Wednesday||8:00 am||Hirotachi Abo||Most secant varieties of tangential varieties to Veronese varieties are nondefective|
|Wednesday||8:30 am||Brooke Ullery||Normality of Secant Varieties||slides|
|Wednesday||9:00 am||Cameron Farnsworth||Secants of the Veronese and the Determinant||slides|
|Wednesday||9:30 am||Ke Ye||Structural tensors of bilinear maps|
|Wednesday||10:00 am||Elina Robeva||Orthogonal Tensor Decomposition|
|Wednesday||10:30 am||Kristian Ranestad||Tensor decompositions and cubic sections of rational surface scrolls||slides|