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
The main goal of this talk is to present a result related to a conjecture
suggested by Catalisano, Geramita, and Gimigliano in 2002, which claims
that the secant varieties of tangential varieties to Veronese varieties are
non-defective modulo a few known exceptions. This is joint work with
Nick Vannieuwenhoven.
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| Wednesday | 8:30 am | Brooke Ullery |
Normality of Secant Varieties
If X is a smooth variety embedded in projective space, we can form a new
variety by looking at the closure of the union of all the lines through 2
points on X. This is called the secant variety to X. Similarly, the Hilbert
scheme of 2 points on X parametrizes all length 2 zero-dimensional
subschemes. I will talk about how these two constructions are related. More
specifically, I will show how we can use certain tautological vector
bundles on the Hilbert scheme to help us understand the geometry of the
secant variety, leading to a proof that for sufficiently positive embeddings
of X, the secant variety is a normal variety.
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slides |
| Wednesday | 9:00 am | Cameron Farnsworth |
Secants of the Veronese and the Determinant
Let $det_n \in S_n({C^n}^2)$ be the homogeneous polynomial obtained by
taking the determinant of an $n \times n$ matrix of indeterminates. In this
presentation linear maps called Young flattenings will be defined. It will
then be shown how these maps may be used to demonstrate new lower bounds on
the symmetric border rank of $det_n$.
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slides |
| Wednesday | 9:30 am | Ke Ye |
Structural tensors of bilinear maps
Bilinear maps are very important in both Multilinear Algebra and Complexity
Theory. For example, matrix multiplication is a bilinear map and its
computational complexity is still a mystery. In most situations,
computational complexity of a bilinear map is characterized by the rank of
its structural tensor. In this talk, we will discuss the study of structural
tensors of bilinear maps. We will present a framework to calculate an upper
bound for the rank of the structural tensor of a bilinear map. Using this
framework, we are able to write down a tensor decomposition of the
structural tensor and hence an algorithm to compute the bilinear map. This
is joint work with Lek-Heng Lim.
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| Wednesday | 10:00 am | Elina Robeva |
Orthogonal Tensor Decomposition
A tensor is orthogonally decomposable if it can be written as a linear
combination of rank-one tensors $a_i \otimes b_i \otimes c_i \otimes \dotsb$
such that the $a_i$ are orthonormal, the $b_i$ are orthonormal, the $c_i$
are orthonormal, etc. Every matrix is orthogonally decomposable because of
the singular value decomposition theorem. In this work we give equations
that cut out the variety of orthogonally decomposable tensors.
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| Wednesday | 10:30 am | Kristian Ranestad |
Tensor decompositions and cubic sections of rational surface scrolls
The variety of multi secant spaces to a projective variety X through a
point P in projective space has been studied as a variety of tensor
decompositions of a symmetric tensor, when X is a d-uple Veronese variety.
For the cubic reembedding X of a rational surface scroll and P general,
the variety is a surface that can be interpreted as a variety of tensor
decompositions. I shall report on work in progress with Nelly Villamizar
and Matteo Gallet on this topic.
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slides |