Linear Algebra/Algebra Seminar

Speaker: Luke Oeding

Title: "Symmetry and Large Scale Computations for the Quadrifocal Variety"

Abstract: The quadrifocal variety is a 39-dimensional algebraic subvariety of the 80-dimensional projective space of tensors of format $3\times 3 \times 3 \times 3$, and arises in multi-view geometry, a branch of computer vision. I will discuss how to use symmetry to study the ideal of polynomial equations vanishing on the quadrifocal variety. Despite being of high dimension and codimension, it is still possible to compute its ideal up to degree 6 in terms of representations of $\GL(3)^{\times 4}$. These computations are performed in Maple by explicitly constructing a basis of the highest weight space for each irreducible representation that occurs in the polynomial ring (in low degrees).  

Further analysis using Macaulay2 (and the package "SchurRings") allows us to rule out certain syzygies, giving a lower bound for the number of minimal generators.  Led by these computations we conjecture that the ideal of the quadrifocal variety is minimally generated in degree at most 9.