50 research outputs found
Fatou directions along the Julia set for endomorphisms of CP^k
Not much is known about the dynamics outside the support of the maximal
entropy measure for holomorphic endomorphisms of . In this
article we study the structure of the dynamics on the Julia set, which is
typically larger than . The Julia set is the support of the
so-called Green current , so it admits a natural filtration by the supports
of the exterior powers of . For , let . We
show that for a generic point of there are at least
"Fatou directions" in the tangent space. We also give estimates for the
rate of expansion in directions transverse to the Fatou directions.Comment: Final, shorter version, to appear in J. Math. Pures App
Random spectrahedra
Spectrahedra are affine-linear sections of the cone Pn of positive semidefinite symmetric n
7 n-matrices. We consider random spectrahedra that are obtained by intersecting Pn with the affine-linear space 1 + V , where 1 is the identity matrix and V is an `-dimensional linear space that is chosen from the unique orthogonally invariant probability measure on the Grassmanian of `-planes in the space of n
7 n real symmetric matrices (endowed with the Frobenius inner product). Motivated by applications, for ` = 3 we relate the average number E\u3c3n of singular points on the boundary of a three-dimensional spectrahedron to the volume of the set of symmetric matrices whose two smallest eigenvalues coincide. In the case of quartic spectrahedra (n = 4) we show that E\u3c34 = 6 12 1a43 . Moreover, we prove that the average number E \u3c1n of singular points on the real variety of singular matrices in 1 + V is n(n 12 1). This quantity is related to the volume of the variety of real symmetric matrices with repeated eigenvalues. Furthermore, we compute the asymptotics of the volume and the volume of the boundary of a random spectrahedron
Algebraic geometry for tensor networks, matrix multiplication, and flag matroids
This thesis is divided into two parts, each part exploring a different topic within
the general area of nonlinear algebra. In the first part, we study several applications of tensors. First, we study tensor networks, and more specifically: uniform
matrix product states. We use methods from nonlinear algebra and algebraic geometry to answer questions about topology, defining equations, and identifiability
of uniform matrix product states. By an interplay of theorems from algebra, geometry, and quantum physics we answer several questions and conjectures posed
by Critch, Morton and Hackbusch. In addition, we prove a tensor version of the
so-called quantum Wielandt inequality, solving an open problem regarding the
higher-dimensional version of matrix product states.
Second, we present new contributions to the study of fast matrix multiplication. Motivated by the symmetric version of matrix multiplication we study the
plethysm S^k(sl_n) of the adjoint representation sl_n of the Lie group SL_n . Moreover, we discuss two algebraic approaches for constructing new tensors which
could potentially be used to prove new upper bounds on the complexity of matrix
multiplication. One approach is based on the highest weight vectors of the aforementioned plethysm. The other approach uses smoothable finite-dimensional
algebras.
Finally, we study the Hessian discriminant of a cubic surface, a recently introduced invariant defined in terms of the Waring rank. We express the Hessian
discriminant in terms of fundamental invariants. This answers Question 15 of the
27 questions on the cubic surface posed by Bernd Sturmfels.
In the second part of this thesis, we apply algebro-geometric methods to
study matroids and flag matroids. We review a geometric interpretation of the
Tutte polynomial in terms of the equivariant K-theory of the Grassmannian. By
generalizing Grassmannians to partial flag varieties, we obtain a new invariant of
flag matroids: the flag-geometric Tutte polynomial. We study this invariant in
detail, and prove several interesting combinatorial properties
Generic Spectrahedral Shadows
Spectrahedral shadows are projections of linear sections of the cone of
positive semidefinite matrices. We characterize the polynomials that vanish on
the boundaries of these convex sets when both the section and the projection
are generic.Comment: version to be publishe
Learning Algebraic Varieties from Samples
We seek to determine a real algebraic variety from a fixed finite subset of
points. Existing methods are studied and new methods are developed. Our focus
lies on aspects of topology and algebraic geometry, such as dimension and
defining polynomials. All algorithms are tested on a range of datasets and made
available in a Julia package