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Series: Algebra Seminar

We introduce a novel representation of structured polynomial ideals, which we refer to as chordal networks. The sparsity structure of a polynomial system is often described by a graph that captures the interactions among the variables. Chordal networks provide a computationally convenient decomposition of a polynomial ideal into simpler (triangular) polynomial sets, while preserving its underlying graphical structure. We show that many interesting families of polynomial ideals admit compact chordal network representations (of size linear in the number of variables), even though the number of components could be exponentially large. Chordal networks can be computed for arbitrary polynomial systems, and they can be effectively used to obtain several properties of the variety, such as its dimension, cardinality, equidimensional components, and radical ideal membership. We apply our methods to examples from algebraic statistics and vector addition systems; for these instances, algorithms based on chordal networks outperform existing techniques by orders of magnitude.

Series: Algebra Seminar

Networks, or graphs, can represent a great variety of systems in the real world including neural networks, power grid, the Internet, and our social networks. Mathematical models for such systems naturally reflect the graph theoretical information of the underlying network. This talk explores some common themes in such models from the point of view of systems of nonlinear equations.

Series: Algebra Seminar

Abstract: Tensors are direct generalizations of matrices. They appear in almost every branch of mathematics and engineering. Three of the most important problems about tensors are: 1) compute the rank of a tensor 2) decompose a tensor into a sum of rank one tensors 3) Comon’s conjecture for symmetric tensors. In this talk, I will try to convince the audience that algebra can be used to study tensors. Examples for this purpose include structured matrix decomposition problem, bilinear complexity problem, tensor networks states, Hankel tensors and tensor eigenvalue problems. In these examples, I will explain how algebraic tools are used to answer the three problems mentioned above.

Series: Algebra Seminar

Error-correcting decoding is generalized to multivariate
sparse polynomial and rational function interpolation from
evaluations that can be numerically inaccurate and where
several evaluations can have severe errors (``outliers'').
Our multivariate polynomial and rational function
interpolation algorithm combines Zippel's symbolic sparse
polynomial interpolation technique [Ph.D. Thesis MIT 1979]
with the numeric algorithm by Kaltofen, Yang, and Zhi [Proc.
SNC 2007], and removes outliers (``cleans up data'') by
techniques from the Welch/Berlekamp decoder for Reed-Solomon
codes.
Our algorithms can build a sparse function model from a
number of evaluations that is linear in the sparsity of the
model, assuming that there are a constant number of ouliers
and that the function probes can be randomly chosen.

Series: Algebra Seminar

For a given generic form, the problem of finding the nearest rank-one form with respect to the Bombieri norm is well-studied and completely solved for binary forms. Nonetheless, higher-rank approximation is quite mysterious except in the quadratic case. In this talk we will discuss such problems in the binary case.

Series: Algebra Seminar

The Brill-Noether varieties of a curve C parametrize embeddings of C of prescribed degree into a projective space of prescribed dimension. When C is general in moduli, these varieties are well understood: they are smooth, irreducible, and have the “expected” dimension. As one ventures deeper into the moduli space of curves, past the locus of general curves, these varieties exhibit intricate, even pathological, behaviour: they can be highly singular and their dimensions are unknown. A first measure of the failure of a curve to be general is its gonality. I will present a generalization of the Brill—Noether theorem, which determines the dimensions of the Brill—Noether varieties on a general curve of fixed gonality, i.e. “general inside a chosen special locus". The proof blends a study of Berkovich skeletons of maps from curves to toric varieties with tropical linear series theory. The deformation theory of logarithmic stable maps acts as the bridge between these ideas. This is joint work with Dave Jensen.

Series: Algebra Seminar

We show that in many instances, at the heart of a problem in numerical computation sits a special 3-tensor, the structure tensor of the problem that uniquely determines its underlying algebraic structure. In matrix computations, a decomposition of the structure tensor into rank-1 terms gives an explicit algorithm for solving the problem, its tensor rank gives the speed of the fastest possible algorithm, and its nuclear norm gives the numerical stability of the stablest algorithm. We will determine the fastest algorithms for the basic operation underlying Krylov subspace methods --- the structured matrix-vector products for sparse, banded, triangular, symmetric, circulant, Toeplitz, Hankel, Toeplitz-plus-Hankel, BTTB matrices --- by analyzing their structure tensors. Our method is a vast generalization of the Cohn--Umans method, allowing for arbitrary bilinear operations in place of matrix-matrix product, and arbitrary algebras in place of group algebras. This talk contains joint work with Ke Ye and joint work Shmuel Friedland.

Series: Algebra Seminar

It is well-known, that any univariate polynomial matrix A over the complex numbers that takes only positive semidefinite values on the real line, can be factored as A=B^*B for a polynomial square matrix B. For real A, in general, one cannot choose B to be also a real square matrix. However, if A is of size nxn, then a factorization A=B^tB exists, where B is a real rectangular matrix of size (n+1)xn. We will see, how these correspond to the factorizations of the Smith normal form of A, an invariant not usually associated with symmetric matrices in their role as quadratic forms. A consequence is, that the factorizations canusually be easily counted, which in turn has an interesting application to minimal length sums of squares of linear forms on varieties of minimal degree.

Series: Algebra Seminar

We prove a result about the Galois module structure of the Fermat curve using commutative algebra, number theory, and algebraic topology. Specifically, we extend work of Anderson about the action of the absolute Galois group of a cyclotomic field on a relative homology group of the Fermat curve. By finding explicit formulae for this action, we determine the maps between several Galois cohomology groups which arise in connection with obstructions for rational points on the generalized Jacobian. Heisenberg extensions play a key role in the result. This is joint work with R. Davis, V. Stojanoska, and K. Wickelgren.

Series: Algebra Seminar

Conductors and minimal discriminants are two measures of degeneracy of the singular fiber in a family of hyperelliptic curves. In the case of elliptic curves, the Ogg-Saito formula shows that (the negative of) the Artin conductor equals the minimal discriminant. In the case of genus two curves, equality no longer holds in general, but the two invariants are related by an inequality. We investigate the relation between these two invariants for hyperelliptic curves of arbitrary genus.