Georgia Institute of Technology College of Sciences

Periodic eigendecomposition algorithm for calculating eigenvectors of a periodic product of a sequence of matrices, an extension of the periodic Schur decomposition, is formulated and compared with the recently proposed covariant vectors algorithms. In contrast to those, periodic eigendecomposition requires no power iteration and is capable of determining not only the real eigenvectors, but also the complex eigenvector pairs. Its effectiveness, and in particular its ability to resolve eigenvalues whose magnitude differs by hundreds of orders, is demonstrated by applying the algorithm to computation of the full linear stability spectrum of periodic solutions of Kuramoto-Sivashinsky system.

A Riemannian manifold (M, g) is called Einstein if the Ricci tensor satisfies Ric(g)=\lambda g. For a Riemannian homogeneous space (M=G/H,g), where G is a Lie group and H a closed subgroup of G, the problem is to classify all G-invariant Einstein metrics. In the present talk I will discuss progress on this problem on two important classes of homogeneous spaces, namely generalized flag manifolds and Stiefel manifolds. A generalized flag manifold is a compact homogeneous space M=G/H=G/C(S), where G is a compact semisimple Lie group and C(S) is the centralizer of a torus in G. Equivalently, it is the orbit of the adjoint representation of G. A (real) Stiefel manifold is the set of orthonormal k-frames in R^n and is diffeomorphic to the homogeneous space SO(n)/SO(n-k).One main difference between these spaces is that in the first case the isotropy representationdecomposes into a sum of irreducible and {\it non equivalent} subrepresentations, whereas in thesecond case the isotropy representation contains equivalent summands. In both cases, when the number of isotropy summands increases, various difficulties appear, such as description of Ricci tensor, G-invariant metrics, as well as solving the Einstein equation, which reduces to an algebraic system of equations. In many cases such systems involve parameters and we use Grobner bases techniques to prove existence of positive solutions.Based on joint works with I. Chrysikos (Brno), Y. Sakane (Osaka) and M. Statha (Patras)

Given a family of ideals which are symmetric under some group action on the variables, a natural question to ask is whether the generating set stabilizes up to symmetry as the number of variables tends to infinity. We answer this in the affirmative for a broad class of toric ideals, settling several open questions coming from algebraic statistics. Our approach involves factoring an equivariant monomial map into a part for which we have an explicit degree bound of the kernel, and a part for which we canprove that the source, a so-called matching monoid, is equivariantly Noetherian. The proof is mostly combinatorial, making use of the theory of well-partial orders and its relationship to Noetherianity of monoid rings. Joint work with Jan Draisma, Rob Eggermont, and Anton Leykin.