Georgia Institute of Technology College of Sciences

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)

I will present a number of stories about some results that I think highlight how results get proved and how they do not. These will span problems from almost all areas of theory, and will include both successes and failures. I hope that beyond the actual results you will enjoy and hopefully profit from the stories.

In this third and last talk on the topic, we will discuss some issues related to existence and long-time behavior of nonlinear dispersive equations on compact domains (or in the presence of a confinement). There, we will try to convey some elegant interactions of this class of PDE with other fields of mathematics like analytic number theory and dynamical systems. Time permitting, we will discuss how such tools can be used to better understand some questions on wave turbulence.

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.

Construction of pseudo-Anosov elements of mapping class groups of surfaces is a non-trivial task. In 1988, Penner gave a very general construction of pseudo-Anosov mapping classes, and he conjectured that all pseudo-Anosov mapping classes arise this way up to finite power. This conjecture was known to be true on some simple surfaces, including the torus. In joint work with Hyunshik Shin, we resolve this conjecture for all surfaces.