Georgia Institute of Technology College of Sciences

The algebraic K-theory of the sphere spectrum, K(S), encodes significant information in both homotopy theory and differential topology. In order to understand K(S), one can apply the techniques of chromatic homotopy theory in an attempt to approximate K(S) by certain localizations K(L_n S). The L_n S are in turn approximated by the Johnson-Wilson spectra E(n) = BP[v_n^{-1}], and it is not unreasonable to expect to be able to compute K(BP). This would lead inductively to information about K(E(n)) via the conjectural fiber sequence K(BP) --> K(BP) --> K(E(n)). In this talk, I will explain the basics of the K-theory of ring spectra, define the ring spectra of interest, and construct some actual localization sequences in their K-theory. I will then use trace methods to show that it the actual fiber of K(BP) --> K(E(n)) differs from K(BP), meaning that the situation is more complicated than was originally hoped. All this is joint work with Ben Antieau and Tobias Barthel.

These are two half an hour talks.Evgeny's abstract: The most useful approach to a classication of 3-manifolds is the complexity theory foundedby S. Matveev. Unfortunately, exact values of complexity are known for few infinite seriesof 3-manifold only. We present the results on complexity for two infinite series of hyperbolic3-manifolds with boundary.Andrei's abstract: We define coordinates on virtual braid groups. We prove that these coordinates are faithful invariants of virtual braids on two strings, and present evidence that they are also very powerful invariants for general virtual braids.The talk is based on the joint work with V.Bardakov and B.Wiest.

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)