#### Abstract

Pg 1 of 12 Monday, 9 July 2001 Compactification of orbits of hyperbolic elements in semisimple symmetric pairs Jiro Sekiguchi Tokyo University of Agriculture and Technology Let g be a semisimple Lie algebra and let σ be an involution. Then we obtain a direct sum decomposition g = h + q where h and q are the +1 and -1 eigenspaces of σ, respectively. Let G be the group of inner automorphisms of g. An element x ∈ q is said to be hyperbolic if x is semisimple and all the eigenvalues of Ad(x) are real. Let p (x) be the direct sum of eigenspaces of Ad(x) with non-negative eigenvalues. Then p (x) is a parabolic subalgebra of g. Let P(x) be the parabolic subgroup of G with Lie algebra p (x). If H is the identity component of the fixed point subgroup G of G with respect to σ, then ⋅ Η x is naturally embedded into the flag variety G/P(x). In my talk, I will study the H-orbital structure of G/P(x) in detail when the eigenvalues of Ad(x) are 0, ± 1. Acyclic groups, large and small Jon Berrick National University of Singapore Acyclic groups form a remarkable, yet little-studied, class. The “large” examples occur throughout mathematics, as automorphism groups of mathematical structures. The “small” examples occur more sporadically, yet even here patterns exist. In this survey, with an emphasis on the history of the subject, I shall try to indicate where acyclic groups may be found, and highlight their importance, their properties and some interesting patterns. NUS-JSPS Workshop on Algebra Singapore-Warwick Workshop in Geometry & Topology