We discuss definable compactifications and topological dynamics.

We study the action of G = SL(2, R), viewed as a group definable in the structure M = (R, +, ×), on its type space SG(M ). We identify a minimal closed G-flow I and an idempotent r ∈ I (with respect to the Ellis semigroup structure * on SG(M )). We also show that the "Ellis group" (r * I, * ) is nontrivial, in fact it is the group with two elements, yielding a negative answer to a question of Newelski.

We initiate the study of p-adic algebraic groups G from the stability-theoretic and definable topological-dynamical points of view, that is, we consider invariants of the action of G on its space of types over Q p in the language of fields. We consider the additive and multiplicative groups of Q p and Z p , the group of upper triangular invertible 2 × 2 matrices, SL(2, Z p ), and, our main focus, SL(2, Q p ). In all cases we identify f -generic types (when they exist), minimal subflows, and idempotents. Among the main results is that the "Ellis group" of SL(2, Q p ) iŝ Z, yielding a counterexample to Newelski's conjecture with new features: G = G 00 = G 000 but the Ellis group is infinite. A final section deals with the action of SL(2, Q p ) on the type-space of the projective line over Q p .

To each of the Johnson, Grassmann and Hamming graphs we associate a lattice and characterize the eigenspaces of the adjacency operator in terms of this lattice . We also show that each level of the lattice induces in a natural way a tight frame for each eigenspace. For the most important eigenspace we compute explicitly the constant associated to the tight frame. Using the lattice we also give a formula for the product of the Norton algebra attached to that eigenspace.

No abstract