It is about symmetric and skew-symmetric bilinear forms on vector spaces and vector bundles, Witt theory for algebraic varieties, Pontrjagin classes, Thom operators, and orientations on cohomology theories.

H.Bohr's theory of almost periodic functions has numerous applications to various areas of mathematics. Two branches of this theory, both extending the classical setting of almost periodic functions on reals, were particularly rich on interesting and deep results: holomorphic almost periodic functions on tube domains and almost periodic functions on topological groups. This talk is devoted to a natural link between these two concepts - holomorphic almost periodic functions on coverings of complex manifolds, their function-theoretic properties and the `sprouts' of the theory of analytic sheaves on the corresponding Bohr compactifications of the coverings.

This is joint work with Alexander Brudnyi.

Let $G$ be an affine algebraic group and let $X$ be an irreducible, affine variety. Assume that $G$ acts on $X$ via $G \times X \to X$. The action is called stable if there exists a nonempty, open subset $U\subseteq X$ consisting entirely of closed $G$-orbits. The action is called observable if for any proper, $G$-invariant, closed subset $Y\subseteq X$ there is a nonzero invariant function $f\in k[X]^G$ such that $f|_Y = 0$. It is easy to prove that "observable implies stable" but the two notions are not the same for general groups. We discuss a useful geometric characterization of observability. We then discuss some of the following questions and illustrate them with the appropriate examples.

(1) When is the action $H \times G\to G$, by left translation, observable?
(2) Does the characterization simplify if G is unipotent? solvable? reductive?
(3) What happens if $X$ is factorial? reducible?
(4) Is $int : G\times G\to G$, $(g,x)\mapsto gxg^{-1}$, always observable?
(5) Can we generalize to the case where $X$ is projective and $G \times X \to X$ is linearizable?