We show that Arnold tongues for the family of double standard maps fa,b(x) = 2x+ a− (b/π)sin(2πx) are connected. This proof is accomplished in the complex domain by means of quasiconformal techniques and depends partly upon the fact that the complexification of fa,b, has only one critical orbit taking symmetry into account.

For a real symmetric domain GR/KR, with complexification GC/KC, we introduce the concept of “star-restriction” (a real analogue of the “star-products” for quantization of Kähler manifolds) and give a geometric construction of the GR-invariant differential operators yielding its asymptotic expansion.

3 Tori 13 3.1 Compact tori . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Representations of Tori . . . . . . . . . . . . . . . . . . . . . 13 3.3 Complexification of tori . . . . . . . . . . . . . . . . . . . . . 15 3.3.1 Complex Lie groups and complex algebraic groups . . 15 3.3.2 Algebraic varieties and localization . . . . . . . . . . . 17 3.3.3 Complex tori . . . . . . . . . ...

We introduce an object which in a sense extends the notion of a covering space. Such an object is required to understand the “group” generated by the exponential functions of quadratic forms in the Weyl algebra, and gives in some sense a complexification of the metaplectic group.

We study the free complexification operation for compact quantum groups, G → Gc. We prove that, with suitable definitions, this induces a one-to-one correspondence between free orthogonal quantum groups of infinite level, and free unitary quantum groups satisfying G = Gc.

A PT symmetric complexification of an exactly solvable potential in one dimension leads to a paradox. The set of its normalizable solutions proves larger than one would expect on the basis of its point canonical transformation analysis. March 13, 2008, file ces.tex

We describe a maximal exceptional collection on the Cayley plane, the minimal homogeneous projective variety of E6. This collection consists in a sequence of 27 irreducible homogeneous bundles. 1. The Cayley plane Let O denote the normed algebra of (real) octonions (see e.g. [Ba]), and let O be its complexification. The space

The complement M of a complex line arrangementA in C gives rise to combinatorial data, namely, the intersection lattice L(A). We prove that if the arrangement A is the complexification of a real arrangement, the characteristic varieties of complex rank one local systems on M are determined by the combinatorics of the arrangement A.

One-dimensional scattering problem admitting a complex, PT −symmetric shortrange potential V (x) is considered. Using a Runge-Kutta-discretized version of Schrödinger equation we derive the formulae for the reflection and transmission coefficients and emphasize that the only innovation emerges in fact via a complexification of one of the potential-characterizing parameters. PACS 03.65.Ge