  We study the bifurcation loci of quadratic (and unicritical) polynomials and
exponential maps. We outline a proof that the exponential bifurcation locus is
connected; this is an analog to Douady and Hubbard's celebrated theorem that
(the boundary of) the Mandelbrot set is connected.
  For these parameter spaces, a fundamental conjecture is that hyperbolic
dynamics is dense. For quadratic polynomials, this would follow from the famous
stronger conjecture that the bifurcation locus (or equivalently the Mandelbrot
set) is locally connected. It turns out that a formally slightly weaker
statement is sufficient, namely that every point in the bifurcation locus is
the landing point of a parameter ray.
  For exponential maps, the bifurcation locus is not locally connected. We
describe a different conjecture (triviality of fibers) which naturally
generalizes the role that local connectivity has for quadratic or unicritical
polynomials.
