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We study the bifurcation loci of quadratic (and unicritical) polynomials andexponential maps. We outline a proof that the exponential bifurcation locus isconnected; 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 hyperbolicdynamics is dense. For quadratic polynomials, this would follow from the famousstronger conjecture that the bifurcation locus (or equivalently the Mandelbrotset) is locally connected. It turns out that a formally slightly weakerstatement is sufficient, namely that every point in the bifurcation locus isthe landing point of a parameter ray.  For exponential maps, the bifurcation locus is not locally connected. Wedescribe a different conjecture (triviality of fibers) which naturallygeneralizes the role that local connectivity has for quadratic or unicriticalpolynomials.

Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity