Wolff Lecture

Part 1:  The MLC Conjecture asserts that the Mandelbrot set is locally connected. It would give us a precise topological model for this complicated fractal object. It can be reformulated as a Rigidity problem intimately related to the Mostow-Thurston Rigidity phenomenon in the Hyperbolic Geometry. We will present recent advances in the problem based upon new machinery in the Conformal Geometry (moduli estimates for nearly degenerate Riemann surfaces). Based upon a joint work with Jeremy Kahn.

Part 2:  Fractal sets in the plane can be roughly classified according to the following Geometric Trichotomy:

Lean Case: HD(J)< 2;

Balanced Case: HD(J)=2 but area(J) =0;

Black Hole Case: area(J)>0.

We will overview this Trichotomy in the context of Julia sets. In particular, we will show that all three cases can be realized in the class of Feigenbaum Julia sets. Based upon a recent joint work with Artur Avila.