One of the most well-studied parts of complex dynamics is the parameter space for quadratic polynomials. The Mandelbrot set is a subset of this parameter space corresponding to polynomials with certain nice dynamical properties. Despite a relatively straightforward definition, the Mandelbrot set exhibits astounding fractal-like properties, and a great deal of ingenuity has gone into understanding this set over the last few decades. In this talk, we will begin by introducing the Mandelbrot set and describing some of its fascinating characteristics. Then, we will prove a few of its basic topological properties, including giving an overview of J. Kahn's proof of connectedness. Speaker(s): Mark Greenfield (UM)