Chebyshev polynomials in the complex plane

A classical problem that goes back to Chebyshev is to approximate $x^n$ by polynomials of lower degree on some compact interval. The monic polynomials $T_n$ of least deviation from zero on some infinite compact set $\mathsf{E}\subset\mathbb{C}$ hence bear the name of Chebyshev. In the talk, I will discuss results about the zeros of $T_n$ and their asymptotic behavior when $\mathsf{E}$ is connected. I will also discuss bounds on the norms $\Vert T_n \Vert_\mathsf{E}$ and present some open problems.