Title: Boundedness of Chebyshev Polynomials over Irregular Sets 

Abstract: The link between Green’s Functions, which solve the Dirichlet problem on a set, and Chebyshev polynomials, which have minimal norm on the same set, has been well-established in Potential Theory. A Green’s Function must be zero at quasi-every point on the boundary of its domain.  A  2018 paper by Christiansen, Simon, and Zinchenko showed a bound for the infinity-norm of the Chebyshev Polynomials on regular sets. In this talk, we will show how the norm is affected by the addition of irregular points. We conclude that, on a Parreau-Widom set, the Chebyshev Polynomials are bounded if and only if the series created by summing the Green’s Function’s values at the irregular points converges.