The Runge approximation theorem is one of the fundamental results of complex analysis. The theorem states that we can approximate holomorphic functions on compact sets using rational maps, but it doesn't give us any control over the critical points of these rational maps. At the beginning of 2023, the American mathematicians C. J. Bishop and K. Lazebnik published a paper, where they proved an improved version of the Runge approximation theorem, the so-called Runge+ approximation, which does give us control over critical points. The proof is based on a technique called quasiconformal folding, developed by C. J. Bishop in 2015. In the thesis we first present the theory of quasiconformal maps, then derive quasiconformal folding and finally, we prove Runge+ approximation in the case of connected Runge compacts.
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